Modern Physics [2 ed.] 0805303081, 9780805303087

Citation preview

MODERN

PHYSICS S E C O N D

E D IT I O N

tegr.'. Useful 'n

( ) J , , (/I.". ') J ,

\

,,0'"'

'' "' -

," s,n.�

I.

, - -I.- ,\(\ �/l1f l

(h -

I.

"

4

(1\ -

-

.00

e-a\
t>. the pi.."l.. I' ,.!lIn.wl), .... I1 I�,II, 110 In cod;o. ....:' al ililrcrcm k auon.,- ) tNt ili-'4U....'h'fl I'"� Illne \l11.IIIIm th , Ih.u. n,�'" 1.ll.llIg k' , � d'IoII ltw lUt'lC' mtc'f\.aJ In \llnII ) hUnC'- "lll �' k!O. tUDe ILl PiIl-n. lhe rtu\� b .tkvWf Ih:UlIl.1l11.1 h') \nl'll.l ttuUl 11 I' 1Ii.\\lnhllll hi It,¢. "1l.lL.. -'"" tk'l\ 11,,1\ Itli Ih.tII \\ h..'11 ,1,\11\111011', The- ('UnI. an \1bjC'\1 �"'('UI'� In, I P ,,�ncd knKlh hlnll1&\lion l..Mh.·r \\ \' "Ill 'IUd"III, thl\ 1 . ..lh •.d \\ ,11",ul ....I"I\� if ,1ft tllllr !.hutlOf\. Hut ,,,l lItlUt..., " \' .1'1'f\�'h 1\, 'h' IIUhl II',' " nlt'II",,1 ,II \ \.\hlle II " IkII...II� ""','111&, Ill'I.U1n aUI'Rln, . kn,lh III an ,>hl'-' '1"'-"\\ )( ' I' as 1"'-""' all) I t " Ir,lIn 111..' log lit III m/!li l\"l!il'(" �\ 111 lUll " II'i kn!1lh "' �IUC3otJUltatoh woO III 1he- An'IC' .... alRI"' .prlle, hl lh... ell\" ..:I ." hI th..' 1\\\1 PIt'\I\II!' l'il ..'d' 11 I' • ,Ol'C' nm,w��' III dl '\UIi '\'i kn�lh l'Ulltt.ldl\lfl '" an \11'11..·.11 Ilhl'lllll l'.IU'l·d t>� !.kl .., '\ III h,hl Ira\('hn, Itl the \>h"" ",., IHIIll Ih\.' 1Ih1\ 1I1g \1"'ll'.;I The dl,-d " n.-al Wr{lOSt" ' lhe pl.lII\. " \11 kn�lh f ":\. tlntlng lu HI.t>, \\ hll hI li..!, II \I In ,\nna', Jr ..ulC, \fII'.1 ami hl.'! """1.1111 \Ill � l". h h.lpren II) ". ;lh�n\.·d \\ Llh .1Il (oJ \11 Itl.: ra-,'lOg pl..n\' _" /"1'\"1 Idl rll" \dm� t,,"I'. th"'n III Ih"'ll 1fi.lllw. "nlHI !Ukl .\",) "III t"t' a U"'lan..'\.· a" ..n It'\\ lhull l ..

1",,\

tune

'�Y'" 2.' Wh('ther Iln(' Ilhl�·t Ii" uhl\lc ,l1\llllll'r l.kJ"l:'lIIh I'll Ihe:' 1.t>'>('1 \\'1'" II"l1\e ul rl'lc.-cn..·('

1he I�ith Ilr an 'It'lje(1 In .. lrame Ihl,.ugh .... hlt,;h Ih.: \lhll.'\:1 1Il,1\,', . ' lI11llll('r Ih:!n Il'ri kn,lh IIlth.: Imllle 111 " h....h 1 1 I' III re I .-\�aLn, lIiJ I h.lt 1\ rc,-!uu!.'d I' rel.llt\'.: nHltllm If Anna \lR h!.'f 11",,;;\r .... '·1\· h'lld In, u pl.lnL. til kn)!th r� II \.\ llulJ ,x'\.·up;.- a di,tanl.'e h:" than f i.lI.:n)rdill� hI \1t'l\('f"\ I.'r-- In

"vh', Irame ,II n:ll'rcn..:e. tlc�au,� Allllil', plJnk mUH', rdativ!.'

\111 Pllrlcd

10 8"b

We nt'" ,,:ull'ldl'r " "Iue,uun thai In \lnt" hlrtn \)r anulhl'r " II ;':OIlHUIlII '1IUn.'l' \11 ;.:,m,lt'rn.tltl\n lor thl' l"ol'glOnll1!; "udenl (It "pcl:lal rda! l\- I I)' I n '0 J"ing. .... e "'ill t.e�1R tll '>('1.' till' inll'rJcpc.·l1lknl:c \11 th..' l'on,cqucm:c, \Vhal ap�ar III he paruJu,l" arc UI,O"II) JU�I nample, of applyi ng. (lIll' l'Un\equcn\:l!

01 'pe"'IJI �lall\'lI) and IglI\lnn� tht" olher,_ They arc a pa\,.'!..agl' ,",cal, and wc mu,1 buy till' ",hole p.ll"I.agl', Anna hi.!' dc\clopcd a re"olutlollJry new plane capahlc Ill' 1.I11aulL ng 'pt.·cll, near Ihal 01 I Ighl It i .. -lO Ol long \.\hen par\'cll lln thc runway. Bob ha, a 2() III

(b) A,"eonlm� 10 Bob

Illng iAlrplane hang.u- With \Ipcn dtxlrn ay .. at ea....h e nd. ;1\ LIl hg.un: 2.K(ill. ,-\nna. a yuung and lbnng Iype. t.lke.. oil. "I.'ce:!cmlc.. It) high ..peed. then '''(lOP' throu�h the hangar AI a high enough "�cd. Anna' .. plane fil'i entirely ""tthin lhc hangar all al on..:'(' a;':l,.'llnJlOg to Bob. (lCclIp)ing only 20 In .., ,huwn

tn Figure 2Jl(bl. In other ",onh. Btlh ..cc, the lail (II Anna's plane ill (JOe dour ....ay al pre"'J�ly nOlln on hi .. W.lkh. \\hiJe Bob Jr. ,.:C\ the p lanc ' .. no..,c al thc olherdoot'\l,'uy al�o pred,ely at nlxlO. Nnw ('()n�idt:r Anna... pt'n,pe,,·!lvc. Anna •

" at rt\t rtlati\'c to her plane and thu, sec .. It'> kngth 01.-" 4() m, She \I.'l''' the

lei A':l'NUla III Bob, anJ thc-),' \hllUldn'I, I"" thC'� d(J 'lnke 0 \lmultaneou,I}' 11\ A.n",,·,

!t

a.;>,:\",'\,bn): II' Ann.

(e) R\,b ,,«, IhI! (I!nler oJ Anna\ ,hip mme t4n.:� ovcr \peed, II> thus

Jt

The lime- according 10 Ollll, ctl\·

�, -

·W

It,n To find the rtadtng!. B\,b and ht� a"I�Ia.nh \� on Ann3'� doc..... al l - 10 n". � c\.lUld u� (1-1�b) Ilgaln, bul lhe fa...ter ....a)' il> Ilm� cldallon. Bob mU\1 "'et' Annl'� l clock &l.h.nclng hall J.� f�hI a� hi, ov.n (")'.. - 1) For c\cr) 20 n� (In hiS dt,lI\.·b. u.:h of Anna', mu�' til! 'e \horter

L � A,

I'

approiKhc,

rna) go 01\ fat

c.

a�

40 Iy

Ihe dl'IOln!;e appr03l'hcs O. The conclu�ion

\he hl..es II)

a.�

hule time

as

is pro\'(Icalive.

Anna

,he pJea�es by merely travelin.!l ;)1

Ii

-

e:�!J> {}

,,1Q \eolie' Ih.1I Ihl' ('\en" ,.ul'fy Ih(' Llln'nt.l Imn,f\lonallon t!'quuli()II�.) .\ t'lu1"1 lIt h�hl IlI:nll" ilt thl!' \lnglm Jo,t ,h they cl\)". The pul'ot!' Ihell �pread.� in

a1 4() n,

-lO=D

--

ImJ ""....lUl"t" In Ihl'" ,,:.I"'e. th�' ...hll"· n'n"..r-. Ilre tnt frame ori�ill),. ( E :(C'n.:ise

...hlp.u 1,!lllt-rent tllnn, II n'Jl'he' the ('nd, 01 hi' ,tJIiOnilf)' ,hip "'lIllUllllne(\Usly

11

Evrnt 4

•

24

Ch.pte, 2

Special Relalivity

sufficienlly large fraction of c.2 Here, the " object" must pass by in 30 yr of Anna'

FIgure 2.1' A Iighl pulse appears 10 spread the same dislance in all direc­

\

time. Using distance and time according 10 her.

tions in the �ame time according to both observers.

speed =

distance . lIme

Dividing by c, we have

�= \'

Bob

40ly 3 0 yr ' c

_

\, =

40Iy/y" 30 yr

� � = ,, - � ,, � 1

40

30

-

1

We have used the fact thai 30 yr ti mes c is 30 Iy. Squaring both sides and solving fOr v

yields

v =

0.8e

It is helpful to confirm that everything fils together in Example 2.5. As the reader may easily verify, for )I = 0.8e. r\. = �. According to Anna. the Earth-Planet X object is only 40 Iy -:- � = 24 Iy long. At 0.8e, it would pass in only 30 years. But how can Anna reach Planet X in 30 years when it takes light 40 years? We must be careful about who is doing the observing. Accord_ ing to Bob, light does take 40 years, but Anna takes even longer, for she moves at only 0.8e. Bob says that Anna takes I1ts = 40 ty/O.8e = 50 yr to reach Planet X. Still. he must agree that Anna is only 30 years old when she gets there, Whereas Anna's explanation is length contraction, Bob's is time dila­ tion. He watches a clock fixed in Anna's frame regiMer IltA' and the interval I1ts on his own is longer.

50 yr

=

3.lfA 5

A question to ponder: Becau�e Anna moves relative to the E:mh-Planet X system, though still at a speed less than c, she t alsu l"'!nmr: that Bob agc... 1 8 year,. If Bnb iii 1 00 when ('arl get, to Earth and , l< · ..:d 1 X year... dUring the night, thcn Bob mU!t>t be M2 year, old. uccording

l

.

l. when Carl pa"..c'> Planet X Thb i... truth

t t

m

"u

Carl's frame'

) return, Anna must accelerate: that i,>. she must c.:hange from one inertial another. (It make� no difference whether her ,�peed eould be kept con" I \1\ inertial frame ha� COn�tant I'e/ad,)·.) So far. we have not considered IS ollght affect Anna·... perceptions, but Carl doe!iiin 't accelerate. so we :'J

v nn hi.. observation... Figure 2.20 demonstrates the main point. Anna i'om one frame. where Bob is 18 years old, to another, where Carl

u: H l"

1

.:r that Bob I ... 82 years old. Although on the return leg of the journey. does determme that Bob is aging more slowly than herself, it is 100 late, )

•

'0; hu�e forward leap in age (according to Anna) has settled the mIder. Ju they reani.1e. l� � 2 1 Illu\lrales, Bob will be much older than Anna when

I v.o questions may still haunt the reader. ( 1 ) How is it pouible tbIl whee wbIIe CIII A'., lnd Carl are both at Planet X. Anna says Dob is 18 yeon old. lIIpJ4* says Bob " 82 years old? (2) Can Carl ",U Anna about Bob', fuIuIe?

A_. n" tOO ycar"lI gld IIut JutIf IooIc at you

yCHI·... GIlly 601

All tgg tnMl!, Bob What 0 differetlCe o little C'lCceleratign mokes!

Bob sends a light signal (rom Earth toward Planet X so that it arrives at the precise inSlanl Anna and Carl pass Planel X. If according to Bob il lakes light -1-0 years ro reach Planer X and it lakes Anna 50 years. then Bob had better send rhis signal when he i� 10 year� old. Let the light signal be an image of Bob blowing oul candle� al his 10th birthday party. When this signal reaches

Planer X. both Anna and Carl will �ee an image of Bob as a JO year old. Each

will ha ve infercepred all earlier signals. �o they will have knowledge of Bob's

prior activities. but none beyond thaI. The answer to question 2 is. No, because neilher has acruaJly seen any of Bob's life aJ!er his 10th birthday. However. we don't answer the question "'What is Bob's age?"' by saying,

"'The age he appears 10 be in the light signal that just arrived." Neither Anna nor Carl will say that Bob is 10 years old when they reach Plane! X, because each realizes thaI Bob wiJi have aged while the ligh! traveled. But Anna and Cart are

scrupulous record keepers. From their observations. they can calculate what Bob's age must be when they pass Plane! X, and the conclusions they reach dif­ fer markedly. Le! us investigate this and. in so doing, answer question I .

Figure 2.22 shows how each of the three observers sees the situation as Bob's light signal leaves Earth. As we showed earlier, Anna sees the distance befween Earlh and Planet X as 24 Iy. She sees me light signal and Planet X

both moving. and rhe separation bet ween them is decreasing at a rate of 1.8e. (This value is a relative veloci ty according to a third party, Anna. She doesn't

Figure 2.22 A lighl beam traveling from Earth to Plane! X according to three differ· ent observers.

4Dly

D.8e Anna

Bob

(a) According 10 Bob

fo Ihe righf at c C

E==> Anna

J.

24 1y

D.Sc

q Carl

and Planet X mo\·jng to the left at OJk _0 they approach each other at 1.8e·

Annil .se(';s the lighl �jgnal

mewing

o

0.80@ ,I

(b) According to Anna C.trl see� borh the light signal and Planet X moving with a fe/alive \'clm:iIY ofonl) 0.2('

W fhe right.

c

241y

6)

0.80

(c) According to Carl


ter Figure

Special Relativity Stands Up to tke Test Cla.,sical

from one frame 10 another mo\ing relative 10 the fir\t at about 60 I..m/s. AI such a low �pced. the effeci 01 the frame motion wa" expected to be ..,mall. but they ruled OUI eflech almo�t 5 orders of magnitude '>maller. Even the icon E =

lI/e2 receives regular attention, A team of ,cienti�" from

uni\'er,itie� and lab,> in ,e\'eral countrie' recently mea,url!d the ma'>s change .....hen a nur.:leu� captures a neutron to become a different nudeu� and Ihen emi" electromagnetic rJdiation (NlIIlIrt>. 22-19 I)e"emhcr 2005. 1096-1 097). The emitted electromagneti" energy was found to equal the "hange III IIIC2 of the neutron Jnd nuclei "IIhm 0.00004";( Eimtein\ cJaim� seem 10 be holding up quite well Gravitational lenses: Applying the Bending of light

n

If il gra\Ilational field "an bend Ji!!hl. then light ,>hould b.

T '

to. ween it and Eanh,

no

I-

, (

1•

.tblt" to tIa\'cl from source to ot-o,en'cr by III/litiph' palh cUound a lIla�si\'e nhJe,,1. Indt:ed thi, happen" and II i,

I

kUlI'" n a, A;ra\,llatLOnal lcn�jng_ Figure 2 .40 " a superb

l

�xamplc Tal.cn b) the Hubble Space Tcle,copt'_ it sho..... fIlultlrle Im.1ges (I' the ,arne gala,y. The massive

. One- application is in \artin� lIU h

krJling massive objects

.htft u':.uclatcd with lhe expan'>lon of the unh

r. :lie,e disturbances are

With lar�e fI:dshifb can be '>cen len\cd by obj

df

e'

Spectral energy density via cla-'i'�ical wave theory

Something i� certainly wrong here, for as Figure 3.2 shows, this parabolic function diverges as f increa...e ... without bound. If true, all malerials would radiate infinite power.

.\,21bc PhocoeJeI:IriC '"

Planck found that he could match the experimental dala with a I•.'Urious assumption: The energy at frequency f is somehow restricted to E = "hi, where " is an integer and h is a constant. The specitic error in cla.'.;sical wave thi..,(lry i!io in the average energy of a given wave. which b obtained b)' !Ote-grating O\'cr an a, is a characteristic of the particular metal. Table 3.1 lists some values (subject to variation. depending on impurities and other factors). If light were lOtrictly a wave, this effect should have several specific traits. First, if light of one wavelength is able to eject electrons, then light of any wavelength should be able to do it. Independent of the wavelength. the mte at which energy arrives (the intensity)--and therefore the rate at which electrons

Figurw 3.3

The photoelectric effect:

Light liberating an electron from a

metal surface.

•S(5

71

iIIiI.!'t�

.. . ...... t n " . ..

... .

-

2.3

7

...... P+niac. PIrtideI

Eo- Seooad.

if ... _.,. is low, then """" Ibough ....trons mighl sIiIl be .jecIed, a _

r

...... time lq should arise. Because • wave is diffuse. considerable

tUne

miPt be needed far enough energy to accumulate in me electron's vicinity.

1.9

2.2

•

_ .clOd couJd be ...... orbilnriJy buJe simply by _ing

• • oVl

..

.

(See &erciIe 16.) F'maUy. at any liven frequency. if the intensity is increueca.

die � electroas should be more energetic. A stronger electric field

sbouId produce a larJer acceleralion.

Imagine the experimenter's surprise when weak light of 500 om wave.. length ejects electrons from sodium. with no lime lag. while light of 600

3.7

wavelength cannot, even al

Drn

mon)' lim�s me intensity. Moreover. the energy of

the electrons liberated by the 500 om light is completely independent of the

4.3

intensity. Classically. this cannot be explained! In 1905. Alben Einstein proposed the following explanation: The light is

4.4

behaving as a collection of particles. called photons. each with energy given by

4.>

E = hI

Energy of it photon

(3-2)

h

where is Planck's conSIant. A given electron is ejected by a single photon. with the photon ttansferring all its energy to the electron and then disappearing_ multiple photons very rarely gang up on one electron. If the light's frequency is too low, such that the photon energy hJis le!'>s than the work function c/J. then there

is simply insufficient energy in any given photon to free an electron.

So none DrF

frrt>d. no moNer hoM' high 1M inlemily: no mailer how abundant the photons. (The phalon energy becomes internal energy or reflected light.) However, If the frequency is high enough. such that hJ > q,. then electron.., can be ejected. The

kinetic energy given to the electron would then be the difference between the

photon's energy and the energy q, required to free the electron from the metal.

(3-3) The subscript " max" arises because

cP is the energy needed 1O free the least

strongly bound electrons. Others may al ...o be freed. but less of the photon \ energy would then be left for kmetic energy.

Einstein's interpretation of the photoelectric effect explains not only the observation that a cenain minimum frequency i ... reqUired but also the other classically unexpected result\. I f a single photon-a particle

of concentrated

energy rather than a diffuse wave--doe\ have enough energy. ejection should be immediate. with no time lag. AI'iio. the electron's kinetic energy .,houJd depend only on the energy of the single photon--the frequency-not on how many !OIrike the metal per unit time (the IOtem.lty). In all respects. Einstein's explanation agrees with the experimental evidence. and the achievement earned him the

192/ Nobel Prize In phy\ics.

EXAMPLE 3 . 1 Light of 380 nm wavelength i\ directed at a metal electrode. To determine the energy of electron!> ejected. an oppo\ing elcctrO'ltalic potential difference I!. e"'lablJ"hed belween it and another electrode.

a,

"hown in Figure 3.4. The current of photoelec..

Iron, from one to the other i\ \topped completely when the potential difference i.,

3,2 n. �

1 . 1 0 V. D�tenninc (a) the work function of the metal and (bl the maximum-wave­ len�th light that ('an ejt.'Ct electron... from thi... metal. SOLVnON

(a) In the region between the electrode.... the electron... lose kinetic energ)' a' they � (1.6 X 10 1'Jl Cl gain potemial energy, II a potential energy difference 01" ( \ , 1 0 V) "'-' \ .76 X 10-111 J = 1.10 eV i, the most they ('an ...unnount. their

qV

.. ..... 'It

...... 1.. Cunonl_ .... 01..,..-. _ _ lbo _ilia potenlial encfl)' diffmmcc equIs 1be muimum kinetic eneru of the photoelectrons.

\..inetic energy k;\nng the fiN electrode mu,t be no larger than 1.10 eV. The potential diff�rence that barely ,tops !.he now is known a... the stopping po,rn. tiat Using equalion (3-3).

1 .76 X 10 1 9 J = (6.63 X 10-'� J . s)

�

q, = 3.47 X 10 IIl J

=

(

' X IO' m "

)

_1 -_

_ _ _

----=- __

380 X 10-Q m

2.l7eV

.- '"

(b) I f the wavelength o f the light were increased to A', the frequency-and thus

the photon energy-wou ld decrease. The limit for ejecting electron ... is when

an incoming photon has only enough energy to free an electron from the metal. with none lell for kinetic energy. Agai n u..ing equation (3-3), 0 = lif' - q, = (6.63 X lO-H J , s )

(

3 X 103 m1s

A

'

-

.

'

� A' = 573 nm

)

- 3.47 x lO-wJ

Wavelengths longer than 573 nm have insufficient energy per photon, so no

photoe lectrons are produced. The maximum wavelength for which electrons are freed is called the threshold wavelength, and the corresponding minimum

frequency is the threshold frequency.

The central point in Einstein'S explanation of the photoelectric effect is that electromagnetic radiation appears to be behaving as a collection of parricles. each with a discrete energy. Something that is discrete, as opposed to continu­ ous, is said to be

quantized. In the photoelectric effect. the energy in light is

quantized. EXAMPLE 3 . 2 How many photons per second emanate from a 1 0 mW 633 nm laser? SOLUTION

For each photon, E =

c

hf = h-A =

(6.63 X IO-" J · ,)

(

3 X

10' ml'

)

633 x I 0 9 m

=

3.14 X 10-I'J

To find number of particles per unit time, we divide energy per unit time by energy per particle:

number of particles time Clearly. photons are rather appear COnll PUOUS.

=

10

10-3 J/ ' -,-"" :-:-'= 3 . 1 8 X 1016 partide!J� 3.14 X 10 I' J/ partic\e X

-

-

"small " and it is easy to see how a light beam CQuid ,

"

t

+

71 CMptet 3 W.ves and Particles I: Elcctromagnetic Radiation Behaving as P'.utides FJgurw 3.5 In a nighl vision device, a light image becomes an image of free electrons. amplified in a multichannel plate and then reveaJed on a screen.

The photoelectric effect has long been used in simple Jight sensors, where light intensity registers as a photocurrenl, bUI it is also used in more sophisti�

cated ways. Let us take a look al one. REAL-WORLD EXAMPLE N I G H T V I S I O N

Incoming

light

By "replacing'" a photon with an electron. whose charge makes i l easier to "amplify." the photoelectric effect i!. a common fronl end on optical imaging systems. One example is the night vision device (NVD). A typical NVD is shown schematically in

Ob}ccti\'e len�

Figure 3.5. An objective lens focuses an optical image onto a thin piece of material. called a photocathode. where the photoelectric effect transforms it into an image of freed electrons. Naturally. the dimmer the light. the fewer the photoelectrons. Amplification is accomplished via a microchannel plate. This element has hundreds of thousands of channels per square centimeter. An electron entering a channel at

:;;::".,�!t". Ptu.!oeh....ul!n� ........ . /.k. •, pn'lloccd ...

.. :; • •

,

•

�."

..

., ••. .

.... �

;H'C

Pl D.

the bottom one as a wave. because A

A simple application of the criterion to electromagnetic radiation is Ute ca!:!> along the patient's body, look.ing at each ..lice. a three-dimensional image of in learning what is inside things without breaking them den),.;ty versu" p6,ilion emerge�. Figure :\. 1 5 "hOWl; (a) the open, probing everything from human bodies to construction basic layout of a PET machine and (M an actual image. materials to superconductors. While the century-old tried­ and-true method of Section 3.3 is still the leader, the conven­ Figure 3.15 Positron emis�ion tomogrolphy_ (a) A tracer emits a tional X-ray machine, with its hot filament sealed in a tube, is rather unwieldy. and the tube's lifetime is often short. The positron that annihilate, with a nearby electron, yielding two pho­ tons that place the annihilation along a line between the deleCtors. i huge and growing field of nanotechnology-applications n which some crucial element is measured in nanometers­ Many lines combine to produce a two-dimensional image of a slice. (b) Multiple slices produce a three-dimens.ional image-in may provide a new method. In work conducted at the this ca-.e, "hawing high tracer uptake in the hver and kidneys. University of North Carolina, carbon nanotubes are allied (a) with the quantum-mechanical effect of '-field emission" to produce a beam of X-rays strong enough to replace the Two phNl)n.. from an electron­ conventional X-ray machine in several uses. A nanotube is a regular meshwork of carbon atoms. forming a cylinder of only aboUi I om radius, and is closely related to many other all-carbon structures discovered in lhe 1980s (see Chapter 10). These continue to surprise us with new, remarkable properties and are a very hoi IOpic of physics research. In the X-ray source application, bundles of naootubes are deposited in a lhin layer on a metal disk. Electrons are coaxed from the layer toward a target not by heating a filament. which wastes power and produces electrical noise, but by field emission, a room-temperature way of producing a flow of electrons that relies on the quantum-mechanical effect of tunneling (discussed in Chapter 6). The resolution of the new tech­ nique is ell.cellent, and another potential advantage is faster response time for tracking moving objects. (See Vue et aI., Applied Physics Letters, 8 July 2002.)

r.�!!t;t·./

...., po,;lwn aIlmhilaliun are

.

(b)

Medical Imaging with Positrons As noted in Section

3.5, once a positron is produced, it soon engages in pair annihilation, simultaneously yielding two photons of a characteristic energy (see Exercise 42). This trait is ell.ploited in an increasingly common medical imaging procedure known as positron emission tomography (PET).

dch:l.:lcd �Imu\l.m�·ous.ly.

Chapter 3 Summary

fundamental daim� as possible. explain to your friend what evidence Ih.i.� pro\'ides for the particle nature of light.

3. You are conducting a photoelectric effect experiment by \hining hght of 500 nm wavelength at a piece of metal and detennining the �topping potential. If. unbeknownst

Electromagnetic radialion behaves i n some situations as a col­ Itttion of panicles-photons-having the paniclelike pmper­ tiC:. Appealing to � few 92

6.

it? (Imagine the two objects are hard spheres.) (c) Is it reasonuble to suppose that we could know this'l Explain. An isolated atom can emit a photon. and the atom's

aspects suggest a panicle nature? 10. A cohere", beam of light slrikes a single slit and pro­ duces a spread-out diffraction pattern beyond. The number of photons detected per unit time al a detector in the very center of the paltern is X. Now two more slits are opened nearby, the same width as the original, equally spaced on either side of it, and equally well

from I� blackbody, t: is not lhe c:onec1 speed. For ,..u.. ation moving uniformly in all directionl, the averap nm'ponl'''' of ,'elociIY in a Biven diftlctioo is

llluminah:d b)' the beam_ How many ph-Nons will he detected per unit time at the center d�tector now',' Why?

Exercises

Section 3.2

Section 3.1

11. For small ;:, ('- i� approximately I + ;:. (al U\e this to

show thai Planc\"\ spe

speed. Using equations (4-4) and (4-5), i l is I'WIlVC

=

fA

E ll

E

P

P

fA does correctly give

= -- = -

II

However. this may or may nol be the speed of the particle. In Section

2.7, we

suw that massless particles, slich as photons. move at c, and their particle prop­

erties E and p are related by E

=

pc. The above relationship then confirms that

electromagnetic wCII'es also move at c. Maller waves would move al

c if E = pc

aho held for massive partIcles-but it doesn'l. They would at least move at the particle speed if E were equaJ to

31 di.�cusses the point further.)

PI'particle' but this too is not the case. (Exercise

Wave and particle velocities. known respectively velocities. formula

r

are

as

phase and group

treated in detail in Chapter 6. The main point here is that the

= fA is of rather limited use for massive particles, because

v

is neither the speed of the particle nor the speed of light. The usual relation­ ships between strictly particle properties (p and equations (4-4) and not

hcl)'

nor

In'panicl/A .

(4-5)

are

=

ml" KE

= � m\.2, etc.) are fine,

universal. But for massive particles, E is

and p is neither

II/Ie

nor

II/Ivpartide'

(Exercise 32

focuses on the correct way to relate energy to wavelength and momentum

(0 frequency.)

4.3 The Free-Particle Schrodinger Equation

Ill!lcrmme the .... a..c fun(ti�ln '4',\. I) �lf a matter "'au" In �MtC ""n..e, all t)pe .. III \\;1H· .... arc the 'ame. h,[ eiU:h, thc� I.. an und«l)inS w.�

H(M 1.1(1

\.H'

equation, 01 \\.hich the VlII\e rundiDn hllln', !;"llnfu-.e the'>\: ten1\!>!) must he II at 1\\.0 familiar "::a'c.. ..oluUlln Let U" Il""-'�

Waves on a String

�

,-

iI�Y(\. t) \1,2

iI�_'{\. I) ih�

where \" i .. the wuve ..peed. The wave function is the ..olutilln \(t, I), which gi\\:" the ..tring', transverse amplitude as 1\ funClilln of pm.ilion .\ntl time. All

Fh4ll.lre 4.9 A "'-Iln" di"t"rh"n�'e on _ 'Inn�.

w;.l\e equation" ultimutely rcst on fundamcnlal laws. [loll 'hi-. one come, (aner mil. A OiIS;C !iinu..oidal solution of Ihi.. of x and I, provided only thai 4: and Cl1 are �laled a.s in condition (-'- 1 1a). It i� left as an exertist­ to YJow thal lhe more familiar A �in(kx - llJ) and A cos(k;r - Ctl)--applicablt 10 waves on a string. for e,ample-"imply don't work the same way. l'be) (A;.T - ta) is equiValent: to are not solutions. h is true by' the Euler formula that Aet A L"OSfU - llJ) ... iA ..inlh - llJJ. ..0 there is a similarity. Bur the compJe" c:\ponemiaJ ha ... (wo parts. an.J they are out of phase by one-quarter cycle. The requiremenl that (4- l la) must hold i" the key to seeing how the SchrOdinger WClI't' equation relates to the c1a..sical physics of particles. We hope rnal it isn·1 al odds wilh the fundamental wave-particle relationships, p >= hJ,; and E = hw. What happen.. if we insert them? Condition (4- l la) becomes

p

' =E 2m

(4- 1 1 b)

� ml·2, Ihis merely says that the particle\ Io.inelic energy must eq ual ih lolal energy, which is the classical truth, because � Given thal

p2/2m

t=

(mv)2j2m

free particle hJ:. no potenlial energy. Thus. rhe Schrodinger eqllalion is related

10 0 dostim/ (lccountinx ofent'rgy.

Although the wave function isn't physically detectable. Figure 4.11, which plots Ae,(b lLr) at t = 0, provides some insight i n to the mathematical

nalUre of a plane wave. The real part of '" is a cosine, the imaginary pan a

sine, and the Iwo parts are out of phase in such a way that the

magnitude is

constant-II varie.. neither in position nor in lime. The direct calcu l ati on of the probability density agrees:

or FI9U,.. ".11 A plane nlilller wave: The real anti irnuginary parh of "\t"O.' 1\1') plouctl ut I O.

/'I'(x. 1)/ = A That the probability per unit length is constant means that if we were to look for ii, a particle represe1lfed by a plane wave would be equally found anywhere.

likely to be

Obviously, a plane wave is not very realistic, but it is still quire useful. In physical optics. we speak of plane waves of lighl, because they are often a suf­ ticien tly good approximation of th e actual wave. The same i s true of matter waves. But the plane wave's importance goes even deeper, for a more general wave can be treated as an aJgebraic sum of plane waves-they are easily ana­ lyzed "building blocks." We will make use of this fact at several points later in

the text. Let us now turn to a topic where we need not know the actual ft i.� one of the mOSI profound ideas in all of physics .

'I'(x, t).

4 . 4 The Uncertainty Principle Th� men: fad that a rhCmlmCRlm ha... u \\ aw nalu� implies tnhc'�nl

".11 Sl.nale- lit dtfaactlob of an C'l«'uun plane wa'¥C

'.....

urk�rt"m·

. thn\Ugh Ii "'ingtC" ..tit I.:au� tic, in II... parth:k pn.�p'·rtie... Fllr (',ample. pa.....mg h. � tn'ma�nd"': P"\llI: W;\\C' t\'l .. ('r.:-.\d \1UI, "l ...t al'oll \:l\U-.c: ""'''('ftai",y i t mu an c _

In the momenta of the pastil"le, l phl't,'n...) delcxttd altcNard The ,lUll(' mm.' aprl) III an e1cdwo plane " .l\C. h�ure 4. 12 dcrll:ls a ..angle· ...ht puucm devcl· opt.-J. (ltle the l'arlier dlluMe ..lit) (me electron at a time. The H:" mp.1Ocnt " f ml)mentum of an elc\:tf\ln afh:r pa"" mg lhn1ugh l' ob'li iI..1u...I)' un,,;crtain. What \\ c mCi\O hy " unl'ertalOl)" in momentum " that if the' ('\penment is repeated many time .. it/I'II(inlih. the momentum d,'I(,I.:I('J ufter pot!'"ing Ihn_lugh the .. ht ..till \,.mc .. o\er a mnge of \'aloe.... BUI hi.I" J" v.c \.luanltf) II? Suppo...e that the P, . ",I\lc!. \\'� record fall wlthlll the r.mgc I \...g · ml.. hl I kg · mJ.... e:-lcept for one .\1 +50 \..g . . mls. What \.\Iuc llll we iN,.i�1\ 1\1 the uncert ainl y'! I kg. ",Js,? 25 \...g . mh;) 5 1 \...g . m/..? The definition of um:ertai nt)' b an ,ut"litrul') choil.:e, hut It lll:w iou..ly I,hould mea!.llre hl)\\ far dC\Hltion... are from the mC';,\n (uwrage) \·"Iu('. In ph)'!o,ics. wc define It .\... standard dc\iation For c)'llmplc. "UPp\N� re(lCllteu c:-lperiment.. are earned oul 10 dClenni ne n quanti ty Q, where Q lni�ht repre�

+

pO!'.ltlOn \. a component of momentum PI' or any other me., ..urahlc '-ttl,lO' Ql I!. obta� lcd " \ time.. the \alue Q� ,... ot"lt.\incd no! time ... illld so on. We find the mean Q b)' multiplying a particular "aluc Qj h)' the nUmN:f of tllne!> it II, obtained . "" I,ummmg. over all value.... then dividing by the hlti.\1 number of time.. for alt value!'..

senl

ut)'. The \alue

.

Q=

:L , Q, ",

L ,. ",

(4-\2\

The !'Ii.\fldard deviation tJ..Q i... defined as the square root of the mean of the squares of the values' deviations from the mean (explaining its alternative name. root-ll1ean-�quare devitlt ion) . For a more detailed. development. Appendix 1. Here we merel), present the fonnula that goes with the words:

ilQ =

L j (Qj - Q)2uj L ni

!>ee

(4·\3)

This definition i .. very welt suited to its role. It is the mosl lraclable one thaI is zero if and only if there is only one value ever obtained, which would auto­ matically be Q. and when values do vary, it gelS larger a... they become more spread out. Although it is important to know that uncertainty has a logical definition. as

we continue to investigate the uncertainty principle in this section, we won't

actua1ly use the definition. The point is that when we say, for instance, that there is " an uncertainty in the electrons' momentum," we aren't speaking of something nebulous but a specific value following from a concrete definition.

So let us return to the single slil. As Figure 4.12 shows, there is an uncer. tainly in the x-componenl of momentum of eleerrons detected beyond the slil for which the symbol is .lp\ . (The symbol 6. oflen means "change." but nO( here. Here it means uncertainly. or standard deviation.) On the other hand were we to conduct a different experiment, designed to establish the POsitio of electrons exiling the slit. [here would be an uncertainty in this quantity. too--.lx. The electron wave front i!i spread over the entire width HI, so there would be a probability of tinding the particle anywhere in this range, and the narrower the slil. the smaller would be thi" uncertainty. Above aU, there is a ljnk between the uncertainties in Pt and x. Because the width of a diffraction pauem, related (0 �.pt' is inversely proportional to the slit width, related to Ax.

�

the uncertainties are im'erseJy proportional. tlpx Figure 4.13 As a w(lve be("ome� more compacl, its ovcml! wavelengfh and momentum are less well defined.

- --

� . --

- ----_.

.b

-

pOSlflll!1 �(llIIrh:fC'ly UOknf)"II: .\ ,mJ p wc:1J ,ldiut'J

�

.

.1, fimte

P(hllil11l tJ�'[hl known: A IInJ p It:�s 1\l·1J ddint'd

� r\ L.-----',..1-

_ __ __ __

rO,ili{ln t'Vt'r, h�Un kn\l\\o: .\ ailJ p 1.'1'':0 Ie,_, \Iell dl'!jnL'J

1

ex -

dx

Although we have used the familiar single slit as a vehjcle. the particular experiment is not to "blame" for the conclusion. Regardless of the circum_ stances. it is an inescapable consequence of mauer's wave nature-whether obvious or not-that increased precision in the knowledge of position implies

decreased precision in the knowledge of momentum and vice versa. Figure 4.13 illustrates a simplified. qualitative argument. The top wave is infinite and regular. While there is no doubt of its wavelength, there would be a probability of

finding the particle at places along the entire infinite x-axis. Wavelength and, thus, momentum hlA are certain (6.p = 0), but essentially nothing is known

about the particle's posiLion (llx = 00). The center wave is regular over only a finite region. The COSI of obtaining a wave for which the particle's probable

whereabouts are narrowed down (LU "* 00) is that the wave is not regular every_ where. In any fair way of taking into account all of space, the wavelength cannot be said to be simply A, so neither can we claim [hat the momentum is precisely IliA (Ap * 0). The bottom wave gives an even better known position, but only by

further restricLing the region over which the wave is regular. Accordingly, it is even less fair to say that the wavelength of this wave as a whole is A, so t:J.p is larger still. The relationship between Ap and !lx is developed quantitatively in Section 4.7. Here we concern ourselves only with the conclusion, known as the uncertainty principle, and irs ramifications.

Because of a panicle's wave nature, it is theoretically impossible [0

know precisely both ItS position along an axis and i ts momentum component along thar axis:

ax and �.px cannot be zero simultaneously.

There is a strict rheorericai iower limit on (heir product:

h

tlp,6x > 2

Often referred to

as

(4-14)

the Heisenberg uncertainty principle, for i ts discov­

erer Werner Heisenberg (Nobel Prize 1 932), it is a shocking revelation. There

is a theoretical limir on the preci sion with Which some familiar quantities can

be known �imuhaneously. If we know a particle's position t'XDCtly. we can .1p. = 00). if momentum is know nothing aboul ils momentum (,1x =

0 :::::)

k.nown e1l3clly, position is completely unknown. The plane wave is II. good

example of the latter case. This fundamental matter wave ha.s a wa"'clcOJth perfectly regular throughoul space, gi\'ing il a perflXtly p�i� momentum, but It represents a particle equally likely to be found anywhere. A property in which there is no uncertainty is said to be momentum IS well defined (j,p( (6.x

::::

00).

-=

well defined.

For the plane W8\'e,

0), bUI position couldn't be mort' undefined

Don't be troubled by the inequality in {.t�I ·n-there is no unl:ertainty

about the uncertainty principle. The � rellecls the simple fael that then.: i)o. a particular wave shupe, called a

Gaussian,

also known as a bell I:urve, for

which the product of lIncertainties is a minimum. Figure 4.14 shows a Gauss­ ian wave form, a constnnt C times a " Gaussian factor" constant. It is mallimum at x

=

e-(,J1d

where

e

i)o. a

0, falls off toward 0 symmetrically as

.\'

becomes large, and the rate of fall-off depends on e. If e is large, the wave form is broad, falling off very slowly; whereas if e is smail, the W Wa\le�

I

I

This value happens to equal the experimentally veritied minimum energy. and the radiu� is also the correct most probable radius at which 10 find the electron (which doesn 't rest on the proton. whose rlldiu� is 10.000 times smaller). That they agree so closely is an accident-we have made many approximations-but it is no acci. i a dent that they are of the correct order of magnitude. The uncertainly principle s powerful tool.

The Uncertainty Principle in Three Dimensions The quaJitative idea behind the uncertainty principle is [he same in m ultiple dimensions as in one. The more compact the wave along a given axis, the less well we can specify the wavelength and therefore the momentum component along rhot axis. The result is a logical generalization of [he one-dimensional

result:

Note that the dimensions are independent. The single-slil pattern of Figure bears this out. Passing through the slil, narrow along x only. produces a large uncertainty in px, indicated by the subsequent detections being spread

4.12

over a large region of the screen. In the y-direction, the aperture is wide. so less is known about this component of position, and there is correspondingly little spreading of the pattern in that dimension. Thus, apy and Il.x can be small

simultaneously.

The Energy-Time Uncertainty Principle The momentum-position uncertainty relation is, a[ heart, a mathematical rela­ tionship. A width in space is inversely proportional [0 a "width" in the spatial frequency physics,

k

p =

= 21Ti"A (see Section 4.7). It is the fundamental wave-particle

lik, that takes it the final step. The same math relates a width in

lime to a width in the femporal frequency (j) = 21T1T. With E =

nw, the corre­

sponding physical consequence is

Energy.·Ime unc)rti inty prinCiple

(4·1 5)

How do we interpret this? If a stale, or even a particle, exists for only a limited span of time. its energy is uncertain. One example is the fleeting life of certain exotic subatomic particles. Their lifetimes can be quite Short-less

than 10-20 seconds-and this leads to considerable uncertainty in their mass! energy. Another example is the state temporarily occupied by an electron as it jumps down through energy levels in an atom. Because the state is occupied for a finite time interval related to

At, its energy is uncertam by an amount 6.£ inversely

at, which in turn gives rise to an uncertainty in the energy of the

photon produced when the electron drops down. This effect contributes to the broadening of atomic spectral lines (see Exercise

72).

4. 5 The Not-Unseen Observer Let us spend a little time ..ummanljng the limitations thai quantum mechanics

quantify

the effcch of places on our knowledge. Although we won't begin to external forces until Chapler 5. if the forces are known. the SchrOdinger equa­ tion may. in principle, be solved for the wave function of a massive ohject, which contains all information that can be known. But this isn't everything we might expect classically. The uncertainty prinCiple. for IOstance, ..ays that a wave funclion of simultaneously precise momentum and position is a theOl-eti­ cal impossibility. It follows that any experiment or measurement thai precisely

determines position must resull in a state 10 which nothing is known about the momentum and vice versa. Suppose we carry oul an experiment on a particle, experiment A. applying external forces in such a way as to determine both its po�ition and its momen· tum as precisely as possible, such that .6.x 6.p =

h/2. Assume. for the sake of A ' We have found

discussion. that ax is 1 00 lAm. and call the wave function '"

the wave function. but we aren't satisfied. for we haven't really "found" the particle-its ·'Iocation." All we have is this mysterious probability amplitude. We conduct another experiment, experiment B. in which the particle reg· iSlers its presence at a detector at a definite location. We rejoice-we have found the particle. However, there are no "point detectors." If the detector', width is smaller than the

100

lAm position uncertainty in \If

A' then we have

indeed narrowed down the possible locations, but we haven't established a

location with complete certainty. Yes. we have reduced the uncertainty in position, but if this is so. experiment B has changed the wave function. At the very least, it has increased !J.p. If we repeated this pair of experiments many times--experiment A to establish lhe initial wave function \If

A and experiment B to "find" the particle­

experiment B would find it at various locations within the 1 00 11m uncertainty

A' and the number detected at a given location would be proportional to j\f! 12. In essence, we would simply verify that I'" I:! is propor­ A A

of wave function '"

tional to the probability of finding the panicle after experiment A. But because

experiment B changes the wave function, we can't "watch"-repeatedly find­

A'

the same particle while preserving a single wave function 'I' The double-slit experiment, depicted in Figure 4.17, is a good example of these ideas. In effect, the slits are an experiment A, establishing an initial wave function

"'A beyond them, and experiment B is the detection of a particle at Figure 4.17 Eltperimem A eSlablishes

'ITA' which repealed eltperiments 8

Eltt""" beam

1111 W

verify.

112

Cftapter .

Wavea and Panicles II: Maner Behaving II!; Wavc�

the screen. By sending in a beam of particles one at a time, we are carrying out experiment A then experiment B repeatedly. Where "'A is large, experiment B registers particles in abundance; where 'IfA is zero, experiment B registers no

particles. We cannot conduct an intermediate experiment, determining which

slit a given particle passes through, and yet hope to observe the interference pattern exhibited by "'A' for this intermediate experiment would itself alter the wave fUDction. (A recent confirmation is discussed in Progress and AppHca� tions.) To observe interference at the screen. we must allow each particle's wave function to pass through both slits simultaneouslY--e. the greater the range of frequenci�

that must be tranurement, is becoming a central idea. In

pursuit, contributes in important ways to our understanding.

the past decade or twO, the topic of quantum informadoD

We claim that if in a double-slit experiment we were to

has exploded. In essence, the tenn acknowledges that we

Keeping a n Eye o n the Double Slit VerifIcation of even

watch whether a panicle passed through one slit or the other. we would destroy the interference pattem-fonnation of the pattern requires the particle to pass through both.

are now becoming able to manipulate much of the tenuous

Using a double-path electron interferometer, effectively equivalent to an electron double slit. scientists have

confirmed it (E. Buks, et aI., Nature, 26 February 1998, pp. 871-874). In the standard double slit, a path difference

infonnntion that quantum mechanics allows us to know-

not classical notions like panicle position and velocity. but

the quantum state. A promising application is in the field of quantum computing. The heart of today's digital comput­ ing is the data bit, which can be in either one of two Slates, usually a 0 or I . A dynamical analog would be a classical

to different locations causes a phase difference, which in

particle that must be in either one room or another, but not

turn gives constructive or destructive interference. In this

both. A quantum particle. however, can be in any of an

experiment, electrons may take two different paths to a

infinite number of superpositions/sums of two quantum

single collection location. but the phase relationship is altered

states. A simple example would be the superposition 'Jr =

by the asymmetric innuence of a magnetic field-varying the

a q.r \

field varies the interference. One path is essentially free, while

whose isolated bumps are far apart. and a and

the other passes through a tiny ··quantum doC (see Chapter 5

arbitrary constants governing the probabilities of being

Progress and Applications), serving as an electron weigh

found at one bump or the other. This more flexible bit of

station coupled to a detector. When the detector sensitivity is

infonnation is known as a qubil. Qubits can be combined.

set very low, thus concealing any knowledge of a passing electron, the interference pattern is clear. As the detector's sensitivity is increased, the pattern's visibility diminishes­ the important phase relationship is lost

+ b "'2' where q.r I and "'2 are Gaussian functions

b are

to carry vastly more information than the same number of standard data bib, giving hope of massive parallel

processing that may revolutionize information handling in the years to come. Many two-state systems have been studied to serve as a qubit-for instance. two different

Quantum Information

As technology marches on, the

energy states of a bound electron and two photon spin

realities and peculiarities of quantum mechanics assume an

states. These topics and some of their applications to

increasingly importam role. What could once be essentially

quantum computing are discussed in later chapters.

WorIdng with the Unc.rtalnty Prindpl. The unc:ma.int}' principle ties together uncerta.intin in Jk1-"iition and momenlUm. but if hu considerably wider applicatlon. The physical world ruibib many pai" of quantities, known as conjugate quanfitia. that al'C' inherenlly linked in the �me

I

I

way To kntlW one quantity e_,at.·tly i\ to be completely ignorant of it .. conJugak'_ ConveN"ly. if one quantil), i� known. an inadvenenl "ob..cI'V4I.tion" of its I;onJu!!ate will disturb mat knowledge. C{lOjugare quantities haying to do With lighl. �uch a.' pha..e and photon number. are of great ,,:urrenl imere,!. ught in which flul,·tuation, in one aspect rune ht.-en redu!,"w at the !,"ost of uncontmlled tluctuations in I fs conjugate i� known a" �ueeud light Wi!h cenain

.... i'"tcri�tKs. �quce..ed light promi�e� to unusually ,mooth cha enhal1l:e high·�pced communication. optical imaging. studies

ruot of jnlere�t in �o-ca"ed quanlum·nondemolilion (QND) mea,ul't'menh, Using QND technique,. a recenl experiment

wa' able 10 delect the �ame photon repe.lIedly ....ithout losing whit'h i, the usual fate of a phown aCluaJly "seen" fG. N(lgue�. el uJ. . Nature. 15 July 1999. pp. 239-242), Progre�.. in QND ..hould aid the advllnce of quantum computing. a., well as gravitation. Gr:hjlalional waves are predicted to e�jll ltJ .II� MIUro·.. Jll'llun ,. an elecbtw'l and

wh>" Hy mUJn. KlRIClhlna UIVfIWJI. hnw«dd you

Thl" ....uan: . ,'l lhe .....1\(' lundlvn', ab'l-.lulc \alue JI\t" the rn.babllll) �r UOit kn!1th 01 findmg the pattldc: pmNtlllll) Jen\II)'

•

I'I'( I. t)j!

to a par1ldc', untkrl)lnll VIoa\(� IWlurc. II� m ....mcrllum and J"INllfOn cannot be rro:l�l) lnow" \lmullaooou,ll_ 1llcrc '\ it thcon:u.:al lu"cr 11mll un lh< pnxiU(1 ,,1th< un.:enOlintle\ 0"'"8

(4 ' 4)

1

Thl\ ,\ LOlw, n J\ the unl.:cnamly prim:lple and \ II con'>6.juenr.:e - M. and the mathematu:ill pmpe:nlC\ of "3Ie, A ....a\e . lundlon for y,hu.:h nwmentum i� kno....n precl'cly loll' - 0) i� onc for I,\-hlCh the Pl-l\IIIOn " compltte ly unkno"n (.l...t .., -I; the pmbabllity dcn"ty I� 'rreild throughout ')pace Convcr.el)'. II I,\-;\VC lum:tion lur .... hu.:h the JXNtlOn i\ kno....n . preci\Cly " one lor .....hlch the momentum i� completely unknown.

01 I'

mdicates advanced que�tlon�

Conceptual Questions 1. E,pcrimcnh df'c..):: ' f ::.J

rrer'l !l ( Id

,·e .w. "

) 1,

,

I.'t'

.J,m,j·

"

... ....;.J:

I'

,

•

,

"

.

('I'

,

,

r

1z2 r1 '!' (..l.. I)

!Jr.

• 35.

",'

'l'lX, t) ]s bv ,Jdimtion "" I t) I 'Ii2(X, I How i� the- cumplt!x appro'".:h l:h"lsin In �eci n -L3 mor.; conwnienl than the a.lternative pmed hert! J In Set·tinn 4.3, WI! claim that in analyzing ek:(;tromdg netl!': wavc!-" we (;oulrJ hanrJle the field� E and B \Ahere

f

36,

37,

G · dl

'0J

= !... c a,

G ' dA

waves would have to obey lhIS' COil). Electromagnetic 0 . , oes thiS change of approach make plex equatIon. and/or 8 complex? (Remember how a complex n £ umber is defined.) An electron moves along the x-axis with a well.deli .fIed momentum of 5 X 10-25 kg·mls. Write an ev"preSSIOD " the matter wave associated with lh'IS eee. descnbmg I tron. Include numerical values where appropriate, A free particle is represented by the plane wave funcll, flJl W(x. r) =: A exp[i( 1.58 X 1 0 1 2 x - 7.91 x 1 016,)' where $1 Ul11ls are. understood. What are the po.' .... minimum kinetic energy.

47. An electron in an atom can "jump down" from a higher energy level to a lower one, then to a lower one still. The energy the atom thus loses at each jump goes to a photon. Typically. an electron might occupy a level for

a nanosccond. Whot uncertainty in the electron·s energy does this imply? 48. The rfJ is a subatomic particle of neeting existence,

Data tables don·t usually quote its l ifeti me. Rather, they

quote a "width," meaning energy uncenainty, of about

ISO MeV. Roughly what is its lifetime?

49. A crack between two walls is 10 cm wide. What is the

angular width of the central diffraction maximum when (a) an electron moving at SO mls passes through? (b) A basebnll of mass 0.145 kg and speed 50 mls passes

taint)' I'or a typical malTOscopic object is generally ao much smaller than its actual physical dimensions that. applying Ihe uncertainty principle would be absurd. Here: we gain ",-nne idcu of how small an objccl would ha\'e to be �f(lre quantum mechanics might rear ilS head, The den�ity \If aluminum 2.1 X HP !tgIml. is

.

typical 01" M)lid� and liquilh around us. Suppose we could nam}"" dl'IWn the velocity of an aluminum sphere 10 ..... 'Ihin an uncertainty of I � per decade, How small ....ould . il ha\e U1 be f\u Ib po�ition un..enainty to be at. lea,t a� largo: a... of ih radiu�'?

-I,!"k

52. A particle i, conne..:ted to a spring and undergoes

one-dimen'ional m(ltioR.

(a) Write an e"pr.:s...ion for the total lkinetic plus

potential) energy of the panicle in terms of ilS pl)�ition .l, i ts rna,s m. iu. momentum force constant I(' of the sprin g .

p, and die

lb) Now treat the particle a., a wave. Assume Ihat the:

product of the uncenainties in position and momen­

tum i, governed by an uncertainI)' relation

� fl. Aho a.\!'ume that because x Is. 00

through? (c) In each case, an uncertainty in momentum

6.pSl"

is introduced by the "cxperiment" (i.e., passing through

a'\lcragc, 0, the uncertainI)' � is roughly equal to a

the slit), Specifically, what aspect of the momentum

becomcs uncertain, and how does this uncertninty com­ pare with the initial momentum of each?

,

typical "alue of �l"\. Similarly assume that ap Eliminate

p in fa

'\lor

;a

\PI.

of x in lhe energy e:xpftSSioo.

(c) Find the minimum possible energy for the wrIe.

50. If things really do have a dual wave-particle nature, then if the wave spreads, the probability of finding the particle should

:::::

spread proportionally, independent of

the degree of spreading, mass, speed. and even Planck's

constant. Imagine that a beam of particles of mass m and speed l', moving in the x direction. passes through a sin­

gle slit of width w. Show that the angle 6t at which the first diffraction minimum would be found (nA = w sin (In'

53. The energy

01' a particle of mass m bound by an UIIUIWIl

�pring io;, pl/2m + b:c.

(a) Clas...ical ly. it can have zero energy. Quantum

m.:chanicall)'. however, though both x aDd p .e "oo avcr.lge" lero. it. � \:.

ted photon doe'i thi, impl)i'? INUIt":

{�

t:1\ce-··;,e., the energy of the photon-but he� it m..an'i the wu-t"fla;rll\' in that energy difference.) {b) Tn what •..tnge in wa\t'length, does this correspond'� {A... m,ted in Exerci-.e 25"7, the uncertainty principle is one C\)ntrihutor III the hroadening of 'ipectral lines.l (c) Obtain a general fonnula rel ating loA to �,.

lal s S

lal > ,

73.

where C is a constant. (a) Find and plot versus f3 the Fourier transform A(/1) of this function. (b) The functionft, ) might represent a pulse occupying = position) or finite lime (cr either rmite distance = time). Comment on the wave number speclrum f i cr is position and on the frequency speclrum if is time. SpedficaJly address the dependence of the width of the spectrum on 8.

a (a

a

69. A signal is descri bed by the

function

D(t} = Ce-l d/T (a) Calculate the Fourier transform A(w). Sketch and interpret your result . (b) How arc D(l) and A(w) affected by a change in 1'7 70. Consider the following function:

-00
ed on the,e three energies adding up. The exception arise.s if we

do allow the potential energy

to Jump to infifllty at any point along the x-axis.

for then the restriction agamst the kinetic energy bemg mfinite at that point fails. Obviously. mfimte potential energy is not completely realtstic. but the notion of a solid wall or barrier is a useful onc. In confimng a partIcle 10 .some region of

are

stoUl �pace, we often imagine that it i!" tlllO' ';lII\f), it 11\ the' I'Cgi\ln bt·twall the wail)o and Ihclt t'lisurilitt thal thc O\'C'r all ..... a\(' lun.. titln .' l.'ontlnul'U' In th\' (\'t:iul\ hclwee wbich I$. never O. WbeD Re � I) is 0. _ '9\x. ., .. whereabouts is a fact in the real situations to which quantum mechanic! i!> muilllUlTl. ItId \'lr;e wna. We miPl va.Iia • applied. We cannot, for example, say exactly what an electron orbiting a lluantllln-� aanding __ &'1 . .. _ lpinning ;d)out!be,,;.....u. ill die Re-1II1 ,.... nucleus is doing. Given these unavoidable limitations in the micro�opic

ignored in our solution of the SchrOdinger equation. The assumed simplicity

all

""

114

CNtpt.r 5

Bound Stales: Simple Casn

world. the predictions of quantum mechanics have succeeded. While those of classical physics have failed. Sight unseen. the "particle" is a wave. The standing wave of minimum energy is known as the ground state 8fId it.. energy as the ground-state energy. It is in the ground state that quantum. mechanical behavior deviates mosl from the classical expectation. Maq Important, the kinetic energy is nol O. A bound particle can�ot be stationary. As we learned in Chapler 4. this would violate the uncertamty principle, for having a position uncertainty comparable to L and a momentum identiCally 0 is impossible. Another deviation from the classical is that in its ground stale the particle is most likely to be found near the center. Classical1y. if we Were t� "rum on the lighls" suddenly and catch the particle somewhere in the course of its constant-speed back-and-forth motion. it is equally likely to be found anywhere within the well. On the other hand, the correspondence prinCiple (Section 3.4) says there should be a limit in which the particle behaves classi. cally. This limit is at the "other end"-large n. Larger n correspond to shoner wavelengths (An = Wn), and the shorter the wavelength, the more paniclelike and classical the behavior should be. As we see in Figure 5.8, the larger the n the more evenJy the probability of finding the particle is spread aver well-the classical expectation. Before we apply what we have learned, we reiterate that the wave func_ tion tells us probabilities of finding the particle at various locations. The inte_ gral of the probability density (probllength) over aU space is the total

th�

probability of I . So to detennine the probability of finding the particle in Some restricted region of interest, we simply integrate over that region.

EXAMPLE 5 . 1 An electron is confined in an infinite well. in the ground stale with an energy ofO.JOey. (a) What is the well's length? (b) What is the probability that the electron would be found in the left-hand third of the wel1? (c) Whal would be its next higher allowed energy? (d) If the electron's confines were roomier, L = 1.0 mm, while its energy remained 0.10 eV, what would be the probability of finding it in the well's left-hand third, and what would be the minimum possible fractional increase in its energy? , ,

(a) In the ground state, n = I . Inserting values in energy expression (5-16), we

obtain 0.10

X

1.6 X IO-L9 =

] 21T2(J .055

J

2(9.1 I

X

X JO-34 J ' sF

10 3 1 kg)L2

(b) The probability density is f¢(x)J2.

, (fL

I¢n (x)1

�

"'''

)

- sinL L

'

=" L = 1 .94 nm

2

"."x

L

L

� - sin'-

As noted above. we sum lbi!. over the region of interest, x = 0 10 x = U3.

probability =

J

pcobability length

dx =

J/� (x)/ n

'

dx =

J o

IJJ

2

-

L

sin-'

(",TX) -

L

dx

'., c.. I , _10 10 . ....-'""' - -

I'"

•

o

I 2n1f -- Sln�3 2mI' 3 I

- _.

�[� 6

L

-

L

';0( 2o"/3) 4,1'11"

J

Thus. for an electron in the ,. - I !>tate, , , I 1 21r probability - - - - !Oin:\

2.".

3

3&

I

3

- 0.137

�

0.196

The probability IS les!. than the classical exp«tation of one-third, and it a,n: would produce higher-energy. \horter-w3velength photom The characteri Mic of having a

ma\imum wavelength would help to di\tingui�h Ihis syMem from others-the

hydrogen alom. for imlance. in which higher quantum energy levels actually gel cloo;er together Spectral " fingerprinls" are mdirdmg III the ....ell Width-a \-�I')' nice: fcaturc=.

is.

Applying the Phyma

Gi,cn a :\ nm ....".k quantum ....ell in Figu� 5. IOb, what is. the w.velmJth of the em1tted phUlon-/ •

TI

N

The ene..,.)' 01 the 11

-

I ,talc i,

w1( I ,055 X \0 :\4 J _ __ :, ')l � � 2 ... 6.1 X 10- 1 J 2(Q, I I X 1 0 .ll I.g)(3 X 10 OJ m)!

-

o'042 cV

Adding thl� to Ihe energy diUerenclC, 1.50 eV, from thIC zero level down to the

valen!.;e level. the neow phown energy would be 1 .54 eV, or 2.46 X 10- 19 J . A

-

(J x IO" m/s)(6.6:l X 1 O- :W J ' s)

._..

2.46 X 10 19 J

-

801 nm

The model we: have: di..cus..a\ herc= shows how characteristics of the quaatum we:1I are e),ploite:d, but II doc\ ovc=rsimplify. In reality, the da:trons in the semkoft. du(tor, due 10 their inleraction� with the !Oc=miconducting material, behave u 1houah their ma.... is ..maller, "'hich would increa� Et and thus the photon erter'JY, Mole­ OH'r. the holes in the: \'aknce ,tate:' are in the Foan1C physical region as the conduetioD de(tron,. '0 they too gt.:1 trapped in the: quantum well and thus also Usume quu­ tiled energie:\, (Semi(ondu(tol"\, induding holes and effective mass, are discuuc:d iD Chapter 10.) In fact. more: ellon i.. now going into lasers based not on the quantum well bui lln the quantum dot (di\(u,ical condition of concern i!i nor­ malizabihty. It must not diverge, and thi

nicely to 0 at one end. but thi" use� up the arbitrariness of the ..lope. and ",(x) still diverges at the other end. We conclude that this E is "wrong" and move on to another, At only certain values of E. we find that for an} initial ",(x) at the "tarting point (unles... the \tarting point is by chance actually a node). we can choose a slope there in alom�. and aloms form halos in WaH! function

producin, ils own distincfive photon of lon8t'( wavelength.

Mol«uJar-based malerials thai talk in this unmistakablt'

molecules. the most commonly !'iludied halos are in certaia nuclei. where an excess of neulrons forms the halo. For instance. the common helium-4 nucleus binds two pl'Olonr,

waY-Ibsorbing one wavelength. exchanging energy.

emittinB another-without somt' kind of garbling � not

easy 10 find. The versalilt' quanlum dot can be tuned to the

and two neutrons In a very light umt. but helium-6.

specific application and provide!'i a nice solution. Halo. In

the CI...iCllly Forbidden Region

depicted in Figure 5.23. add� two more neutrons. which

haunt regions quite far from the nuclear core. Forces in tht atomic nucleus are very complicated. and halos Provide an important view in helping unravel the mysteries.

Wave

functions of bound panide� usually extend inlo the

classically forbidden regilln. sometimes very far. One

Figure 5.22 A quantum dOl DNA nanosensor. Sandwiched

DNA "i�'k� to a quantum dot (QD). " E�ciUltion" energy excite�

the dOl and t'merge\. affer FRET. as the emission (Cy5) photon

.... .. Sandwl!;hed hybrid

•

Sueplav.Idlh· conJugalcd QD

E�(;ilation (488 nm)

�

Nanosen�or assembly FRET

·':;'�ii,:;:r.;;:;'m

iSSion (Cy5)

Emission (QD) (805 nm)

their shatt'd "strong force"-. are more likely t o be found in the

proton bound together by

raoaIJICe CDClJY lranlfer (FRET). In this clever process,

(670 nm)

Figure 5.23 A two-neutron halo

around helium.

"""".. " - .. ..--I by ...._ ...

Chapter 5 Summary h' unJo:Nand

I b.:- "!uantum-n)C\,hmlt.,.'a.I tw:oha\'ior (.f _ masslvc

l¢,.e-.'t. Il lS rn..,:�"UaI) I,. lou.... I� ....,I'C' fUl\\.-tioo

hn!1t'f ("quAi,,", Il\lO ,.r tho: S.:h"... J"."..nI1al cn('rg� (I( 1), 1'1

AII -,"

mc-nl }tC'ldina • "alut b an ot.er...... MII:h • 1'__ GI'

tu�·h. In

...

'f'l.{,

n. I

tulu­

ltwo rre�n..�

"I'

$"hlllg th.. S..:h""hnger O:llualn'o hlr "'\I,tl 1:\ r\lU8hl� anill,... n/ ,/:rlJr: for 1"('), gOU\ 10 the ct ••" •.;••1 pn.:,,:o:lIure 01 \lll\ln� I' SCpaf'JUlln (If \'ari.lhk, )ieIJ.. tho: tlllll:-lIhJl'llCnJent S..:hrodingcr c1l until11l

IIkVnthlUm 'M.1UJd In � �k1 • dlfItftaI vUue 11 III 7J ldc'nti�·.ally The CJ.ptrilnml ("&MOC be .amply • &Mer ...... th'oR of Ihc wne a;ylWm. .. My obaecwdoa dlICurba it nw �ult \It 1lWl)" Idcntk:"aI fCpCbtioM would produce .. ....... knl'''' n liS lUI c'ra:tautln 'alue, IlIkI a .WIdan:I *"Iadon, wbidt lju41Uum rM\:h&ni,,·s dk.'lCa; .. it» deftRllion or w ... .. ... 'linl)'.. Tltc>y mOl)' tit nkulatC\l frum thr- Yra� function "Ia hows. Cla.....ically. should merely slow down abruptly at x = O. its kinetic energy dropping a,> potential energy jumps. then proceed at constant reduced !;peed. There is no classical turning point where its kinetic energy falls to 0 and at which the force would cause a reversal of its motion. The particle would not rebound. But classical physics won't do. To understand the behavior of ,>omething as small as an electron, we must apply quantum mechanics. In the region to the left of the step, x < O. we win have a right-moving incident wave, and we alloW for a left-moving reflected wave. Each solution of ( 6- 1 ) is stH! a solution if it includes an arbitrary multiplicative constant, which will eventually tell us how they compare; so for the incident wave, we use Ae+ilr, and for the reflected wave, Be-ib:. In the region to the right of the step, x > 0, where U = UO' the SchrMinger equation is

,p",(x)

2m(£

O = --C*Ck' A*A k 1 ",lf.i. 0,

Agure 6.'

Reflection and lraruomi!lihOfl

probabilities for a potential �tcp.

LO r---r---'--'_--�

I

O.

o. 0.4 0.2 ()

1

I�

then going. back through to x < O. for there i� no force beyond the step lhIt could reflect anything. We ..imply accept that as long as no attempt is made 10 lind a particle. we merely have an undisturbed wave that. though it reflects completely, happen.. to peneuate the classically forbidden step. The "penetra­ tion depth" Ie; the "arne a.. in Section 5.6, where the wave function outside the finite well wa.' abo ..imply a dying e;(ponenlial. I S = - = a

,

EILu

-;= �== h

V2m(Uo - E)

The clo.. er E i... to Va' the greater is 8, and the �Iower the decay of the wave function. In any case, the wave function is very small for x » 6. As always, eleclromagnetic waves show analogous behavior. Just as a Vo > E step completely reflects massive particles, a smooth metal surface completely reflects light-making it a good mirror. The light wave doesn't propagate through the meta!, and no photons are transmitted, but there is an electromagnetic field within the metal, oscillating with time and decaying ex.ponentially with depth. For an electromagnetic wave, the penetration depth ie; often called skin depth. Figure 6.4 shows reflection and transmission probabilities plotted versus EIUo for a potential step. If E is less than Vo' the wave is totally reflected. When E exceeds Ua' the reflection probability falls rapidly with increasing E.

6.2 The Potential Barrier and Tunneling Tunneling is one of the most important and startling ideas in quantum mechanics. The simplest situation is a potential barrier, a potential energy jump that is only temporary. If a particle's energy is less than the barrier's "height," it should not get through---classically.

E > Uo

Let us first consider the case where E is greater than the barrier height Uo. As shown in Figure 6.5, at.x ::::; 0, the potential energy jumps up, giving the pani. cle a "kick" in the backward direction, and at .x ::::; L, it drops back to 0, giving it a forward kick.

U(x) Figure 6.5 A clas�ically surmountable potentia) barrier.

=

{�O

x < O, x > L O < x < L

Classically. the particle shouldn't reflect; it should merely slow down between .x ::::; 0 and .x = L, then return to its previous speed. Let us see what quantum mechanics has to say. In the region x < 0, we should again have our incident and reflected waves. Between x = 0 and x = L, the situation is ex:actly as it was to the right of the potential step when E was greater than Vo' Thus, SchrOdinger equation (6-3) applies just as before, giving the same mathematical solutions: e+ik'� and

t--"" ,l. In the Cllse of me simple step, we threw out the h:tl.-mo....ing t

a '. (or beyoml \ ... 0 able to renect a wa.. e. Now th�� is-the nothing was there pOCential drop at x ... L Therefnre we cannot jUstifulbl)' thn\\\' l\ut lhi" solu­ tion. Finally, In the region ,\ '> I.. where the potential �nerg) is a�ain O. the SchrOdinger equation i'io the �ame as fl1r \ < O. a... Ilrt tts mathemattcal solu­ tions, t'*/lr and t-Ill . But here W I! d11 1hrow one out, r'- loll , for O\\thing mo..�s to the left in this reg\On. Altogether. we h:\\'e

tII1 not energy quantization. 'ncident particles may be sent in with any energy, bur resonant (complete) tran.')mL�sion occurs 31 only certain energie�. A familiar behavior in light (yel again!) is completely analogous. Nonreflec. tiYe coalings exploit the same wave properties. If !.he wid!.h and refractiYe index are cho.�en properly, a thin film will pass lighr withoul renection. In both applications, rl!llections within the film ("over" the barrier) preci!;tly cancel lhe wave reflected from the first interface. As always, it is important 10 remember that wa\'es. nol particles. interfere. A particle-photon or massive--encoullIering an obstacle may later be detected as a parficfe. having been either reflected or transmiued. bUI the process inyolves a probability that come!> only from the It'(f\'e function. E
C', there ,:- no 'real m�lll\C'ntum in....ide ' Ium::llon.. 110 IlIlt rerott'>C'nt nght- llml le{t.mo\,ing ['W1'­ the b;lrrier. ilnd the '''''0 .:il''' Still, bllth are necl1et.l tll cn..ure ..moIJthne..... It i.. lert [I" an c'erriloc to ohtain thc 'mllo,lthne" I;onditilln... fnlm whkh the rdle..:tlon and lnln"ni�"'lm rorohahilitie... lE).C'rri-.e 1� :-.how thllt find ", eo they l;;Jn ;\l;tu311) he dcdul;cd by a ..impic f('pi;Jl;l'menU R

T

'

t

..inh� \/�m{ Uo

,inh)1 \/2m(Vo

/:')L/IIj

+

''

E) VIII

�(EIVo)l1 - E.'!Vu)

4(£/ Vo)(\

�Vi,,�un

"i nh

E, Vo) E)i�hT+ 4(£!Uo)(l

FIgure 6.1 Wa\'e 'un.;ti�\n� (or pani. in gcneral. nOtl/,ero--a particle can escape through a barrier thJ.t it cle .. �'I dillerent energie.. inddent fnun can't �urmount cla....ically. H ilc'ld, il "tunnels" through. and the principles of quantum mechanics demand that there be such a possibility. The �olulion to the left of the barrier is ..orne t.� -----�c--"'"'--c��_.,,-� combination of incidenl and reflected wa\'es. of po!'oili\'e and negali\'e momentum. At :r: = O. it smoothly joins a function inside the barrier that tends to die lc.J ------'c--�--���­ off (the C in the eKponentially increasing C,e-+a..'being u!>ually quite ..mall). And at.� "" L. thi.; "moothly joins a transmined wave of positive momentum. There ;!. f' l -'""'-­ no physically acceptable solulion thai is identically zero beyond the barrier. Figure 6.7 show!. wave functions (real part) for particles of different enert:, _ .... gies incident on a barrier from the left. Note that the wavelengths decrease a!> kinetic energy increa1ly different.

F&gu.... 6.2' . ..., ,

We have a complex exponential that moves at speed wdko' the phase \elocity central value!\ ko and wOo modulated by a cosine function appropriate to the dwldk. A!\ !\hown ,In Figure 6.21, it is the latter tern, th,\t speed at moves that defines the envelope-the group velocity is dwldk. We see this more clearly bility density. by computing the proba 'I'·(x. I)'I'(X,

t) '= 4A2 cos2{(dkh

-

(dw)ll

(6.20)

The phase velocity disappears. Because a particle must move as the probabil.

ity of finding it moves, we expect the speed of the particle to be best repre·

\'l!F,

dwldk. Our surprising conclusion is that the sented by the speed of depend on the actual central values Wo and ko' doesn't group/particle velocity bul rather on how w varies wilh k from one constituent wave to another.

Although this simple wave group exhibits an important feature-that the

e�d �wldk- it isn't very particlelike. it is peri. probability densit� moves at sp : odic, as is any jillile sum of penodlc funcl1ons. So let us now tum to a more

particlelike case in which the probability density isn't spread out all over space.

A Particlelike Wave

The most general way of expressing a wave group is +00

'V (x,

t) �

100

(6·21 )

A(k)ei{h-w,) dk

This is a sum of plane waves of all different wave numbers, each multiplied by

its particular amplitude A(k), and each including its time dependence. (It is equa·

rion (4·2 1 ) with time dependence added.) Position x and lime

I are the usual

independent parameters, and we will address A(k) soon, but where do we get w'?

This is a central question, and the answer is that it can be considered a function

of k, for each phenomenon has built into it a relationship between w and k, or

equivalently between E and p. Consider two familiar cases: electromagnetic (EM) waves and matter waves. From equations (4-7), we

see that for plane·wave

solutions to Maxwell's equations in vacuum, w = ck. equivalent to E = cpo For plane-wave solutions to the free·particle SchrOdinger equation, equation (4. 1 1a)

showS that w = hk2/2m, equivalent to E = p2/2m. A relationship expressing frequency as a function of wave number is known as a dispersion relation. EM wave dispersion relation: w(k)

Matter wave dispersion relation: w(k)

=

ck

� -

2m

(6·22)

(6·23)

Slmple wave IJOUp.

216 Chaptw 6

Unbound Slain: Sleps. Tlllmding. and Partide-Wa\'e Prttpalafion

Before looking closer at the group behavior, consider the phase velociry in these [WO cases. By detinilion. phase velocity is just the standard fonnula for the velocity of a plane wave.

I'

pha\C'

W = A( = . k

(6-24)

Thus.

EM waves: I'phase

Mauer waves: \I

ph!N:�

= W

k

w k

= C

(6-25)

lik

= -= -

(6-26)

2m

Whereas electromagnetic plane waves. unsurprisingly, all move through vacuum at the same speed

c. matter plane waves of di fferent wave number

move at different speeds. Here is where the distinction between pbase and group velocities arises. Our main object of study is a wave group Whose In the case of malter wave numbers cover a range of values centered on

ko-

waves, the phase velocity corresponding to this central wave number would

Vo

= po/2m = vol2, which doesn't equal the velocity of the par­ ticle. However, the phase velocity is not the important one. (This is quite a be flko/2m

relief, for as we soon see. it can exceed

cO The wave function (6-19) for the

two-wave group bears this out. Its complex exponential moves at the central, or average, phase velocity

wr/ko•

but (hal part of the wave function disap. pears in the probability density, which is what really corresponds to the par· ticle's mol ion and moves at a different velocity. Does this also hold in a more general group? As written, wave group (6-2 J ) is general, but almost always we are inter­ ested in a wave function of a particular shape, and one of the most conunonly considered shapes is a Gaussian wave packel-a single bump. At

t=

0, we

desire our wave function to be of the form

(6-27)

Note that this is a right-moving plane wave multiplied by a Gaussian bump that falls off away from the origin. Because of the Gaussian factor, it is nOI infinitely broad (� *-

)

XI ,

�o its momentum is not perfectly well defined.

However, the oscillatory complex exponential gives it an approximate wave number of and thus momentum of Mo. In fact, its real part, involving cos

ko kox. would resemble the top/starting wave

in Figure 6.20. The way we

ensure that integral (6-2 1 ) agrees with (6-27) at 1 of

= 0 is by the proper choice

A(k). Appendix F, applying aspects of Fourier analysis covered in Section

4.7. discusses what choice is needed, then tackle� the integral. Here we simply study the result. For the wave function given at

1 =

0 by (6-27),

equation t6-2 1 ) gives the wave function at arbitrary lime r. and ils IU\lbabH­ it)' deosity IS

1"'(\, 1)1' where �' -

(b-2Rl

dw(k) \ dk '0

--

and

Given the many factors in (6-28). it is worthwhile noting tiNt that it is cor­ rect at t = O. It reduces to

which is indeed the complex. square of (6-27). So what doe!". it do gresses? Taking it one piece at a time, consider a ca�e where D

==

as ,

pro­

O.

(6-30) This is just a pulse "sliding" along the x-axis at speed s. and given the defmi­

tion of s in (6-29). we see that this is the same result as for OUf earlier two­ wave group.

Note that nonzero D wouldn't change this conclusion. As we soon find out, D governs not the speed of the probability density, but how much it spread!;

in time.

Let us calculate the group velocity for the familiar cases. From equations

(6-22) and (6-23),

EM waves: v group

Matter waves:

\

� dw(k) � !!.. � dk dk Ck k C

� dw(k) � !!..hk' group dk dk 2m

v

\

(6-32)

o

flko

'0

� -

The group velocity of an electromagnetic pulse in vacuum is

III

c.

(6-33)

because all of

its constituent waves share that phase velocity-they must move as one. Equa­ tion (6-26) told us that a matter wave's constituent plane waves move at differ­ ent speeds, but group velocity (6-33) is just what we expect it to be: Maim = Po/Ill = vO' the velocity of the particle. No matter what may be the

Z1. CIwptar 6

Unbound Scala: Steps, 1\mncling. and PmtJde-Wave PropagatIon

Figure 6,zZ Dispenion relations for

EM and matter waves. 'The phase

velocity is the slope of a line from the origin. while the group velocilY is the slope of the: tangenl line. EM wav6

w=d

Mauc:rwavb

w \'pb&M

-

wilt.

/ /�

w �

"iI?l2m

'J ,

•

phase velocity of constituent waves. the region of high probability moves • the speed we expect of the associated particle. How do OUf results relate to Figure 6.20? As noted in connection Wirb equation (6-27), the wave function starts out looking just like the lOp plOi in the figure. As time progressc!\. its little crests do indeed move at a phase veloc. ity different from the speed of the envelope. (The actuaJ 'I'(x. f). which is a bit messier than I"'phere leads to a refractive index that varies according to

n(w) where

w

=

JI

-

b

w'

is angular frequency and b is a constant. (a) Find the dispersion relation.

I.

(b) For a pulse of central frequency wo' detennine the phase and group velocities. (The

reader may quail al seeing a refractive index less than for it implies a phase veloc· ity greater than c. We confront the seeming violation of special relativity afterward.)

SOl\,;,', )N

(8) By definition. the refractive inde� of 8 materiai is the ratio of the speed of liaht In vacuum to the sp«d of 8 pure electromagnetic plane wave in the malmal­ that is. to the pha� velocity.

e

=

n � --

\'phase But the speed of a plane wave is also

w k Solving for

�

wlk, so that e

VI - b/w'

w. we obtain the dispersion relation. w(k)

�

Vb

+

(ke)'

(b) Using (6-3 1),

We now reexpress this in tenns of the given

=

wOo Using (6-34). we obtain ko

= � Vw'0 - b e

Thus,

- b/w"ij Finally, evaluating the phase velocity also at the central frequency, we have e \lgroup

= cYI - b/wij

The example seems to make a very unpalatable claim: The pulse's group velocity is okay. but its phase velocity is greater than

c.

We don't have to look

far for this "problem" to recur. When mass/internal energy is taken into account. the same holds true for matter waves, even in vacuum (see Exercise 44).

Is special relativity violated? 1t is true that any individual plane wave may

travel faster than c. But a pure plane wave-of infinite extent in space and infi­ nite duration in time-cannot transmit

infonnation.

It doesn't vary in any

significant way, in a way capable of conveying information from one place to another. On the contrary, to transmit any intelligence, the wave must be modu­ lated in some way, perhaps varying amplitude or frequency or simply turning it on and off. When modulated, it is no longer a single plane wave, but becomes a

220

CIu!pter 6

Unbound SlalCs: S�PS. Thnneling. and Panide-Wa\'t Proj)a!!lIlion

combination of plane waves-a wave group. The information travels at the group velocity. which is less than c. Under somewhat exceptional conditions which have understandably garnered considerable attention (see Progress

Applications). even group velocity may exceed

and

c. but this requires further

rethinking of what it means to transmit information. and in no case has a vio­ lation of special relativity been suggested.

Dispersion Not only can differences in the phase velocities cause a wave group to move al a speed quile different from its constituents. but if can also lead to the phenomenon of dispersion. the spreading of a wave pulse. Dispersion arises whenever the dis­

D in equation (6-29}--the second

persion relation is nonlinear; that is. when

derivative of w with respect to k-is nonzero. Probability density (6-30) assumes

D = D, and it describes a Gaussian pulse simply sliding along the x-axis, unde_ fonned. at the group velocity. However, when D is nonzero, we have to go back

to probability density (6-28). In the denominator of the exponentiaJ's argument D causes the moving Gaussian to become broader, ultimately a constant of I . In

the factor multiplying the exponential. it causes the probability density as a whole to decrease. so the pulse flattens out. Thus, the probability of finding the particle

spreads over an ever larger region. while the probability per unit length climin­ D is caJled the dispersion coefficient.

ishes. Governing this behavior,

Dispersion would occur for the GPS pulse in Example 6.3, because the (6-34) for the medium is nonlinear; for the same reason, it

dispersion relation

occurs for a matter wave

even in

I'GCUWlI. As shown in Exercise 48, probabil_

ily densily (6-28) becomes

1 'I'(x, /)1-,

=

VI

C'

+

" '/'/4m'.'

Note thar because the factor

1I.212/4m2£,4,

portional to

e-4•

[ -(X

exp

2.' ( I +

-

Sf)'

n'/'/4m'.4)

]

(6-35)

which causes the spreading, is pro.

the narrower the pulse's initial width, the more rapidly it

spreads in time. Exercise 49 investigates the phenomenon. It is worth reiterating that our wave group is a solution of the free-particle Schrodinger equation, just as is a single plane wave, but unlike the plane wave, it is a good description of a well-localized particle. Figure

6.23

shows

the time evolution of the probability density of a moving free particle. Disper­ sion invariably leads to increasing uncertainty in a particle's position.

Figure 6.23 Dispersion causes a matter

wave to spread.

1'1'1'

..

o

'I

.... ... AfPII .... ..,

P ROGR ESS A N D A P P L ICATIONS Resonant Tunneling Diode Photon Detector

,\pplil:ati(ln� of tunneling in modem ek�·tronics are muillplying rapidly 1ll1waday\, One holding great promi�c. tlc\eloped by ;'I:ienti�t!> at To�hiha Re�an:h Europe anti ('ambrid!!e llnivcrp and manipulate microscopic Objects. and the atomic-scale dexlerity of the STM i� increa!>ingly an�wcring the call. Researchers at the University of Berlin ha\e employed an STM to carry out a chcmical reaction with single molecules. As shown in Figure 6.16. an STM is used a!> a source of electrons (a) to loosen an iodine from iodobenzene. leaving a phenyl (b). The iodine i\ coaxed to a convenient. out-of­ the-way location (c) by van der Waals interactions and chemical force\ between it and the tip. By the same

Figure

6.25 An STM with X-ray vision.

,..... 6.26 Sin,.moIccuk SUIJUY via STM

very Impor1anl 10 the pnx:essing and !>Iorage of quUItUIb information. At the olher e�lreme. \e\-eraJ experiments In Ihe pa.\1 dec;Jde or liO ha\'e demomtrated a group velOcity for hght greater than (. The efleet (l(curs in medIa at frequency ranges chanu:teri/ed by anomalous dispenlon ' in which the refral;ti\o'e mdex. contrary to ilS usual bt'ha\-'lOr. del,;realiC' With frequency. Whenever somcthmg-sul;h a� pha�e velocity-appears to mo\-'c at wperluminal \pc:edli, the first queMIon is always whether it can be u�cd to transmit information In aI/ l'aSe\. the answer hih been no, Some feature of a pulse may \eern to travel through a region faJ;ter than light could through a vacuum; il may even seem 10 emerge before the apparently corresponding feature in the inci. dent pul�e cnlen. BUI it� information i", encoded in corn. plex ways. and even the "front edge" of an incident pul�e

force!.. [he phenyl is slid O\i�r Cd) 10 anolher phenyl. Ihen. with e:U'ilalion pro\o'ided by another ..hot of lunneling elel.'lrons Ce), the IWO ar� "wcJtled" into a biphenyl mole­ cule

10. (Saw-Wai Hla. el .11

Seplembt'r 25. 2000. )

M.ny Spe.ds of light

.•

Phy�il;al Re\'iew Leners.

carrie\ information about the whole, so Ihe arrival of any feature " not really Ihe question. When anomalous dis­ per"'lon prevails, II i� Io:ommon to speak of a "signal veloc, ity" dislincl from the group velocity_ and never bas this been found to c,'(l;eed c

The group \'e!ociry of Iigh! has

bet:0lTl( qUite a hOI I{lPU: in m:enl yem. a\ new experimental tcchniques are probing all e\lreme\. In the lale

1 m... light'�

group velocity wa., ,Io\\-ed to les� than I mI� by �ending il

Figu,. 6.27 A /ighl pulse "stopped" and restored.

IOto 8 ruther exolic medIUm. a Bose-Einstein conden\atc

(or BEC. � ChOJp!:cr 9 Pro�rts� and Applicatiom), in which lhe rrfnKIJ\'t' inde'( inl;re.I-\e$ abruptly. Other methods of slowing Iighl pul'o('s 10 petle,trian speeds an:: also being !otudied, Ughl of sUl;h slow group velocily may lead 10 new /ighl swil,he, orother opt�It!(:tronic devices. 1I may also prove applicable 10 nonlinear optic.s. in which a medium doesn'l respond linearly 10 lhe electromagnetic field. This effeel l" exploited. among other lhmgs. to double laser frequencies. Presently, nonlinear optics requires high-power

(. )

mm

'-:::----'

lasel'i. but slow light may provide a low-power alternative. The abilily to slow lighl ha.. recenlly led to a further Iwis!. In \eH!ral experiments carried out at Harvard, MIT,

(.) '----=--'

and the Air For..:e Research Laboratory. using both gaseous and solid media. the group velocilY of a light pulse has been s/o","ed ,really as il inferacts wilh the media, at which POlO! a separate la�er �ignal beginli a lransfonnation of its information from light to atomic lipin �Iates. a\ depicted 10 Figure 6.27. and ending in a final "image" frozen in the "pin slale.\. known a� a polarilon. A later laser signal retrieves the light pulse on demand wilh little di\lortion. Thi� ability [0 stop a lighl pulse may prove "

"

/

U"f f

« ) '----'

10 '--_----'

Chapter 6 Sum mary Just a.'. it has an effect on II massive particle. any change in potential energy ha� an effect on ,II maner w�ve �nction. A ..mall particle subject to II force Will often manIfest Its wave nature. behaving II.. II classical panicle would not. It may be reflected by a force that would not cla!>sically reflect it, even by a force in the particle's direction of motion. Conversely, its wave function can pass mto II region despite a force thai classically etcnmnl' II and [J (the amplitude of the

l . I' eljui\'.tlcnl [{) 'ft' , l ,han(ol nf;cl,I.b)' PUl\'IIJCJ that IA ) ,."" 8" • + d)

Show thul lP'( t I

IjIt_> A' ·

!til

+

A'rA'

t

J(II

14. Verify Ih...t Ihe nan, 11\t.'I\l!! and rell�,�tlon pmoabdJllc, aiwn in cqualltlll II'!,7) udd hI I

IS. C"lt;ulale Ihe rt"IlCClulII pnl",,""JI) hll a � (,\' electnm encountering II ''It'P In whl�'h tht' p"lenI131

lIrvps by 2 c \' 16. A p4J1i�,:le mtWUli In a rt·8 1\lu {If lero fon:e t"IKounle,.... a precipu:e---a sudden drop 10 the jXllenllJI cncrg)

10 an

arbltranly large neSJ1I\1," \;lIUt', What is the pn."abiIHY

dUll i t will "�{l {l\c:'r tht' CtI�C"'1

; f; I' dt" �'fl� h) the waw

17. A beam of patlld� of C'ncrg)' E and ,"{" den! up,m .1 pc.lll"flli�1 Mt'P llf U(I fUnt'li on

(/11M: ( .-)

wa\c 1O'".1e the ,Icpl III the hmll� A

I ,.al'

(.) Detemlinc l'{'ll1pklely Iht' ret1e�'fcd wave and the wa\.'e ,",,,.Ie Iht' ,tep hy enf'()rl'ing Ihe required

if

...

O. and mterpret your re,ult,

...

\

)0 0

0 ;lnd

20. P.Lnldc' �ll energ) 1- an: incidenT from the Icft. ",t\t.·rc

Up)

t:.

O. and ill the orig i n encounter an abrupi dr'lp In 3£. (a) CI"'�lcally,

jltllentlill energy. \\how .

43. Example 6.3 gives me refra(.'uw: index for high.

frequem:y electromagneti(.' radiation passing throop

� in lenDs of. flCtor P ,i...en in Exercise 30. (a, SuppoIC dw in SO11K' system of units, J. Dnd a 3.R'

Earth's iono)ophere. Thc constant b, related (0 the SO­ called pla.�ma frequem:y, \'anes with aunosphcric c0n­

ditions. but a typical value is 8 X IO'� rad1/s1. Giveo .

boIh 2". find two ...alues of 2f that give �sonanf tun·

nel;n., What are lhcsc: dis� in ICnns of wAvelen,gths

of Ill? Is the tenn rt'.JOfUUll 'unn�/jng appropriate']

(b) Show mal lhe ,ondition has no solution if J

= D. and CApJain why this must be so. tc) If a drus;nil parti­

d� wanlli to �urmounl a barrier without gaining energy. is adding a s�ond barrier Ii good solution?

Section 6.4 41. The maller·wave di)opersion relation given in equation 16-23) is COn'Cct only ut low spce- :"�) h'lIo\\ , from (bo21"', th;ll p.... .... · leJ.n1ed in E'anlrok 4.1, In Il GaUM1an fun.:u\ln A, , .,l lhe 1,lnn I$lan WI'''! tun.:h1ln "'",'\lld � prop',rti,'nal l.... � \) ..quareu t'-I "; ! .... ) C\lnlpJ.ring "'" 'Ih the IIme­ IkpcnOelll G'1U�..lan pmhabihty 01 �uJ.hon ttl-'�). ""'t \et. Ih,ll Ihe un.:el1ainly in pt" llion of the hnle-(','ohing Gall"i;ln wave functllln of II free panicle i, gi'cn b)'

"'-1W¥l. il" \\"I'Io'ft1 eneraY ltale would.. of couno. be \be p:t\lUnd ,tate, bul ,.,ould It be bound'? AuuI'M 1hM• •,

letit tur 110 Yohtle. It OI,..,.-upws. Ita "round stale. wbleh l& much I'Met Ihlm 110, and lbal the burien q\lelity as. \1;ide, Sh\'IW

thllt 0. t\l\I,h avenae time it 'NOUld renWR "'-,und i, klm:" by T (mW'/lOOMLl)a2... . where ,r

-

1.\'�mllLvfa

L

11ml ,.., 11 ,u ut. .. ill f: and in.:reru;e, wiln lime. Suppose the wave I"un.:lion of nn electron is inilially delennineu lO he (I Gau....ian oj 500 nm uncertainty. How long will il lake for Ihe uncertainty in Ihe electron\ po�;tion to rellch 5 m. the lenglh of a typical automobile? 50. Show that the qUite genernl \\ave group gi"en in equation (6.2 1 ) i:. a �olulion of the free-particle SchrOdinger equation. pro\ided that each plane \\ave's wdoes sati,f)' the matter ....ave uispeJ'!;ion relation g;,'en in (6-23).

Com prehensive Exercises Solving the potential barrier �moothness conditions for relationships among the codTitienls A. B, and F, giving the reneetion and tran�mis5ion probabilities. usually involves ralher mes�y algebra. However, there is a spe­ cial case that can be done fairly easily, though requiring a slight depal1ure from the standard solutions used in the chapler. Suppose the incident particles' energy E is precisely Uo' (a) Write down solulions to the Schro­ dinger equation in the three region\>. Be especially careful in the region 0 < .f < L. It should have IWO arbitrary constants, and it isn't diffLcult-just different. (b) Obtain the smoothness conditions, and from these, find R and T. (c) Do the results make sense in the limit L _ �? 52. Show that if you attempt to detect a particle while tun­ neling, your experiment must render its kinetic energy so uncertain that it might well be "over the top." (Hint: Apply the uncertainty principle, and note thaI the parti­ cle must be localized within the penetration depth.) 53. A particle experiences a potential energy given by U(x) = (.r2 - 3)e-� (in SI units). ea) Make a sketch of Uex), including numerical values at the minima and

51.

F:\cn:i�e 54 �\\('� il n,u�h Iilelime for a trapped panicle t�l e�(a� an cndllsu� by tunneling. {Il) Cunsider .n e\el.;lmn. Gi\'en lhat W � UK) nm, L - 1 Itm, and ", nd «- Uo&.'\liiumption Uo - � cV. fiN \erHy thllo! the: f.. hold�, then ("'aluat!! Ihe hfetimc. lh) Repeat part lo.), but flu a 0.\ p.g panide, \\'ith W '" 1 mm. L =0 I. �, and a harrier heighl lio tnal equal� the energy the panicle would have if ,t� �pc('d were jUlii! I mm per year. 56. Exen:i\oC )\) gi\'h the: ..:undition for resonant tunnehR8 through two bLlll"ief"\. l"oeparuted by 110 I>pace of width 2s, expres,e!.l in term, of 110 fa..:tor 11 given in Eurcise ?>O. Show thaI in the limit in which thebarrier width L _ :xl, thi� .:ondition become\ exactly tnergy·quanlilaliQR con­ dition (5-22) ,'Of the flRile well. ThUlii. resonant tunneling occurs at the quantized energies of the interve"'R8 well.

�5.

Computational Exercises 57. A computer can solve several equatioRs in several unknown� ea..,ily, and here we study a particular E > Uo barrier problem, where all the values it needs are real, Once the tompUler finds the multiplicative CORSlants of all the function� involved, we can verify equations (6- \3) a� well as 'OCe what is happening when the particles are "over the barrier." Sti\l, it helps to simplify things as much as po��ible. With length. time, and nuw. 1ll our &a­ po�al, we can choose OUf units so that the panicle mIlS m and the "a1ue of Ii are both I. and the barrier width L is exactly '71". Suppose that in this system of units, the energy E of the incident panicles is I.. t 25. and the barrier height Uo is I., Furthennore, because only ratios 1ft ever

7 Qua ntu m

Th ree D i mensions a n d t h e Hydrogen Atom

Chapter Outline

7.1 The SchrMinger Equation in Three Dimensions 7.2 The 3D Infinite Well 7.3 Energy Quantization and Spectral Lines in Hydrogen 7.4 The Schrodinger Equation for a Central Force 7.S Angular Behavior in a Central Force 7.6 The Hydrogen Atom 7.7 Radial Probability 7.8 Hydrogenlike Atoms IJ 7.9 A Solution Examined IJ 7.10 Photon Emission: Rules and Rates

P

erhaps the most profound failure of classical physics is its inabil­ ity to explain the simplest possible atom: hydrogen-an electron

orbiting a prolon. According to classical physics. the atom should be unstable. Any lime a charged particle accelerates. it emits electromagnetic radiation. so the continuous centripetal acceleration of the hydrogen atom's orbiting elec­ tron should cause it to lose energy and spiral into the nucleus. What we observe, however, is that atoms are USUally quite stable, emitting no radiation. They can be induced to radiate electromagnetic energy, but they do so only at certain frequencies. And while easily demonstrated with a simple diffraction grating, this too defies classical explanation. One of the first great triumphs of quantum mechanics was its explanation of these observations. But before we can understand the hydrogen atom quantum mechanically, we must extend the SchrOdinger equation to govern matter waves in real, three-dimensional space.

7 . 1 The Schrodinger Equation in Three Dimensions

to one dimension. the SchrOdinger equation is

a U(x)'l'(x, t) = ih-'l'(x, t) at

231

As we know. this equation

IS

ba."iCd on an

ene� accounting of the t general. a

rur.:.:: I!.I! . ��.... Sec�'" GIIIt

U = E. !qJBliaI coordinates.. U(x. _v, .:}. as Iii me wave fuocllon. 'if! t, Y. to I), In Ihrte dim a product of three functions. each of a different indepe aclaa

variable. and here the ohvious choice is Cartesian coordinates.

o/I(x. y, ,)

=

F(x)G(y)H(z)

Following the u�ual recipe for separation of variables. we next insen this pr1Jd. uet into

(7-3), then divide by the product Expressing V2 In Cartesian cOClf'Ib. U IS 0 Imide the well. we thus obtai n

(

nates, and noting that _

�' il'F� X)G(Y)H(:) 2m

+

" F(x)G(y)H( ,) ill

ax'

+

a'F(X )G(Y)H('» az'

F(x)G(y)H(z) =

)

E F(x)G(Y) H(z) F(x)G(Y)H( z)

Canceling top and bottom where possible and mUltiplying both sides by -2mih2• we arrive at

-=o'-:F.;:. (x",) F(x) ax' I

+

g

o'G(y)

G(y)

01

+

-� = H(z) az' I

a'H(z)

2mE

(7-4)

("

Each of the spatial variables is now in a separate term, so the usual argument applies: If the equation holds for X ' YI' '\, it can hold for x ' Y I ' z \ only iflhc I 2 term Involving x does not vary with x. Each term must be a constant. Thus,

d'F(.1,ntly oqall�_ W('re \,h(') I»" h�e. the sollJ·

t,om wou.ld be UponeOllal nther Ihao "nu"",,,jal Ellcmw 17 dl!'mon\lnlC1.. hov,·c'cr, thai C\por1I!'n·

lial� cannot be 0 al both wall\

function IS 0 outside infimte walls, continuity demands that it go to 0 at both

walls. These are exactly the conditions that gave us standing waves in the infinite well-the solutions are essentially identical.I

F{x) =

nx7TX Axsin L ,

G(y) =

ny7TY A sin -Y

Ly

H(z)

=

nz1TZ Az sin T

,

ID

(7-6)

'Tbt only rn.1 difftf'ml.� is thai �ach dirnens.eon is lIadependeni. 10 the COD­ ,WII' • ond L I1. ·

be 0, f

(1. 1. 1).

The corresponding wave function is . 1 1TX . 1 1TY . I1TZ

sm-- sm-1/I1 1 1(X, y, Z) = A SIO -2 3 1

With three quantum numbers, it isn 't quite so obvious what state would have the next·higher energy. It is left as a quick exercise for the reader to verify that itis (n..., r1y, IIz ) = (I, I , 2) .

Chepter 1

236

Quantum Mecha

TABLE 1.1 States in

the

3D

fr�� / £�,J'.#I, 2"';/) 6

2. I. I 1 . 2. I I. 1 . 2

6

.

11

3. 1 . I 1 . 3. I 1. 1 . 3

11 12

2. 2, 2

14 14 14 14 14 14

\ . 2. 3 2. 1 . 3 1,3,2

2. :l l 3. 1. :!: 3, 2. I

2 , 2:: ., . .:.

.J. I, I

1 , 4. 1 4 1 ..:

1 , 4, 2

2.4. I 4. 1 . 2

4. 2.

1

2. 3. 3 3. 2. 3

3. 3, 2

4. 2. 2 2.4. 2 2. 2.4 1.3.4 3. I . '"

1 . 4. 3 3.4. 1 4, 1 , 3 4.3. I

3. 3 . 3 5. I . I 1 . 5. I I . 1, 5

nil · m

18 18

19

-

19 19 ...:.:,

_ _ _ _ _ _

21 2' 21 21

21 21 22

22

22 2' 2' ,.

25 25 25 25

25

25 27 27

27

27

bers corresponds

10 a

. 51Tx

17r)' h rz sin -s' 10 sin L sin ---=L L = A sin L L x 1 7r)' 57TZ . 1 1T l-rr z 57T)" SIO -- sin--s' A l-rr x 5 sin L L L L sin= A sin L L coordinates.(x, y, Z). mum values at different maxi their have these else. enero-y-IS call d same g the If nothin ce-different wave functions. having The coincidenand energy levels for wh.ch .t .s true are said to -rrbe-h2dege�ial' 11 ) .IS The energy 27( ' /2 degeneracy, l). contex this in enl judgm l 2 ) are mora L 2f. no '/2m 1 2 ( -rr ' plying and sa.d (im -rr2f,'( 2mL ) degenerale. Levels 3(onds funclion. wave said to be 4-folderate single 10 a to be nondegen -each corresp }1TX

37TY

3-rrz

= A SIO-"'5. 1 . 1 L '" 1 . 1 .

:.-.. " ::

_ _ _ _ _ _ _

1 . 2. 4 2. I . 4

•

17 17 7 1:: ..:

_ _ _ _ _ _

3.3. :: '

energies. Of those ers for many allowed numb um quant 2 of sets WS j2rnL2) correspond 10 l 2 ( 7T2f't Table 7.1 sho the energies 3(-rr2fl2j2mL2) and . , 2 2 I ) and (2 I , ( I , only y shown of quantum numbers, respecuvel (- The different sel,' four sets 0 from quan. que lts uni ce, resu /2mL' ) , for instan set of -rr '-h ' e(lch qUOnt 27( y, 11m energy ng Ihe same. energ BUI despile havi ers. numb . tum cflon. The wave funclions are different wave fim

II

-

1.3.3 3. 1. 3

real three-dimensionaI World '" our in ons cati appli any to m e the same energy and the number of equaI-energy An idea crucial s can hav metrY 0f Ihe syste state ' m. that multiple . sym the w.th s ease possi ,Iales incr the box is as symmetric as ble: a cube. Suppose Deg eneracy

Energy (7-7) becomes

9 9 9

, " ,. I

I.

Hydrogen Atom

6

1 2. 2 2. I. 2

3, 2. 2 2. 3. 2

well

3

I. I. I

nsions and the

Dime nic� in Three

10_

=

0

•

EXA M P L E 7 . 1

'

e

cubic 3D infinite well �f I nm ( I . 2, 1 ) state of a the (n.r' n�_, n�)on's energy n. (b) Where is I e P1(4)) sinusoidal, the

,




'

��

:

ell

is

(7·23)

� mteger.

shows that if this functi n is to meet itself smoothly whe n cjJ must be an The smoothness condition changes by 21T, then _' this angular situation. a periodicity condition-has therefore given us a qu tum number associated with the 4> coordinate/dimension. We choose the s Exercise

bol

36

Vi5

m[ for this integer. Attaching a subscript to distinguish one

function from another, we thus have

For each

� uantum

� ber

nu

me

�

tha

=

0, ::t l ,

::t2, ::t3, . . .

I�

an allo�:�

(7·24)

arises-and two dimensions are yet to

come-a physical property IS quanuzed. The property quantized according to

Figure 7.9 St.mding wll\'es on me ..�·llXis...

I

,, I, , , ( , ,

,,

,,

,

,'\, -

'----"

"

"'t -

, -�....

,

,,

mj

-

%. 0

,

,,

�

r

\

,

,,

,. ,,

,

,, I

,

) � ",f' ,I I, , I

,, ,

�"

-

- � ..

,

no,

,

,�

,

, -�

IIIl == 3

,-

,, ,

,,,

0

2

,,\ ,

�, ,

,\

; -Y

'

the value of tnt is the z-component of the electron's angular momentum. Lz: This may not strike the reader as even remotely intuitive at first, but consider Figure 7.9. which crudely pictures the real part of einl/4>. The dashed circle repre­ sents a "cJ>-axis." For mf = 0, eiQ4l is simply 1 and is thus the same distance "above" the cJ>-axis around the entire circle. For mf = 1 , eil4> is cos 4> + i sin 4-. Its real part passes through one whole cycle, varying from positive to negative and back. as goes from 0 to 27T. Similarly, the me = 2 wave. ei 24> = cos24- + j sin 2cJ>. passes through two cycles in an interval of 27T. In general, the circle's circumference would be an integral number of "wavelengths": 27T1" = meA. If we make the independent argument that a "wavelength around the 4--axis" should be inversely related to a tangential momentum mVt in the usual way (A =

hlp), we would have A = hlmvt' where m here is the electron mass. Combining these two observations gives us 27Tr = mth/mvl' which if rearranged becomes mit = mv{. and mv{ is simply the classical ex.pression for angular momentum (r X p) in circular orbit. The orbital plane for the azimuthal angle is the : ' ''' " nnt \i f t . \ Il1l l

".we

. ties

1 ul lhlDl . we clll\llOl ", r.siuered Ihe radial � pe< 00. bul we can II \cui dI.­ rlccu.m would be 100 c As ell has a um qu e wave function mc , l (II. set h eac (JII �"" "'t ' Thus. because arc degenerate. I n fact , ShOWn the ground ..rale pt I'f! ce ex els ' lev y T Ie 7.5 de mo all energ nstr.'e' Increases as n-, lab acy ner th ge de Exerci"c 46. the Increases. . . b er 0f states as 11 ' qUlcidy grow mg num Gi ve n its spherical sy m mct We do is rather special . Actually. hydrogen entation-a componen t ori al ati Sp an , gular m( pend on not expect E to de ears i n eq uation app C? of at wh t Bu (7,31 ), 't be a factor. momentum-shouldn The fact that they do n ot e it? on nd depe es d energi pend Shouldn't the allowe In general, the allowed energles y. rac ene deg l nta ide acc fOr on f is known as nger equation before the h dro 6di Scnr the of t par ial gen. equation (7-30). the rad " is only in Ihe special case of e sim e. on nd depe do . rted ple specific U(r) is inse entally" does not A II)' de ial energy that E "accid ent pot via m ato . en rog IIr hyd simple case would lead to qu m this en m fro rgy ergy ene ial tion of the potent ctrons orbiting e nU In particular. additional ele Cleus levels that depend on e. s destroying the accidental de�eneracy thu , rgy ene ial ent pot . would alter the ental to chemlslry of energy on C are fundam e enc end an dep and d y rac Degene . apler 8. are dIscussed further 'In Ch

V��I �

I

;;,sf

e

07'

It

�

,:

�

:

Normalization

complex square of the wave funcl'Ian. The tern· The probabil ity density is the ' . . S drops oul In the usual way. As we noted m ee· poral part of the wave functIOn . . r the pola part IS and out, s real drop also . Now . 7 .5, the aZimuthal part tlon arrive at we , real also is h whic al. radi incl udin g the

1'I'(r. 8. ed by 5%� (c) Make a !>cale dia­ gram, loimilar to Figure '1.3, illustratmg the energy splitting of the previou'lly degenerate wave funclions,

(d) Is Ihere any degeneracy left? If so, how might it be "destfoyed"1 23, An electron is lrapped In a cubic 3D infinite we\!, In the states (II... "I' II�) = (a) {2. I . I), (b) ( 1 , 2, 1), and (c) (1. 1, 2), what IS the probability of finding the electron in

)0 l"

t) lble? (b) To how many differe", statcs do these

•.

What lun!';li,'n, w�luld be I:tc!,;('pll1� le ,tundmg,wBve

.".� \, \', .:)

boundary condillons, An eleclron " confined 10 a cubic 3D mfinite well 1 nlll on a 'ide, (a) What al'(' the Ihree lowest difft'rr>nr ener­

, Iln\l ('

II\linlle- \\('\1.

\\'\\I. 1\1\'("1 I\'(,U 'I\ll,,'lhh ",h\'n 1\

\"ml'he, i\ 1\)11\1,1

lin I

\.I!ti.:'

"I

m;h', .11\\1 Ih\,' ""'I.I�': \,11 1

" I\,,\\o C" 1\\\1)

1 ,hl'uIJ tIC" Ihe

Ih('

,um

�

I}

�� II"

Nt·\ � I)t.!.\'.� I )/('>, ..h,,,,..

..hl'ul,1 h\1'1... til\' 1\\cnt�(' hut the

" 1\\

I Wh� Ih" I' 1\\1\

\1t'lI·cll'/im·a \,I\lll' ni l � IcaIU\I\" Ilw l'i�llnlu, "PPf\,\\\'h ,d.:rrcl1

41,

hI III S"dill,, 7,' )

In 'NII.· nJ". to. Ihl' "\....r,llIlr hn the "lUM'\! \11 Ihc ;'1\Ij!ulur 11I11lllCIIIIUll ,� ,hllWfl l\l tlI.:-

What

'L " (

:",''lInPlnlCnL� 01 .U\gulllr momentum ,... 1\ J'O',ible tl> nenl of angular momentum, ttk' alUm I' equally likely to be found WIth any llll(lwed \·alu(,' of L,. Show that il the probabilIty dcn�i� lies for lhe'ol' dillerent p!)\\ibJc ,tale\ are added (with equJ.1 weightin!:), Ifle re,ult i, rndependent of botl! ¢ MId (J. 5J, A wave tundlOn wilfl J. nonrnfinnc w;l.\elength-howe\cr appl'IJ)(imale it might be-ha\ nOllLero momentum and Ihu\ non/em kineti.: enc'l!). E,·en :l 'ing/c "bump" h,t\ "ml'ti\." enc:f!!l- In eltiler Col.\e. we nn \3y !lUI thc function hOI, kincli.:: Clk.''l1)" hec:lu\c it ha\ cUl"\alUre-\l ,econd demalne. Ind(,'Cd. 1he !.anetu; energy operator in any WOrnJnolle \}�tem imohc_\ a ,econd deri\ati\e. The onl)' functi�ln Without kinetic energy would be a ,tr.ught line. A\ a \p:t;ial ca\e, ,hl\ Include, a con�U!nL whi�'h OJ:ly be thllu�ht ot a' a fUnctIOn with an infinite wa\elength By I�l('king at the cun·atun: In 'he uppmpnau' dim,mioll(.{). an\wer the following: For a given n, I� the kinetic energy '>Old) (a) '.JdJa1 JO the \tatc of I()\\.e,t t-tha( I�, ( "'" 0; and (b) rotational in the \Iate of high('.\( (-milt j�, ( '" II - I ? 52. We h,l\·e noted thai for a given energy. a,> I increase� Ihe mallOn " more like a circle oIt a comtant r.Jdius. with ,

the rotational cnelllY im:rea\ing 3S the radinl energy cor· n:\pondingly dcaea,\{'\ Bul l\ the mdial k.inelic energy o Illr the large\1 � valuc, l Calculate the ralio of c)(.pccta· '

tinn value\. rJdial energy to rotational energy. for the

(n.

f.

111/ )

(2. I , "" I ) \tale. U�e Ihe operator, KE"" KE""

'"";Jilr a photon and Jurnp� from the ground "ale to ii' n = 2 level. What wa!> the wavelength 01 Ihe photon? 68. Roughly. how doe\ the ':I'J m".., p 81. l \Crl'I"C />in tJl''':U''':'' IhL' IJea "r l'C\Iu':L'tI 111.:1\, \\ hl'n I"ll oh)t.'>.:b mll\': unJer Ih(' mnuence of lhelr mulu;IJ /111\.'(,' alonL'. \\e I:an Ire.1I Ihe "/11111'" motmn ..., "

it Ilnl"-I",Inll"ll" ..),Iem of m""J-l "' 1m/lint " m,l. , >:l'ounl ror ihe .-\mnn� (1Iht'r Ihmg'. Ihl .. ;1110"'" u, III I I....:! Ih ..t lhl' nudt'u" 111 ;1 hyJrog('nhke alom l)n'l pc::-r­ ledl)- " ,umeu inllnlte. '" In the rhapIl.'r, Ilul " of m..,.. ttl I , I:.-hile 111-; I' the mass orthe (.rhlung negallie charge, (a) Wh;.)1 pen:entage t'rror is

lntnxlun:d 10 Ihe hyt.!rogL'n grounJ-..t::"e energy by

a"ul\llOg thn •

crl'I'" i Ii.ul,k' J'l"'\tl\'("

cbIu'1:" , 1l>4 III h",tfllit!:". but 1'Ift('

\Jll bo.� � \� IhI: '-1.1\\\ ' , ... th.11 1,1 the \'I("ll�'" by Ihc ('mlk'mh ttlr..:e I' �'\l·n b) l \'f'�i·'h1't·llmr {d (1" ...l1 lhlll r _ /11 uti. i, \h,� \tI • In!!u\;u rl'\.�\ll'n..:)' e'II!(II IH lhe lllill1t1l�Ull'il,lnlp pholtl\l Irn{u...nr) al l.'Hhl:r �LlI.I 1,1 hy\ll'\lgco', allowc\! encfg'�'·)

s.c. Ik e'pcl'\Ullun ,,,lUI! I.llh�· c\ed"'"" ).,I11C\1..: �ncrg) in Iht' hydftl��1\ };I"unti 'lilll: 1'1111111, Ihe ""'�IIIIIIIIt' ,'I the 1I1t,11 en�fgy (,ee I',el\: ...e flO). What mu', be Ihe \\o,l1th 1.1 II ,,:l,Ib,� 1I1tinll'" well. ill IeI'm, Ill' "0' for II, �round ,IJlc h) hale Ih., ,amc cnt:'rgy'l

86. �f'C''':lrill llllc" urt' IUlly due In lw,1 ellcd�. Doppler bnlJlknillg >Inti Ihe unr.:�n'\lnly prull;illle, The rdJII\e e e ",,,e!cngth l1ue to the flr'l ellel:t ( .. ) " gl\en by E,cl\:"(' Z,57

\Mt;IIIOn

HI

.lA

A

\i )'/"flTfm

c

",here T I' the lempcrmurc ot Ihe ,>ample lmd III Ihe light. The Vllriation due 10 Ihe wcond cHel'l (�cc F.xen.:bc 4.72) t" given by

m.l" of Ihe parllc1c� e1ll11lmg lhe

\

_

1(111) �a. w1lh I" nunu,,\lm urtN.t 1Wbu. � .

rmhtol"___ �II" lOl l m aNS ,,� IMlUmum. or IPheboG.

1\10 unlt'i .. fal. Wtwn ... tbole minimum '" mul· mum radII. II� I'MIIU' ", lit ,,"1W"Ioe, noli chaftll,q. MI '� rilJla' �U�lu; t'MlV) "O, .no.!. \I ""�,tc eneIJ)' I ('Ilurch "".II,,"�I hunl "" ."'11:,,1 mn:hanl\:' rob.lioaa' ,,'Il('Il) " ¥I\CIl h\ " �/. ""here I I", lhc mumcnl uf U'ICI" llll, ",hl\,t! hI( . "'''Int .. """,," Iii. !IoImpl) ",,.2 ,.) 1bc �" '''\l'n, '1'Ot'C'\J ,1.1 f'('lIh1:"hu" I� f'I !�� 104 mil. ('111";\1 \3\\' Ih II.I\v;uhu fl\l\n\('l\Ium 1M \lenh that the ,um of Ih," filil\ 1\;lllIllIa\ ,""I('l\lla\ ('nt'lJ) lind n.t.tlunal t'l'IC'raY 111'1.'

(,,\11;1\ .II \"C'flhC'ht'lI 1111,1 "phdl"n

,Hl'ftIf'mltf'r'

\1I�tlh" tl\t>mC',,\um I' n'n"f'''N \ (d ('1I1I;ul.lt Iht '1,11\1 ,,( Ihe l/-,a\U,l\l\llllll ,,,'tentl,l.\ C1\('I)I' 01"" nltll.lI\lml.l \'II':: \�� ",h,'n Ihe ,llt'Jb,IIII lm­

tier of p;l111Ch:� and leading to the \>arne C(ln..;ilNOQ. .... n as the Slatc.'t detcrmi.rwnl i L It � no

IV ' .. 04

�

"' ,., AJl1Iff"

. ' and. e le tron The .I
11 "

II

"

anl t. sy mmet . the nature of frol11 ric tly direc . .. ( 8 . 22) 'a cause; ,ha' ple 0 If " S SIgn pnnci ,he Oli OU . "a'e (8·21 !U,'0n etnc IficallY ), symm , eVc w pec h s ICh I'f elC ,Ia'C perfec'lY acccp . I, ,., tt le par . ua , d,\'I eq al. 0' are Ill< IllS. bOSl 10 s 110' applY .0 ,, 0", 0, /P/' doe ll(' p,i Ixc/llSiOIl able. The

an e anl, • Th ,he Slale is os' an m ons os b , er IP , " ' " c mu lle l HO d u m me sa and occupa'ion of s Boson

be

ing an d Pair les rtic composite ' pan'ocl" osite Pa or boson" bu, Com p fermions r o,he · lisled i n T are e es able I q' I are 8 maS! af t he part f ' compo d of uarks neutron. WhIch are e " olo and lso be r � lC Hlr '1' ' addlllO g ular - mon n,un f an the fru11l . a are I ru "

Fundamen,alonparorttIchelesodoer. In ac!n. c,n a ..ites. even IIvbeu' c0I11poha (see in de,a,l in Sec"on .7, are p adeyd nee'a dnlno'ge be 'd I r' untl-,h . h or , er e mlCg can addand odd number a bo,on, S u th the ng atnl

" b liN e Ihen h C 2p e1t!clron In Z h��h ." aIlt.lth(r hJ he "\:h

•

thtrc

'\1

� 5 bo

i."LIfT'C'IJlI\ln \10\.'(' llthcr ('Iel'u·on ... are added, beg lnn l "g , roll. SUI \\ nh pul\l\ c energy bet \vecn c.ubon \; fk)(l....t In SnUun 8. :!. the re ' 6 eleetra l ' n� I Wh Ich require, anu,,)mmctnc. � 10" I' �I�I(' I/J(JIWJ th'l .. t thelf tl't\l Ihe U \. • I n l u e\ 1.... � � the antl\YIllIll ' ' ( 1 0 $('(1100 t:d .. i.:UV ''' ,fIIlcrt'OI f\lr tc: one electron (0 complete the ()Ih...·r hand. fl)r "hlch 7 == its I ow �el1ergy 1/ • � ,hell. "0 il prefer. to capture a weakly bound electron of a orher n . . atom rJlhcr than !!l\e up rul) of I'" Ill.my ughtly bound OUler e lectron s L'ke SOdium, � it I llul1nnt' I' quilt' chcmll'ally active. bUI in the opposite sen'\e e tak r i valence of be d 10 sa . I thu!> ( " Lih. and ewi s nllher Ih;m the gi\Cf e 2-8 gen. t\\ l\ t'iecU'On.. �hon of a complete shell. i") said 10 be valence . There IS - 2 and one of +6. nOr betw no f'C.1I dl,tinclIon bel\\oeen a valence of e - I and . 7. Valcnce.. ll.I'C' 'pecitied '\imply according 10 whether the atom u:ua Iy [end� If a sodiu ll1 at to n.,\'CI\C or donmc electrons in a chemical reaction.) Om and Ou , atom Will seile �odium ' · flnne Jwm are broughl logelhcr, the nuonne . ' s danglin g. its g more completin tighlly bound \\ (" tlh . bound valence electron. n = 2 sh " . ell. \lthl)ugh pulling the electron from sodium requires energy. the "hoie n O i uonne'" " """ 2. shell is deep, and the net result is a l ower�enero eoy SIale. A n , Ion' . Ie bond ,.. fomled. in which the ions are held IOgether by electl"O!>t3tic attr t ac IOn. CompJcling the comparison. Z == I a neon differs by just one eleclr n ,odium and fluorine. but it'\ behavior is vaslly different from bOlh lIS � 2 I

•

8.9,

,

_

8.1-1.

ahrupt

' I

9.

�

�� e � ox�. �

f N

N

•

from

e

. fI sh l! I' full. It has no valence electrons to offer. as does sodium, and no hOles t i f 11, as

. (k'IC, Iluonne. It thus joins helium as a zero�valence noble gas. only gru�glng ly partil'lp.llmg in chemical reactions with other elements. A\ Z increases beyond .

10.

Ihe

v.�ence (now excludmg both fulJ "

==

3s and Ihen the 3p slales fill, .

and

�

1 and II = 2 shells) Increases exaCI IY as n.

"

'

.'

"

1,. "

---

,

'­ ,

-

'­ ,

'" "

--- '

,

"

•

�

,

+

*

•

1\

1\,

2

-

-

+'

..'

>.

*

L, 1

Bo

B

4

5

N

c

7

6

o

F

No

N,

M.

8

9

'0

II

/2

does during the filling of the /I = 2 levels. And just as the chemical behaviors of Z = 1 1 sodium and Z = 3 lithium are similar--each having one electron beyond a lined shell-so too arc the behaviors of Z = 1 2 magnesium and Z = 4 beryl�

limn, each with two valence electrons. This is the periodicity of the periodic table! Shown in Figure 8.15, its first column is valence 1 , containing Z = 1 . containing Z = 4 and Z = 3. and Z = 1 1 : the �econd column is valence Z = 12. The eighth and laM column does lIot correspond to a filled /I = 3 shell.

2,

The 3p Siales are rull, but the 3d states remain. Furthcnnore. as Table 8.2 indi­ cates, the 3d states fill only after the 4s. However, the energy jump from the 3p to

tbe4s level is large. Thus although Z

=:

1 8 argon does not have a full II

\1 is relatively tightly bound, so in common with Z

=

=

3 shell,

to and Z = 2, it behaves as

a noble gas of valence O. By these same arguments, Z = 17 chlorine, in the sev­ enth column, is chemically similar 10 Z = 9 nuorine. Both arc valence - I .

Si

13

"

.

•

§

•

'"

J!

I ,

.

,

•

•

•

•

I

• - -

•

, r . -

• I •

•

,

J

•

,

• •

i ·� I : w

N

,

,

: •

�

•

, •

•

'

. �

•

· ';

�

• • · -

r;

�

· -

,I

$

- -

•

•

·

• • •

· • .! -:

! � 1 • � -

,

•

�

tates.

II == fluorine art in in n S "" V " e (;U I e 4(0.053 0.050 nm =

valence electrOn Sodium'!> lone

Zeff = 4. 2

nm)/Z,"

has 11

::=

3.

3 nm)/Z,"

9(0.05 0.180 nm =

theless about the same size as Z =:. flourlne is never . 9 (Note that _ of fluonne s " = 2 valence eleCllons each . l e ma l l s e 0 ou, Acco,dmg 1 ned . We would expect the interveni ng I hydrogen.) so 4.8 are scree 4.2. y nl o of e t the �even II = 2 electrons should would see a charg hly twO charges. b � roug en scre fI as SIX others orbiting in the same range I electrons to e extent. Each som o t other al�o screen each about 3 IS reasonable. screening of tive effec an letely scree ned .by all lower·1I elecirons. II. of radii. so comp were ons == 3 electr ' elliptical orbit s·state, however, IS a highly If sodium's II only I . An ' of Z . tive . should see an effec nce InsIde the lOner clouds. increasing liS prese le iderab trOn cons gl\lng this elec 2 2"1 etfccti\e Z. approximately 3 / = 2.25 times an 11 == 2 is s radIU rbit II � 3 ? of fluonne because sodium's In hydrogen.. an than three times that rodlus I� more Um sodi effectively .,maller charge. The an rodius. higher n. also orbits es beJng In a beSid tron. valence elec . Z Fluanne has

_ _

.

:::;

�

,

8.5

Characte ristic

X-Ra ys

lowest·energy states u P IO completely fiJI the . alOms. electrons In ground.state . d vaI ence e Iectfons can Jump around am ng xClte . um energy. E n some maxim olve no more Ihan le s of ecwnward jumps inv Do . els lev ed fill un higher ns in the ult ra violel nge i ndicales, producing photo 6 8.1 ure Fig as S, tronvoll glh" In Section · much shorter wavelen made to emit be can ms ato r, Howeve

� r: 3.3:

Figure 8.17 Characteristic X-rays are produced when an inner-shell hole is made.

All Jl;i!"eter.ucd

°

o

... ..

". °

b

o

.,/'

•

•

n=1

..... eje ..:tmg an

..... orb lt , illg electron

and U¥king a hole

° 0 . whl�h i� filled b} eled tpn orbiting 0,�....!. an In a higtj"er shell " o ,/ creaullg an )V'ray photon.

we learned that electrons smashed into a target produce a

continuous spectrum

of X-rays via bremsstrahlung-but something else also happens. As Figure 8.17 illustrates schematically, if an accelerated electron knocks an orbiting

electron out of an atom's

illller shell, a "hole" remains. An electron in a higher

energy state can then jump down into the hole. producing a phmon. Of course.

this leaves anmher hole, so the process can repeat. Inner-shell energies are

often thousands of electronvolts. and photons produced in transitions between

them are therefore in the energy range of X-rays. 103_1OS eV, with corre­

sponding wavelengths of

Figure 8.18 An X-ray specmtm. Characteristic X-rays

�

1 0 nm to lQ-2 nm.

Because atomic energy levels are quantized, only certain X-rays can be emit­

ted by a given element, and each element is different, so these

characteristic

X-rays serve as tingerprints by which we can distinguish the elements.

Figure 8 . 1 8 depicts the X-ray spectrum for a particular target elemenl. Super­

Wavelength

imposed on the continuous bremsstrahlung spectrum are some of the target's

characteristic X-rays. We use distinctive notation to refer to the X-ray photons

generated in inner-shell transitions. The II = I shell we refer to as the K-shell.

the n = 2 as the L-shell, the II = 3 as (he M-shell. and so on. These letters are A subscript also used (0 indicate the shell at which a LTansition

tenninates.

Figure 8.19

Ka

X-rays.

advancing from cr on through the Greek alphabet designates how many shells higher the transition began. An cr LTansition begins one shell higher. a {3 two sheJls higher, and so fonh. Thus. a transition beginning at the L-sheLi

(n

•

= 2)

•

and ending at the K-shell (II = I ) produces a Ka line. An LfJ line is from the N­

= 4)

within

a given shell are not down to the L-shell. (Transitions shell (n energetic enough to be part of an element's X-ray specrrum.) Inner-shell elec­ tronic structure is largely IOdependent of the behavior of the vaJence electrons,

with their temperamental periodicities. so the variation of characteristic X-ray

wavelengths with Z j.:, relatively smooth. Fig ure 8.19 illustrate!l. the smooth variation of Ka lines.

•

•

.'

50

, tiO

z

EXA MPL E e.'

. otbInng a cha1geofZarr given in Section 7 8 as neodymmm. 1bt efdPS tor ll1 � b Of . K X·ray Z = 60 gt

_

zt,,;t fSDIIJ* lht W � the

�1I�J

r ON

�

�Jl)

The pboCon t

1\

b;fTOII

otMf clouds, and thai bef re o 5. � ilboul halfoftM L-�hell clouds add 10 the scree 15 o

[55'

he - -IJ6. eV 22 A _

y minus us fimal.

A

_

-

-

59'J

'

.,

/-

3.7 1 X 1a' ,V

=

:z

3 .7

1X

Iran\j.

/ ()4 eV

O.033 nm

between 0.033 om and 0.034 om Wave1e 1 eod)'mJum KK X·-ys n_ 8,, for many elemenlS using simil 1. engths wavel ar e ) e�lJm:lI apprO)(IJ:n 59 Excn:ise h a· f F' dilla 0 e I 8. fils Igure model 9 1 Ihe t w well Actus n

,

" ,

Iron) and �ho"s Jus

h0

ore

,I.

IJ

.

ways. Those produced by a "'-J . . X-mys are used many ,_ CharaclensllC IO \V 1l X-ray source for crystallography. Con common • are ·aJ ren rna verse rarger ly. ,amples somewhat unknown are also very usefu tho� coaxed from l. B y X-mys With tabulated values and measUring matchmg chamCteostic re,ali.V , e s elemental compositio . learn much about the. sample n and \4-e , leS, mtenslt con. . . electron IS not the only way to exci te chara . centralJon. An accelerated cIensh. . ' l c ic art p ) es are (alpha also comm X.l'3ys. Prolons and helium nuclei only use . . d ' X-ray emiSS . as partie "" e-mduced ion (PIX and Ihe leehnique is then known '" . E) . . techmque IS liS sensitiVity, this due to a low O N J advantage of The pnm_ e r back . contl The X-rays. uous ung brem bremsstrahl sstrahl ung of ground level spec . of the charged particles and has [rum is caused by decelel'3l1on nothing 10 · . do · · with the specific larger. Given comparab'e kmellc energies, a proton ' w· . Ith a much larger mass than an electron, has a correspondmgly s maller a ecel era· " . " lion. so il emirs less "braking rad latlon. _

�

lJ D V A N C E D

8.6

The Spin-Orbit Interaction

In this secrion and several that follow, we will study Some of the Way" . In which the atom's various angular momenta interact We start with the sim . P �t case, returmng to the one-electron hydrogen atom. In an atom, the electron, with its intrinsic magnetic dIpole marnenl . . " essentiaJJy an orbiting magnet, so it should interact with any magnetic fi . ,� . le present Our simple solutIOn to the hydrogen atom assumed thal lhe only POlen· U·aJ energy In · Ihe alom IS · due to Ihe Cou'am b attracuon · belween nucle us and electron. This is certainly oversimplified if there is any magnetic field pr ese . nt. ' We need look no further than the alom uself. MOVing charge!ii produc fr e ma g. nerie fields. Because il orbits, the electron produces a magnetic field . , and om lts

�

fel�

-

-

1240 eV nm

t

Illog

the tkdron'lj Inllial energ

I l:,mi 1240eV ' nm forhcglve�

I

Assume- 1;�tV� ---.... ;:

In

orb.lS a charge of 59, due 10 screening by .

X-:II�COCIrnbution� from

K-shell eJcdrUl'l and

tJOA. llortMb "

.

�

t\,() The Spin.()fbit IntcncbOl'l

rehiCh ....

Figure '.2.0

pictured in Figure 8.20. the lIucleus is in orbit. producing a tield in !">pe'tive. the electron's intrinsic magnetic dipole moment has an orientatIOn energy.

The t"lectron feels a

netic field. dut" to the ··ortnting" in the ...arne dirt("-tion a!o l..

A detailed study of this interaction is beyond the scope of the text. But

319

mal·

proton.

'th fairly simple considerations. we gain a good qualitative unden;tanding

.... ld a reasonable idea of how much the interaction might alter the atomic 'Y levels. Strictly speaking, this "new" interaction invalidates our naive an "

e�'b . . . h drOgen atom solution of Chapter 7 . However, we WIll 'iee that the mterac· ergy is small, so the effect can be considered a minor perturbation on an li n en ption. eS'ienlially correct descri

�

The orientation energy

of a dipole

IJ.. in a magnetic fIeld 8 is

U=

-

• - - - -- -

'" What the prolon '>tt!o

f.L ' B .

Here the dipole moment i s the electron's intrinsic ""s' related to i t... spin. and

c"4 '"'

due to orbital motion-hence the name spin�orbit interaction. Thus the fIeld is we write

u

=

-""S ' Bdue lo L

8 (due 10 prolon "orbit"')

(8-23)

e

To obtain a rough idea of B. we assume that the electron "sees" the prOlan

I'

orbiting il in a circle. as in Figure 8.20. The field at the center of a current loop is given by 8 = J.LoIl2r, and the current J we relate to as follows:

L

Here we have

assumed that

whether the motion

is

viewed

as proton

orbiting

electron or vice versa. the orbit radius and speed are the same. Therefore, by multiplying and dividing by the electron mass, we relate I to the electron's

orbital angular momentum. A s Figure 8.20 shows, the B felt by the electron is in the same direction as L, by the usual right-hand rules, so we have

(8-24)

Finally.

inserting this and equation (8-8) into (8-23), the orientation energy

U= -

-p.. S · Bdue toL

(

e

- --S me

)( .

I'oe

41TTI'I..r3

)

L -

-

is

(8-25)

The energy is high/positive when the two angular momenta are aligned and low/negative when antialigned. This makes sense, for if S were aligned with L. Jls would be opposite L and thus B . and we know that a dipole's highest energy is when it is opposite the field. As we might expect from Section 8 . 1 . the spin has only two possible components along the internal field. However, the values,

�S -�': t:J - - - ­

_ _

�

_

_

What the electron two. indicated in Figure 8.27. differences can be written as

A.. the

AE = (E,p, , - E" ,)

+

reader may verify. Ihe energy

:

B 2 .., " ",

There are six equally spaced photon energies. (d) We see that the spacing

,II

2

B = 2m, ", 3 =-

�;\}

is

( 1 .6 X 10 " C)( 1 .055 X 10'34 J . , )

2(9. 1 1 X 10 31 kg)

3 . 1 X 1O-15 J = 1.9

X

(0,05 T)

2 3

IO-6 eV

te) The splitting is very small. The transition energy is still essentially

- 1 3.6 eV

- 1 3,6eV

2'

I'

= 10, 2 eV = 1.6 X 10'" J

•

t

or L"

_ 00' • ... .. � ...,..

10 dIt local met¥) I Ui:al.lU1 !l,I X 10 thel �J 0 � . n... &k' ,.., oI lbt COftJY lhift Thh tiN nj1h. ''tltIme .... the In ,,, ..tUft lM , Z ;I( 10 cauuDl - umi ,00 , 1), ",I I� l'\lem,11 fiC'/J ,PIII LfII'ID tcI'IC'J II .t-.JUl U': MI ' iCC' \0(\.1 II alacIuC Z " ;I( 10 ' nm .pan •

IDIO au

� � tJ*.'aJ

Effect Strong Field: The Paschen-Back

Jcgcnt'rah.' !C",t" . ' mud (';Ncr if BC"\"t " ...fron rwlrnn& lhe' r.phllln' �If 8-g Ii field (l\e-rn helm" Ihe in!COlaI on the' onkr of ,mil 1�1,j \ll ,tnln ti Id hl�l�lht'r. and each , ... then qua mal woukf oc.hc',.... L\c �'()l.lrk I• •,nd S Olt/f"(j 0 luptnJcnd) ."Jong R ," as tkpldl'd In hgul\� i'\ 2X U...lOg (8patial ...tate.

quantum dots, the nuclear ,pin.. of the surrounding atoms

requiring, of cour..e. oppo.,ite ..pin... Via external c¢ntro\s-

the r'llh:nll�1 "IICJ} J\ .lIlctn/. ;and oot' d«trolI " nullied Mlin w,,,,,... , , !.l.lt k�� b\ ;t NIner through "hilh "';ill' fUIKII'1f!1 un f"'S§/lunnd .h JC'pl([eJ In Figli/e 8 .ll flIto �mlu,ndud"f nudC:1Il .IlliXl the \.ep;!f3Il-d

'r'"�

t'le\:lnm\ I.h"��rlll.t. aJJmll .l roll\J B-field line� point roughly in the )'-direction

would be Z ror the first nobl; gas?

18. As indicated

of charged con�tituenl� hal'e a magnelic dipole 6, SupPlhe thai at the channeJ'� outgoing end in the

What i f electrons were ,prn-� in,tead o f \pin-t What •

separate them from one another_ One (ea�on can be

20.

seen

10

Figure 8.16. Explain.

Figure 8 16 C' an mtinite

36. \cn/}' Ih.Jl lhC' normalilatlon eon�tant gnen in E\ample � Z j, nlrrCCI for both ,}'mmelric and anri\)'mmetric \loJle, and .\ mdl."pendenr of II and 1/' �17. The gener.. 1 fonn for �ymmcrric and anli,ymmetric

11

I

(b)

overlap--and functions A and B approach equal energy. as do function� C and O. Wave functions A and B in diagram (b) describe essentially identical shapes in the right well. while being opposite in the left well. Because they are of equal energy. sums or differences of A and B are nOW a valid alternative. An electron ina sum or difference would have the same energy as in either alone. so it would be just as " happy" in A, B. A + B, or A - B. Argue that in this spread-out situation. electrons can be put in one atom without violati ng the exclusion principle. no matter what Slates electrons occupy in the orner atom. 41. What is Ihe minimum possible energy for five (noninleracting) spino! particles of mass m in a one­ dimensional box of length L? What if the particles were spin- I ? What if the particles were spin- ? 41:. Slater Determinant: A convenient and compact way of expressing multipanicle sillIes of lIntisymmetric character for many fermions is the Slater determinant:

44.

45. Exercise 44 gives an anti�ymmetrk multipanic\e state for two particle� In a box wilh oppoIate with spins opposite and the --.arne quantum numbers i�

!

0/1" (xl)m,1

iJI",(x!)m,\ iJi",(xJ)m,\ 1Iign). or neither--of multi­ particle states t and It with re!>pect to swapping spalial slates alone? (bl Answ'er the same question. but with respect to swapping spin state alone. (c) Show that the algebraic sum of slates I and U rna) be written

(X)nlJj.

columns).

What property of determinants ensures that the multiparticle state is 0 if any two individual­ particle Slales are identical? (b) What property of detenninants ensures that switch­ ing the label� on any two particles switches the sign of the mu!tiparticle state? (a)

43. The Slater determinant i!o introduced in Exercise 42.

thOt if state, /I and n' of the infinite well are occupied and both �pin\ are up, the Slater determinant yields the amisymmetrio.: multipanicle state:

Show

where the left arrow in any couple reprC!.enb the spin of particle I and the right arrow that of panicle '2 (d) Answer the same questions as in pans tal and (b). but for thi!> algebraic sum. (e) Is the sum of ..talh I and II �till anli�ymmelric if we swap the panide..total-spatial plus spin-states".' (f) If the two parti­ cles repel each other. would an)' of the three multipar­ tide states-I. 11. and the sum-be preferred" Explain. 46. Exercise 4S refer" to state... I and II and puts their algebraic sum in a !>imple fonn. (a) Put the algebra!.: difference between these state:> in a "imilarl) §imp!e form. (b) Repeat part.. \dl and lel of Exerci-..e 45 but for the algebraic difference. "Upy ... 47. A lithium atom h3.!o three electrons. � roo.: indi\idual-panicle "tate" corre...ponding to th. First

= L! + 51 + 2 L 5 to eliminate L · S " �,ll,, L = Vi(1 + l )fi. S Vs(s + 1j� J= V iU + I)", obtain equation (8-32) from th�and

7S. Using!!

82,

electrons occupying the Is and 2p ir LS coupling were ignored. A.. i.. done for helium in Table 8.3, determine for a carbon alom the various states allowed according to LS

coupling. The coupling is between carbon's two 2p elec­ trons (its filled 2s subshelt not participating), one or

which always remains 1 0 the 2p slate. Consider cases in which the other is as high as the 3d level. (NOTe: Whell both electrons are in the 2p, the exclusion principle re�tricts the number of states. The only allowed states are th"t in which STand (T are both e ...·en or both odd.)

Comprehensive Exercises 83. Relati"i�ti(: effects are rather )mall in the hydrogen atom, but not so m higher-Z atom�. Estimate at what value ofZ relativistic effec� might alter energies by about a percent and whether it applie!> equally to all orbiting electrons or to some more than others. For this crude guess, it

i� acceptable to combine quantum-mech,lnical results

you have leamed, related to energy, angular momen­ tum, and/or probable radii, with some classical relalionship�. �. la) E�timate the repulsive energy between helium's electrons. Do this by comparing the energy required to remove the first electron, 24.6 eV (see Figure 8.16), then the have equal energy? Can you c' �otume and lhe numbers of different Lim!> of parudlOSl

chem,cal �p«ie.. h migh' be in'� b} hc.llm8 at co..,....n ",tum. (.1£ > o. .1\. = 01. b� fltt

upiUlS'oo r.1£ = O • .1V > 0). 0< in a chrmical ndtllt, tilt 'oJu","" DOl Iht

reacUOfI that chanse.

Internal «nelllY

- � -

T

(9-4)

A glance at equations (9-2) and (9-4) raises an obvious question: How on Earth do we take a derivative wilh respect to E of a number of ways? We

'In hr..1 Inn,fer hom ')� 1 10 sy>lell\ 2. tht

d;",nkr of..

rf.

O_11� 0 1'" N ·- 1 0

001�

AI "" 50

O,O_�O O,O!5

I----�--=::::oe ::: ;o...

o

...... .. ...... . .. . � . .......... ..-I"__

�

"

40

30

50

An a..erage Q\cillalOr occupies its fifth energy level. The probability that parti­ cle

i ha, Ihe

avecage energy-that 1// =

(

quanlum numbeJ'\ add 10 45-would be

� -s

= =

45 + 9 - 1

�

5

and Ihe remaining nine panicles'

)/(

8.86 X 10'/1.26

X

50 + 1 0 - I

10 "

50 =

0.Q705

)

The probabilities given by (9-9) for all values of 1/, are shown in Figure 9.6. The

cune drops ..harpl)' as 1/, increases, which leads to an important conclusion: Varying the energy ofJU�I one particle causes a sharp change in the number of ways of distributing the remaining energy among the

other particles. The greatest freedom to distribute the remaining energy occurs when that one panicle has the least energy. Therefore, Ihe more probable state for a given particle, the state i n which the number of ways of distributing the energy among all particles i s greatest. i s one of lower energy. Equation (9·9) is cumbersome and limited to the special case of harmonic oscillators. In a system of infinite wells. for instance,

squared, and the expression

En

is propoilional to n

would be entirely different. However, it may be

_�hown that in the limit of large systems of distinguishable particles. converge to the Boltzmann probability.6

n

p

all cases

b�

"'I'btdtn.m;;,n and lboJc,;,f lilt forth.:,>!mng

"qlllltum l diSU'lN..l"" Ii delennt 10 App:I).l,\

H. � II U. be,1 Itl _ tao. Lilt �h'hty 1\ flSty/ til'll :and t>eeause KICID, the den, al,OO,

IOfdIIU .Iantics dIt,tllmdll'iuc, 1Uld dJtlcrrOl.�'.

This is the probability that in a large system at temperature particle will be in stale n of energy

T.

an individual

En' where 11 stands for the set of quantum

numbers necessary to specify the individual·particle state (e.g._ n, e. Probability drops exponentially with energy.

me_ fils)'

II i� important to note that the Bolt7mann probability and a1l the di!>tribu­ n�)'et to come can be applied to '/(�T)d)' y= 100e-(mg>/k8T)dy

mg

Tht: result rnake� sense qualitatively. At a higher temperature, panicles are more energetic and so should rise to greater heights. Conversely, if either fI/ or g were larger. we wouldn't ellpeci particles to rise as high. Inserting the mass of an N2 mol­ 1 ecule. m = 4.7 X 0-26 kg, and assuming a unifonn temperature of273 K (although 1101 of uniform temperature, nor is it all nitrogen). the result i� Ihan the elevation of Mount Everest-a fair approllimate value. 8,2�. a bit less

the aunosphere is

The Maxwell Speed Distribution The distribution of speeds in an ideal gas is

an

important topic in classical

thermodynamics. Known as the Maxwell speed distribution, it is just one of the many faces of the Boltzmann distribution. We begin by considering lhe average of a yet unspecified property fly)

that depends only on the gas particle·s speed, v. Examples would be the speed ibelf and kinetic energy. The energy is assumed to be solely translational

kinetic energy. dependent, of course, on 11.9

We now average, as ill Example over the volume V cancel,

9.3.

but in this case the spalia/ integrations

J j(1I)e-!m,;J./k8T J e-�m,.2/k8T

dvxd\l>d\'�

dl'..dl' dl' J :

Sincefty) depends only on speed. rather than on velocity, we further simplify things by using nor rectangular velocity components bUI spherical polar coor­ dinates:

I'x' l'y.

1'_ --) V. e. c/>. By analogy with dV = ,2 sin 8 dr d8 dcjJ in spmia/

�ngular integrals

coordinates (Se�tion 7.4), the "volume" element in I'elocirv coordinates is tN, dIll dl': --)

v2 sin 8 dl' dH dcjJ. With this replacement. the cancel. and integration o....er speed l' alone remains:

"PoJ)al"""� ,deal JUC' 1\;I,'e ,n!emal degree. of f-oom_ rotalJON-t and vibnlJJ>lUJ. wh>eh dqIend ... "'.-�r,>m< in tb,

·

\

\

'

•

!

! : /!

(

j

and disuibullOO 11 r - 0 The ftnni·Dirac ,." ,.... }-emll rft'l!(i}' t.. \lIUe ..... ltmpcnturer.. the � kJr 1oW r 0 mil( ,ink from ,u ..r d l EfO

E '!!

,

,,\

r � I,t

II

9,6

",I lOw T ,.�� Ptila

I).fa e....

o ,� L

0

'.

_

--

,

hNU,h H AI wne " urn T: t" c••

" AI

f. F1

� f

-

f

•

Tne Ouantum

Gas

n\ to a \ituation more reaI'I�h. c ''-A_ um dl.,tnbUlio UI4l\ . '. the quant annl .... no n we do, we enter an area 0f ph)"" \\e mmon ongln. A" ilco III c J c f"":llIatl\f' J,ltrol..:! of Ihe mo" aeli'e In r'''ar h ' mvohmg . h., bsumjng equilibrium between atoms and photons' the rate at Whlch · . . atoms make upward transHlons from level I 10 level 2 must equal the d ward transition rale his is known as the "principle of detailed balance." .. ply put, the two emISSIon rates must add to the absorption rate.

�

�:�

9.8 � l.&ter

A\pon N2

+ B�um N2 Y(a£) = Bab\ NI Y(,1E)

Solving for Y(.l.£), Y(IiE)

�

N,

N2

Babs

'[he ga5 atoms. being diffuse. obey the Boltzmann distribution. �o the ratio NI ' \'2 is e-EII*BTje-EJkBT = etlE/ksT. Thus. Y( liE)

�

s j --'Bs '-:::--= A po Bab

,'Elk.T

stim

- --

B,I»

Here is where the "intriguing conclusion" arises. We defined the quantity of photons of energy 6E, but we know what this must be! uE) H as the number energy density (9-47), with d£ (i.e., the photon spectral is the photon gas the atomic energies £1 and £2) replacing hi in that fonnula. energy; not one of The t1Ei*oT is in place where it needs to be, and the only way the re�t of the denominator can be correct is if Bs1im = Babs' This means that if the numbers the two levels happened to be equal, a stray photon of the of atoms occupying as likely to induce absorption as stimulated emission! just is ncy proper freque don't want these rates equal. We want stimulated we laser, a in Of course. emi��ion 10 predominate, but how? As the reader might have guessed, simply making N2 » N1 would do it-Rwm would be much larger than Rab�-bul such a situation is certainly not !he way things are in equilibrium, where number drops with energy. (Note: We a.>sumed equilibrium to prove that B\lim = Bab�' but they are indeed constanle,. so they are equal even when the situation is far from equilibrium.) In fact, such an overpopulation of higher levels is crucial 10 laser operation and is known as a population inversion. Being unnatural, it must be established by some e�temal means. Bul so long as it exists. one phOlon of the proper frequency. perhaps the result of spontaneous emission, may become two, which become four, and so on, very quickly resulting in a large number of coherent photons. Before investigating how a population inversion might be established. let us take a look at the basic elements of a laser, shown in Figure 9.23. Because spontaneous emissions could iniliate separate "outbreaks" of coherence in dif­ ferent directions, the medium ie, tuned to amplify the coherent light injust one, Parallel mirrors are placed at the medium's two ends. Photons not parallel to !he axis have relatively few opponunilies 10 induce stimulated emission. while those parallel renect back and fonh. providing many opponunities. The length along the axis is tuned 10 a standing-wave condition L nAn, so that waves moving in opposite directiom constructively interfere. In the re do _ ...,.Jlcide ..

either uncontrollable or

40'

n. ..... II a device tJw produces coht�nt liabt

r:::�:�

.. .. _ If ,.,.. . ..... .. ' tilL .... . ... . poIIII ... _ , , g ... " .. ... tI . " 'c ..-., A " ., ' .,.. 11 .. ... .. _ . .. _ _ ... _ of .. " _ ... $ ' n, ..,· ' . ... " ui.. . ..,. ... . " '\ ..... ..., .. .. ndd .. ... " '11 ., .... .. __ ...., .. , 4 _ _ IWD .. .. _ bE p ' •d

d e

,$ ,

"

(9:2)

(9:1) 10 . ......._ _ .. � .... IIUIIIber 01 .. , . ' '. • . .,. ...... ..... ...... ...... .,. ... .. .. � . ..._ ... ....- ooIy .. .... _ oI ..... n 71 e! t til � ., 1M ."nn diIb'itu­ _ _ ... _Bh ' , ... _ .... _Dn:.

X(£!.. -

1

1

(Ila - ...,." " '''' ( '"

•

(9:) 1 )

...,..'

1

...,."

_

1

+ 1

(9:321

(9:)))

' -':y ill all .. .. s t" .... .. rI .. .. 0 � ... 1M ,$..... ....... COGYGae '" Ibe "dIasicar' 7= indhti,..tisbehiil ty r .. . .".1- •• • ' ... .. ...... ..... . &IlDCb .., .. .... . .t 1000 " 1 AlII 1·. ...... _01"" _ _ _ af .... . . _ ... 11oc ... . ... Mu... .. « n I'_ di8Ir .. II low ...... 1IIdIor hiP , , _ _ _ .... oII p111i: ...... .... .. .. 7'11 • d . .. .... __..., ........ .. wIIi)c Ihr: _ a.. ... ..... _ occ. ' ..• .... oI l � ,.u. .. .. 1M _ _, • _ Il1o """"'"'"' • ..

.

_ .. ......., .. 0, C '

en

I

I

-.

.. . . ..... c6II iIMiIhc a _ GWa' .. if .. ..... ...... .,a I pMidc

I _ II _ _ .. V, ... _ _ _ ... • ....... .... .... -- - - - "" "' . ...... ...., of _ � _ afr ' _ .. _ ... . .... .... ....... . ... .,-. I _' " I ....... _ � .. ... I • ..... iz . ..... . ..... ... .. .. , rI _ I , ' "' .... - . _ "" .. ... .""'ip ' I ac 7 , 7 .. ..,. " • • •, . .. .. 01 .... . : .. .... ... af- , .... . ... .... .. 7 z

...

Vi.I

. m 7';'" A aoncquilibrium population i .. Ifj W in lite Ii....-producin' medium. in Whit.: ... ... 01 cIecaoas in . hiJher..enelJY §Iate is greater than IbIA . � s&IIe. Ir the: hi,her Slale IS meta......ble. e ....., 10 It by ulemal optical pumping or electric diKt\arat .. JR'CDted � quickly returning lind fNlolabhshing eqw, ...... A pauln, photon wtKKc energy equal." the dlff� r and lower state!> Ihen precipilales Stimul ..... � •• OM PI which the newly produced photon i" cohertn widI die fint. With . population in\"erSlon, the process quick! t � '-. two. two prodUCing four. and , � '"""'"' . prodUCIn, SO CIIt---Ud . powerful coherent hcam re,uh�.

.....

�

•

:

IndICates advanced questions

Conceptual Questions

I. Coo§ider I �ystem of two identical objocb headmg

SU"light IOWan! e4U.:h other. What would qualify and what would di�qualify Ihe �y.)lem iI!> a IhnmodYllamlc syslftn. and how. if al all. ,",ould thi\ relate to Ihe e1a.�, tieity of the collision·� 1 WhaI informalion would )"ou need In order to \ptclfy the macrostale of the air In a room'" What Informallon would you need to \pe

17. Cla" I�all). what woulJ he the a\er,.. energy of. par.

\\hether the den�lt)' or ..tale, ,h(lUld be Independent or

tide

f, llfl ln(rca! isn·t a

thennl)dynamic �)'stem, but the �eneral idea Stl\l

applie\o, and the number of combinatIOns is tractable.) 21, Con..ider a room divided by imaginary line.. imo three equal parts, Sketl.:h a Iwo-a,.i� pill! of the number of way, of arrn.ngin� particle, vel'ou, N.,I and N .... for the cao;;e � n.. �� Note that Nm,Jd1t: " not llldcpendent, being of N � I(}"

Nlch

Nn&ht

and - Nlcfr Your axe.!- 'hauld be COUM N Nn¥M' and the number of way' �hould be repre.,cnted by den\lty of Shading. (A form for number. of wa), applu,:­ able to a three-Sided room 1\ given

Appendix J. but

the queslton can be an,wered Without II.}

22. The Stirling approximation. JI

•

III

Virr J't tt2 e-J , I�

very handy when dealing with number.. larger than nbout

100, Con�ider the following ratio: the number 01

ways N particles can be e,'enly divided between two halvc\ of a room to the number 01' wayj5)\' for large N.

(b) E�plain how thi\ fih with the claim that a\'er;tge behaviors become more predictable in large sy"'\cm,

Section 9.2

23, The entropy of an ideal monnt(lmir.: ga, i, 13I2Wks In E +

NkS In V - N.ls In N. to within an additive \:omtant

Show that lhi\ implil!l> the com:r.:t reiation.ship bet\I.'ccn

internal energy £ and tcm�rature.

... ChIp.."

SiIbec"CII MIll_ ."

K. The dIOpOm show, .... .y"""" lhaI mlY ..cIwIg< _ tbmnoJ and .. e d lU llcaI lt,

....onductm' .. puII.Oft. Btau.. boIh £ and I'mol chlnp. we ,OnSJdtr 1M entropy of eacb sy«m to tit a (Ull(1lon of bach' S (E. VI. Considenn, the e..\change of lhmnal entfJ)' unl y. we lIIuN In SectIon 9.2 lhat II ..... J"h.5OOlble to define 111 1.1 d..SIa£. ln the mOlt �n· craJ �ar.e. PIT ,s abo dcfinN as �mcthlOg. tal Why huuJd pr6\Urf \:"URlC" IOltl ria),. and II.' \Iohal mlghl PIT ht equated.' IN()I�: Ch«k In 5tt whether the unil\ make �n.'it.) Ihl (Jlven thl\ rcl,UlllO'Jup, oJIow that dS " dQIT tRelTlC'mt'C1 the fl",1 h&w llf Ihennod)'nOtmll" .)

I_ I

28. In a large! ..ystem of di\tinguishable harmonic oscilla.

ttlf', how high doe.. me temperature have to be for !he probabiht)" or occupying the ground slate to be le\\ than 29. In a large ,,},tem of di�[inguishable harmonic oscillator how high doe� the temperature have 10 be for the pmba tile number of particles occupying the ground state [0 hi; Ie" thlfl j '] .30. ObtaIn equatIon (9- 15) fro m (9-14). Ma�e use or the followmg sums, correct when �tl < I :

�?

I

�

2:X' � I -x 11,,0 � x-:: :"-" 2:1Lt" � -'( I x) ' " 0

31. Show that "'Iualian (9- 16) foUows iTom (9-15) and (9- 1 01 ' temperature and M and .v given m (9-16) and that between E and n in (9-6), obtain equation (9-17) from (9- 12). The first sum gi\en In Exercise 30 will be useful. 33. Show that in the Iim.it of large numbers. the exaCI prob­ abdity of equation (9-9) becomes the Boltzmann proba­ bility of (9-17). Use the fact that KU(K - k)! .= }(A, which holds when k « K. 34. The euct probabilities of equation (9-9) rest on the claim that the number of ways of adding N distinct non­ negative :ntege� to give a total of M is (M + N - 1 ) [/ IM!(N - I )!J. One way to prove it involves the follow­ ing trid. It represents two ways thaI N distinct integers can add 10 M-9 and 5. respectively. in this special case. 32. Usi ng the relationship between

\(10 K. I� bndl)' put in conlBd · wllh a ·'hut' ubjC1.·I. I"l - .IUO K. amJ 60 J of heal flow, Inlm :he hU1 111'IJ�l ln :he \:"old (mc The objecb are then K'polI3ICd. Ihen temperature, ha\in8 l."han�ed negligibly duc 10 1MI' large 'IIC\ la) What are the \:"hange� in cntrup)" t'll ('a�'h ol'ljt'd and the ,y\(cm a\ a "hQle'� tt'll Knuwlng \ml) Ihal lhc,c (lbjel."1\ Mt' in l'ontact Jnd lit the 81\en temperature" \\ hat i\ the ral io or the proba­ hlhtlC�' (Ir thclr being Jilund In the ,ec{lnd (final) state hI thell �lf Ih�lr 1'IcapC' ii' il.\ C'nergy i� Um-rhlll is. if

ih JlI.,\il'lC' KE 1\ cquul In magnitude 10 [he negative PE

hc)ld,"� 11 10 thC' �urfll(e. Suppose the particle is a gas

mc'le�:ule to iln atmo"phere, (a) TC'mpc:rarures n i Earth's

�Im{hphcn' lIlil}' R!'arh /000 K RefC'mng to the \-alues

�

lI t lnc:d in hen:ise.j.'i and gIVen that R

Earth

= 6. H .

X

W " In ilnd �'f:Mlll c 5.Q8)( J()2� kg. should � f:"nh ..hie' 1\1 ··hold on" 10 hydrogC'n (1 g/mol)? 10

Mmgcn (28 glmol)" (l"'III�: An uppc:r Iimil on [he

numl"ot-r (If nm/t'\:u!(', ,n Earth'� IItmosphere is about

lO.h I (bl The-ffltlOn\ mass i� 0.0/23 times Earth's. ils

rmhu, 0.16 tifTle\ Earth \, and its \urface temperatures

K. Shoold if

�

floor is proportional to a shelf's height, how do

the 10131 energies of distributions (a), (b), and

...

fbI A'5umlnB II rC'fflpc'llIfure of 300 K, how much less

ri� III .170

able) ways are there now? (el /frhe energy you expend 10 liff a volume from the

Ilble 10 hold on [0

(f) Use these ideas to argue thaI the relalive probabili_ ties of occupying the lowest energy states

silO/lid

be higher for hosons than for classically distin. guishable particles.

(g) Combine these ideas with a famous principle [0 argue thaI [he relative probabilities of Occupying (he lowest states should be lower for fermions than for classically distinguishable panicles

51. There is a simple argumem, practically by inspection. that distributions (9-3 1),

(9-32). and (9-33) should

agree whenever occupation number is much less than Provide the argument.

52. Equ3tion (9-27) gives the density of stales for a system of oscillators bul ignores spin. The result. simply one s[ale per energy change of fuuo between levels. is incor_

reel if particles are allowed different spin Siaies al each level. but modification 10 include spin i s easy. From Chapler 8, we know that a particle of spin s is allowed

2s + I spin orientations, so the number of states at each

level is simply mulliplied by this factor. Thus,

D(E)

"

(2' + l )/�wo.

(a) Using this density of Slates, the definition NfllurJ(b +

I)

SKtion 9,5 49. \eritJ.' Or.ll ihe rrobllbihtie� shown in Table 9.1 for four

N"

Ji\lingui,\h3hlt' oscillators �hnring energy 2liE agree

the floor, and si� volumes oran encyclopedia. A.

0, E, and

B. C,

F.

(a) l.i�t all the way� )'OU can arrange Ihe volumes with 6\·e Oil the floor and one on the Sixlh/fOP she/f.

I ABCDE, -, -, -, -, -, FJ. Li,t all Ore \�"l1ys ),ou can arrange them wilh four on One way might be

(bl

II\(' floor and IwO on the third she/f. fc) Sho\lo

Ihllf l�re are many more ways, relative to p&1S (3) and (b), to arrange the sUI: volumes with

IWO on [he floor and two each on the firs! and sec­ ond ..helves. (There are SeVeral ways to answer

-

5>, and

�

J N(E)D(£)d£ o

14'llh Ihe ("'1.1\("1 prob3bililies gil·en by equalion (9-9).

50, You ho\"e ..i, shelves, one above the olher and all above

I.

calculate the parameter B in the Boltzmann distri­ bution

(9·3 1) and show

that the distribution can

thus be rewritten as

.N(E)SoIU = (b) Argue that if kaT » 0, the occupalion number is much less than

I for all E.

53. Using densily of Slates D(E)

(2l·

+ I )/hwo' which

generalizes equation (9-27) to account =

for multiple

•• 1$$ .UOWN ,pin ,tat�' t�e� Exen.:I'>(: 52), the doeflnition

�l21 +

Section '.6

I ) - l\ and

N-

II1II

SL ShO'\\' tI\at, u"inl equation (9-)6), density rJI ....

J .'·(E)D{E)d"� �

(9-381 follows from (9-)7), Dcn,ilY of Mates (9·)9) don not dcpead. 01'1 N,_ total number of particles in the sy'tem; neUher docs the .... ,il), of ,late, in equation (9·21). Why bOt7 60_ For ;l panide in l one-dimensional (to) box. Ell is pro­ ponil1nal tll a ,inglc qUlntum number n. Let .. UmpIify thing.. by ignoring the proportionality facwr: Ell ,.1. For I :\D 00:\, f... " , " = , the 20 and + " +

59.

•

",he f\lr B In di,tribution\ (9-32) and (9-33\--- Ot£) Cor a 2D infinite ....ell

(ignOring ,pin) 10 which

E

""

.."

-

In'.

II&. Humpat 9 .. 1n\o'Cllilaied one cnceriOl'l fw quantum IndtllUIJUlthoIbthlY "" b mCI'eftR: 10 "�ril' nitro­ JeD Heft "'" IIIWS1I,aIC me other, (.) CakulalC' lht � � -.WtnI IIllfDJO'J moJ«ub n i lhC' '" Auurrle' a t.crnpcntun u' }OO K. a prn.�Uf'C' Q' I "Im, .aad AI, 1I'uI, ., If'" nllNJftl. 'hI Calrolak lhC' �"l"('­ kn6rh oI. '),J'U/ ftlrroacn molecule.- In the." . ,el H,,� cb )'ou, rnulb �1.uC' III E\amplt Y,.,,· 66. Tht r� ...., ... 't .. tklilX'd b) IF '" ¥obr:rc 1:, .. lilt fermi C'lX'lJ),. llJeo h:mu C'rJ('rgy for e(lndue-

;""1'

11011 C'lt!,;lhlm If) It-d,um

Fc-mu wl,,'II),

a'll \\fhA,

,�

J I C'\". (a) ('aku/alt' Iht'

\looM bt tht �.nrkng'h of an

C'l«uon "'ilh Ih,� �dil,;it)", 10:1 If each ,odium alom

�-onlnbult� nne l'nntJUl'!i,ln dC't:tnm to Ihe d�crron ga.�

10. The maumum wavelength light that will eject electro

dit

fmm melal I via the photoelectric etreci is 410 nm. f:n� II)ct;ll2, it i� ::!80 nm. What wou ld be Ihe potelll;al J'eren.:e ihhe:,e: IWO metals were put in contact? 11. Copper hJ� J den�ity 018.9 X J03 kglm\ and no photo­ electrom tlfe ejected from If If Ihe wavelength of Ihe incidenl light i� greater than 275 n m (in the ultraviolet r.rngeJ. How deep i� Ihe well in which i ts conduclioll electron�---Olle per atom-are bound? 12, Deri\'3t10n of equation (9-W): Our modd for calculat, ing E i\ �quation (9-26). whose denominalor is the 10lal number of panicles N and whose numerator is the total energy of th� system. which we here call U IOIaI' , StJtl wilh the denominator:

lind �I-d'um 'I'"m, all' \p.!Icc-d ruu8hl} 0..17 nm lipan. i.\

it n«tUllry, h)' tht \'rileria of' t'qualielO \9-.$]), 10 treitt 1/1" l'nntJUl'II,Ifl d(·..-rnm !ld\ a\ il ll'I/JI/tU!II gli�',1 67. Til .IIl'nin CtJllllliun (9-41), we l'akulilled a inIal numher ot krmillm N;" a funl'lInn ,1( E.'f. as�uminF r ", 0, �tltninJil \llih Ct{ualfnn 1\)·4/ t. Bul nOIt' that (\)--l)) IS the Jcnl)mlf)ltl,!r u( nur nwtJd f(l( (alculM,ne ,1\ erage pani­ d... ...n('f¥Y. C'\Iualmn (9·101 h� numenllor " the 10131 (Ib oppu..w hI a\ela,!!... panidej cnerg)'. I4hil'h l4e'll caU {llIuJ he'.... I In mhcr l4'onk the Iiltal �)\tt'OI c:nctg)' U,\ 1� II\COrasc: panlde �'ne�_\ E tlm(, the /(J/al number oJ' P.V1,�·Ic:, V ) (ukuI.de {'"U/ iI' ;J fun�:tioll ofEf ilIId u�e Ihl� ... �ht'\l Ihotl lltt.· min' mum IT .. 0) ene(8), oJ' a gll� .., �pJII � 'ennJtln� "iii) hI,' nnllen

t>7 ";Jkulalt'� Ih�' minimum total energy in a !(omliom ;tIlIj i\ applicable 10 conduc­ li un l'/cl'tnm\ In lI IlWI,./, The (II't'IlI,i:e panicle energy i� the Mal encrg)' Ji\'iJl"1.l "y the numner of panides N.

68. E\cl\:i,,�

-j

N=

o

In�ert the quantum gas densilY of Sillies and an expres_ sion for the distribution. usi ng j: to distinguish the

Bose-Ein�lein from the Fenni-Dirnc. Then change vari­ J IIble�: £ ::: yl, and factor B,,+\ /(sT OUI of the denomina, lor, In the integrand will be 3 faclor

C)- I

== I :!:: c, a .sum of two inlcgrals results each of Gaussian form. The integral thus becomes two terms in powers of liB. Repeat the process, but i nstead lind nn expression for UlOldl III tenn� of liB, usi ng

U.�ing (I

+=

�Y" em ,II' \PIII

.�hl)n' th.." hc: ..\cruge panidc energy E ofII conduclion

cl«" nm al 1,1\4' tl'l1lpt'rJIUre IT iit 0) is (J/S)£r' This (vIm I' ,'llm'...niC:nI. br:-in,l! rulher �impJe, and il Clln eas­ ily t'IC' put ," Icrms Ilf N. I', and m l'ta equation (9-42). 69. Thi\ pmfJlem imc�ti�ltle\ what frJclion of the anlilable chtUJc mu�1 he Iran,terreJ from one wnductor to unulbr:-r ItI pn,Ju,� a IYPlcal c(lntao.:t potential. (a) AS:l rough "rpn):{imafion, freal lhe I.:OOOUl.:tors as 10 em x 10,'m .;qUiin:' plale, 2 I.'m 3pat1--3 parallel-plale capae­ ;1II(-!j(1 thai q 0= n'. where C ::: £0 (0.01 m�IO.02 ml. How mUl'h dwgt' musl be translerred from one plate

hi lht other 10 pmdu,'t' II polential dift'erentt or 2 v·, (bl Approx;mllitly ",'hal fTUl.'lion would Ibis be or the lotal number of �'ondul'lion electrons in a 100 g piece of copper. whi"h hi&!> one conduclion e1cclron per alOm?

/ Jf(£)O(£)d£ wilh a re\ult derived In Example 9.6 1hould go to 7ero and obtain a rough ..nlue. (a) Starting with ..N'(E)FD eltpres\ed as in equation (9-34). !:Iho..... that the slope d.N"(E)ro'd£ at £ = £F is - 1 I(4kaT). (b) Based

on part (a). the accompanying figure is a good approxi­

normal ga!>, all molecule!>. on

""O when T is small In a mation to N(E) ..

such a.. air. when T is raised a lillie.

average. gain a little energy. proportional to kaT. Thus. the mternal energy U increases linearly with T. and the heat capacity. aUlaT_ is roughly constant. Argue on the basis of the figure that in this fermion gas. :b the tem­ perature increase.. from 0 to a small value T. while some particles gain energy of roughly kaT. not all do. and the numtxr doing so is ale;o roughly prop!..lttionaJ

Section 9.8

80. The fact that a la!;.er\ resonant cavity so effectively

to T What effect does this ha\'e on the heat capa�it)·"

(e) Viewing the total energy increao::.e as simply

.lU

::;:::

sharpens the wavelength can lead to the output of

(number of particle, .....ho'e energy increases) X

�veral closely spaced la�r wavelengths. called

(energy change per particle) and a'>Summg the density

longitudinal modes Here we see how. Suppose the

of states ie; ,imply

'ipontaneous emission �rving as the seed for stimulated emission b of wavelength 633 nm. but

particle energies. e;how that the heat capacity unJer

somewhat

tional to (ksRIEF)T (TI)'ing to be more preci,e i not

fUllY, WIth a line width of roughly 0.001 nm

:1

con..tant D o,'er the entire range of

the� lowe,t-temperature conditions should be: propor­

410 � ' _ IUIly _while, rar Ill< �') .,.,..,.", 'I IUbjcd 10 .everal corrediocu from etfC(li we

,port.)

di,tributeJ unifonnly throughout a sphere of radiu\

7 x 10 ' I � m?

(bl How Joe' this result fit with Exercise 87'>

89. When a !>tar ha� nearly burned u p its internal fuel, it

may become a "hile dwarf. It is crushed under Its O\Iin

1

..

enonnou� gravitillional forces to the point at which the

e,clu�ion principle for the electrons becomes a factor. A smaller �ize would decrease the gravitational poten_

0

lial energy. bul as'mming the electrons to be packed into the lowest energy ible Quantum states of a single electron in the pres­ ence of two "atoms," represented by one-dimensional finite wells of width L

and separation a. We choose the finite well as a model because it is capable of holding an electron bound, as is an atom, and it allows wave functions to wan­

der through the classically forbidden region, as the unrealistic intinite well would not. We must allow for mingling of wave functions among atoms.

413

stales. which molecular . pairs of ve states. ha . lS" d atOm" P " atOIl _ of "olate ' 10c O' wO " T ' differe .. ,0. � rns and p." e: 10 su rg nve lion co

at I"ge 5Op",.

(e) (b)

L

(ol

energies for the fOUf 10West nctions and fu e . ' II. They somewhat resem· OWS wav tion IS sma 0.1(a) sh ' ose separa wh / Figure 1 s ' g e fi'ntte well. (The .op light a sm s/atom 0 f en Ons w . func.t. states in .hree, the ligh. red below" gy wave energy below " has . est �ner red low ur dark distorted antmode. ble the fo odes, the ) only one badly ur antln one has f o has red a larger separalton, A. dark gy. red one ener m ow O'to . a' h'tgh and . and the b m palfS has .wo, for are essen.ially equal, and 'hei, palf eS . of .he lower these s.at , r eve How rg,eS nt in the right atom and 0ppo. y coincide. ), the ene uall I(b virt 10. ilar. also become closer. At Figure very sim . er pUlr have tions are of the upp func eS gt e r ene wav converges 10 a single energy. atom. The each pair left (c), the . . l 1O site in ' equal energy. The low·energy paIr on. Figure . approach rati epa oer s . pairs ,.ill iaro algebraically added and sub· ce that eS tha" ' f cOinetden stat no .wo . is 11 to It ds, the two In o.her wo separauon ted '·alOms." at large 0'0 isola .he of converges 1 atomic teS s of .he two " � I s.a combination r ' are the ea lin .ed trac combinattons of the different states that are stateS are nverges to co r molecular pai wells, there would be three upper there .hree larly. the es. Were states. Simi . stat erge to three m ato ration would conv 2 lSola.ed. large sepa a. ich tWO " � · ncome together, theU stateS, wh n N atoms olecular general. whe n � I m In related m es. N olecula of sta. r I band, � or mn a se', isolated.ato ne 'o form and b' b t , and molecular com so te . her N,slll m sta.es form anot ISola.ed.a.o iC stateS m ato 2 energy of the isolated-atom state n � around .he . ster stateS; their clu rgies Figure 1 0.2 Illustrates band's ene n deereases. n ratio give sepa A on. atomic wells!atoms to produce apart as .he states of N and spread of atomic ning e atoms nbi the cOt and do belong to separat ' . electrOns can schematically e WhIl ntrate in thIS chapter cular bands. e 26), we conce N.state mole (see E"rcis apart far e. atoms are by the whol when those are shared that s state iatom on trUe mul.

I

.

:

'

I

1 0. 2

Mol ecul es

energy than the se(>'state is of lower . the molecular hen w s molecule eleetrons IS crucial. As they Atoms form atoms' valence the of The behavior lower e nergy may resuIt. But araled atoms. . atom orbits, a + ated· separ molec le, two ve to their u possible case: the H rearrange, relao st imple the s g iderin � conS e rgy may be en 1 by t01ll begut e Th . ? e re 1 0.3 how . W depicted in Figu tron, elec I � P us the neg· 0 .y : eUC ener "ts I.�n protons and one on's energyelectr the ( I ) ( " ) he positive twO parts: ' 'ded mto ' dIVI wt·th the protons-and " . potential energy il shares . ... t , onS1°der f!.Pil the electron s ns attve, attractive proto two . ' energy shared by the tial repulSive poten "

C

10.2 Molecuta.

FI9"r. 10.2 At small separations, each atomic state becomes N-�tate band.

FlguAi 10.3 A simple molecule.":t+­ an electron shared by two protons a dis­ tance a apan.

an

At large atomic separation. elec::trons occupy 11 '" I and

�;:��r

11

'" 2

molecular

Slates

�;:�'�r

��... '-} .1f := -=- \ (�ee Exercise 40), we obtain from equation (10-4) thc pos l eV) than those involving molecular vibrational transitions (>0.1 eV), which are in

energetic than molecular rotational energy differences « 0.1 eV). not to proper scale, Figure 10.21 illustrates the relationships. The curves represent two different electron states-perhaps the ground state and the firsl excited state. When an electron is in an excited state, the atoms are still bound to each other but are farther apart on average, and the interatomic potential energy well i" wider and shallower. Even so, for each electronic stale, there are closely spaced vibrational levels and even more closely spaced rotational levels. Thus, while we might expect a molecule to produce a simple speclr3.1 line when an electron jumps from one of its allowed energies 10 another, vibrational and rotational energy changes break each line into many, yielding a much richer structure. Note that symmetric molecules. such as H2 and N1, lack an electric dipole moment, so they do not produce spectra due to pure vibration-rotation transitions. However, the distinctive structure of Ihe':.e levels is superimposed on l)pectral lines arising from transitions between elec­ trOn states. (The oscillating electron cloud provides the necessary dipole moment in these case�.) tum more

Although

Fig'" 10.21 �lol«ular \lbratlOnal and rotational le\-els ror

1....0 . different dettrun k\d\.

HI�hcr

ela:lf\ln If:o\el \'Ibr.ltllllUl

le,d' II

I I I

l :; ' = ) " :� ��' ,

Lower ek.;lrun

:2

, 10 � tol 11;' ,

D_

ll

ROlallonal e�e \ J

Db....

1 0.4 Crystalline Solids lei u..

now

lum

from mol�ule... the bondmg of a relati\'ely 'imall number of

atom.. into \OmewhJI larger unil", to caCd matter Indude.. ,tudies of 'iohd!l, hqulds. and other syslems in which the beha\'ioT'i of man)' alom' together must be understood (such as Base­

Ein,tein conden..ate". di-.r.:u."c,ed in SectIon 9_6). In the following sections, we restritt our attention 10 cl)'�talhne wILd... All elements and compound:. form :.olids at sufficiently low temperature .. a cr)stal lattice, in which certain alom� and/or high pre\,ure mml often a

1-a,:t-(o!1lltmi lubK II�d

.

are found at 'per.:ific location, in a mlcro"copic unit thai is repealed identically countle.... time, in all dimen,ion"_ (Some materials form amorphous solids. in whICh anglc� and bond length.. are \0 Irregular thai the location of one atom i� cs..enlially unrelated to that of another only a few atoms away. Familiar exam­

Side �itw HC'O\Itln;d dOMCSl l'3'l.etl 4hi:p>

0 •

----

'

.

.

n

0

-;;

c.n

Simple ,.;ubic .. ,)

ple� are gl:c>, and rubber,I TypIcal atomic "pacinI! ID a cT}lotal is 0.25 nm to

0.5 nm_ or 5 to 10 Bohr radiI Geometrkal con'ldemtlon, ,how that In three dlmen�lOn.. there are only 14 pcMible lattice 1)pe5. or wa),,, in which atom'l can be arranged in a regular geomelric pattern. (Exc:rCl..e �5 gi\'elt a 'Imple example of"geomelrical con�ld­ c:ratil,m" in two dimen,]Iln�.) Figure 10.22 "how .. �\'eral of the rnO:.t common. The bod)'-ctnltred cubic I' a repetItIOn of cube'i wllh atom... at the comers and (enler. It ilt found in the ,olid form, of the periodic table-s fir-Holumn ele_ .. in a number of tntn..ltion metal", iuch a." ITOn. chromium. and menl'. a., well 3. tung,ten The face-centered cubic i.. aha a repet itive cubic sUllcture. but ....11lI alom, at the center. of the t:ub.t:/II("tl lD,tead of al the center of the cube. The:

10.4 Cryltalline Solidi ept helium) solidify in this structure, as do many transit.ion ele­ DObie gases (exc g copper, silver. and gOld. The hexagonal close:st.packecl includin . ments ,tr\lc(Ure has a six-sided symmetry and shares with the face-centered cubic the of resulting in the smallest volume per atom for a lanice of idenbcal distinction found throughout the periodic table­ �pheres. Eleme ts wi Ihi st cture

�

�

� �

ar�

helium. magnesIUm, ZinC, tttanlum, osmIUm, and many rare earths. Compounds same crystaUine structures as elements. For instance, sodium chlo­ form in the ride is a face-centered cubic structure in which a sodium-chlorine pair replaces

the individual atom. In facl, the sodium and chlorine ions each i.ndependently ntered cubic arrangement. because the atoms of one are a fixed form a face-ce those of the other. Cesium chloride. though resembling the from displacement

bOdy·centered cubic, is an example of a

simple

cubic structure, fonned by

each ionic type independently. Besides their lattice geometry. crystalline solids are often categorized according to how the valence electrons are bound in the solid. Four categories are generally recognized. Covalent Solid

ill a covalent solid. such as diamond, each atom shares covalent bonds with those surrounding it. resulting in an unbroken network of strong honds. Such

solids are relatively hard. due to the inherent strength of the covalent bond. and have high melting points. They are

poor

electrical conductors because an

\a1ence elecLrons are locked into bonds between adjacent atoms. The crystal lat­ uce assumes a geometry determined by the directionality of the covalent bonds. In the case of diamond, it is face-centered cubic. Ionic Solid When atoms with nearly filled shells meet atoms with weakly bound valence electrons. the fanner may seize electrons from the latter, producing an

ionic

solid. The solid is held together by the strong electrostatic attraction between the

ions. Thus, ionic solids are relatively hard. with high melting points. Because the transfer of electrons leaves both positive and negative ions with noble gas electronic structure. electrons are not free to respond to electric fields, so ionic

solids are poor electrical conductors. The ionic bond lacks directionality. and

the lattice geometry is therefore determined by whatever arrangement leads to

the lowest electrostatic energy. which depends on the relative sizes of the ions. in any case, lowest energy results when signs alternate-that is, the closest neighbor of an ion of one sign is an ion of the other. Because ionic solids depend do not fonn ionic solids. on asymmetry between atoms.

Metallic Solid

elements

Except for noble gases. al l element!; have valence electrons. But in most cases. no arrangement of

covalent bond!; can join all of a given atom's valence elec­

trons to those of the atoms surrounding it. An element or compound with "leftover" valence electron ... forms a

metallic solid.

When bound to their i,oiated atoms. valence electrons have relatively high

energy. In a metallic sohll. however. the atomic valence states mix. much as in a diatomic covalent bond but I'nl.:ompassing all the atoms in the crystal.

This

.a9

to • . ne. ... beoor1 bY potassium .I"ne. b) ,hl"nne alo and b) �.. put . 7 IOIid (anned -.w ttl. I;iod _ _ 11CA1I 'LE

.

__

.... ,,1

__

qedL _

.

--':�. - I. When -valence .7'1 " dIIorine

+. "'II '711 "-

logeth". chlorine will . . ........ _tng complete ....hell po"'llwe and

l('IM. 1be: kIIU �houlJ tht-n f\1fffi an K)ftl,;' ",>W1IunH.'hloridc whd. In..thid­ � � IC'MkI d\k1rmO not potassi um .;u fonn an hlflll; dtd. I.' ilJrrul('al ar('l4n ..III .,. dIIf(' tk(trOOi �)mmttrk.lIl. I). i\"�,iWQ h.n. illJ� 'J t�1ron In. phm

.#. I)� ,..tilt. "'-' 11 �11I n..-c ItnJ 1,1 � o:�·\akntl) Wllh mullirk .1then

� Il In a 1.11tK'('. \\ Ilh lC'fI" \'ef C'lt,,:trvn.s, I"lIlill1�m �ll the 'MUlti' t )I'IC '

.;I.nJ l '

N.'l m" dn\,:(', IhtIllO\\lf the nc'\t. Thu, Inr, we hU\lc di,cU',sed only Ihe nature of Slales QI'ailabll' 10 an elec­ tnln In a c£),\\(11 We have not udllressed the question of how many electrons might be pre..ent to fill those Mutes, and it is this point thaI go"ern� whether n materi,l! will be a conductor or an insulator,

N

I

Electrical Conduction Before we study the rclation..hip between energy bands and conductivity. it i\ \\orthwhilc 10 review aspect!> of electrical conduction Ihat may be under�tQO(.I

from a classical point

01 view and 10 identify those that require u quulltllm�

mechanical approach. Figure 10.26 Beh;\\i\lr of allowed electron SillIes as atomic spacing varie\. MInimum N alomlc

�hlle)

B�d width

AtomiC 'P' ing

•

, "1 ... .. � . •_ .. Ck;tl l' •

BOlIld, and �ar' 10 a one­ ,... 10.27

di�1 crystal

. .

..

. .....

.

I

) .....

-'1,-----,1f�r¥) 1!4p , II c

2 bud

!

" -.

:�

are free 10 move aboUi i n ondu o piclure. electron, Cl ,. In It\( da....IL:al ions_ This i s a th nec ti s ary , , Ingrt_ io", " Ith the po" colh, "flee 001 th,) ma) electrical current is proportional to e applied anding why the ndcN 10 dlent Were the'e no call'ISlo u , ns, asing function of time, being an IOcre . l'idd ralher than < constantly would ncce erate, field c electri proant to a con�l electron.. ..ubJed represent a retard'Ing force ons Colhsl nt. " Increa>;lOg curre - In dUl,;\Og B ..h:adily ously attmn lermmal speed in whIch a bal· nc tanta m .. �t >; almo . e..-.en�e, electron and the decelerating effect 0[ acceleratlOg field the n bet\\ee anee I " reached , ' clllh",on... Speed ; moves randomly with an avera typical electron ;: CI�...ica\ly. 11 with a positive i n , !I IS ) "t ", After a co\lisio the tempcralu chaneten,lIe o[ - n as another. With no externa e ectr.ic _ onc d-lrecllO monng 10 But when an e,ectric a.. IiLel) to be the electrons is zero. \'elocit)' of all . . . a compOnem field, the B\'erage collisions between the tLme . electrons gam, 10 the nt, pre!\C I' time field average belween colus define 1" as Ihe . lle the field. Let given electron of \'elOClt) oppo� a may have time. in inSiant n time AI any but an average eleClrOn Ii,ions. or collisio lime, [ . span 0 - g [or an arb-Itraty 1eraun eee a ' 1 ree 'i been 1". It WIII thus have acquired an adde •

I

,

1 I

_

_

'

found Il!; folloVo-'!i:

j•

charge

charge

time . area

-=

distance · area

distance time

:::

e X number distance volume

time

field.

the

It" I'J '" the number of free charge carrier:. per unit \'olume. We "ee that ..tic 3 current den,it)'. is proportional to the cause, an electric which I' (/ftc!. . .. , . y l constant IS . own c i l . 1m d " y 0\. the as t e con pr oportlontu lt h t v u \, law The reciprocal of ils resistivity p--and is given the symbol u.

::rial-the

j = uE

( 10-7)

vity decreases as the collision time 't decreases. One way ecrease the collision time is to increase the temperature, for faslcr-mo\'ing

(I � �ical1y. conducti

��

c1es collide more frequently with obstructions around them. Thu\, the conductors decreases as temperature increases. ductivil)' of o l Cl Much of the classical view of electrical conductivity is valid quantum 'banically. Collisions are still viewed as the origin of electrical resistance.

:e � �

er. we find that in metals. though Ohm's law still holds. the collision larger than can be explained by a theory in which electron\ may m ' e is much llide with all positive ions. The quantum-mechanical explanation i" that near edges of bands) the states allowed an electron in a periodic crys( Ctpt 1. are essentially those of a free particle. The electron is a wave. upon which the \'

posith'e ions have little effect. It is not the regular array of positive ions that up�elS the otherwise free electrons. Rather. deviarions from regularity perturb

the electron wave and thus determine resistance. The most important devia­

U(lnS at ordinary temperatures are vibrations of the positive ions. These increase with temperature. contributing to an increase in resistance. (The facl thai conductivity in metals tends to vary as T to the first power is further evi­

dence that ionic motioll. not mere presence, is key. See Comprehensive Exer­ cise 77.) At temperatures below about 10 K, vibrational motion is so

diminished that the predominant sources of collisions are microscopic lattice

m i perfections. These may be either point defects, due to atomic vacancies or lmpurities. or more widespread disruptions of crystal regularity. In any case. the quantum wells at certain locations are altered, and the electron wave is perturbed. Unlike ionic vibrations. the abundance of lattice imperfections is largely independent of temperature, and so is the resistance lhey cause. In all solids. electrons fiU bands from low to high energy in accord with \hI! exclusion principle. In a conductor, the topmost band is only partially filled: this band is where lhe conduction electrons reside. (Lower full bands do DOl participate in conduction, as we see in Section 10.6.) To a good approxi­

mation. the conduction electrons behave as though moving freely inside a macroscopic well (the entire crystal) with no ionic potential. Their energies are those of an electron

gas (Section 9.6), and the energy of the highest·occupied

siale. measured from the bottom of the band, is the Fermi energy, EF" As

depicted in Figure 10.28, with no external electric field, as many electrons

" I.... .,, �

..... .. Os- to

'l\idl

ickf pllCftYl (k�(� 1nOl'C" da.1J,l(tl Ji ...,nf':Itn •

;S �

\

IIra:.J Jt.Ift'.

""'"

,

,

.... .t

.

.

'

,{.., > 0

0

hlled

....-- Eleclric field

tJt,..-!n( ',tId · 0

/

direclion as in the opposile direction· Theke{ffe in" � in onc Ct of � ukl � mel\ �''1 enta toward Ihose in wbich 10 shi fl clectron mom I' field mal e\le =: P/h) ,;tn the mo menta of all the conduction elecl fl�iI� [he tield. Although I� ()pn ' r tOns only among e higheSI�energ W't' ,h ttt , � Ii« mal IC I_� a net Shill .

e

�

1'.

Y stales. . £f' where the momentum IS 10 the direct'IOn belo" y l h l , t g i ,["Ie' ' of '" fnlm . l.Ie � where I" I IS O�posl le. Classically or qu ' above l:.F' 0 to the tOlal number in the valence bifid at T 0 IS

( 1 0- 1 1 )

=

Equtllion ( 1 0- 1 1 ) is a very useful relationship, Band!'. tend to be of order eY, we ,ce that the 10 eV III width (see Ex.ercise 50), and with k Tmonl =o muluplicative factor is roughly of order 1 0 �, However, it IS the ex.ponential factor that dominates. (A realistically varying density of state!'. would change

lo

only the less-important multiplicative factor.) The difference in conductivity

bttween conduclOrs and insulators is one of the mo�t stark contraO

T=O

acceptor state.) Thermal excitation readily bumps electrons from the valence

.

band to acceptor states, leaving freely moving holes.

Again there is always some thermal excitation across the entire valence_ to-conduction gap, creating electron-hole pa irs in both n-type and p-rype semiconductors. Both thus harbor charge carriers of the "wrong" sign (i.e.. valence holes in n-type and conduction electrons in p-type). known as

minority carriers. But far more abundant are the majority carriers: valence holes in p-rype and conduction electrons in n-type. What makes doped semi·

a

conductors unique is the ease with which we c n vary the sign and abundance

.

of the carriers via impurity type and concentration. As we see in the next sec·

tion this is central to modem electronics.

A Closer look

Effective Mass

a particle moving Ihrough a medium. and

Section 6.4 shows Ihal the velocilY of a panicle. called group

\Ie/DeilY. i� related to its mailer wave properties UJand k by

I'

.

paruclc

: 1'

8"'"P

dw dk

=-

As demonstrated in Example 6.3. Ihe dispersion relation rnat gives (J) as a function of

k

accounts for rne effects of a

medium on a particle moving rnrough it. Thus. we may write the acceleration of a particle in any medium. including a cry.�ralljne solid,

d dl

as

0 = - 1'

.

parodc

( )

d dW dl dk

=- -

J2w dk dk2 dl

= --

Effective mass is defined as the ralio of rne exlemsl force to this acceleralion. Ignoring losses due to frictional forces. it is the external force alone that does rne net work on

the force times

the panicle's velocity is (he work per unit time (power). �

dE dr

-

Using E = ItW, we then have F" ,' _ " " " � -.

=

dw dw dk � h -- = dr dk dt

11 -

or, canceling vpMI,cle on both sides,

FUI

=

dk h -;;;

dk

Ill'parllC . ,c dt

1 0 . 8 S emiconductor Devices

how doped semiconductors may be exploited to produce twO \..et U� nOW study important circuit elements. most e of th

The Diode

applications, we wish to allow current to flow only one way. The ill many

most basic is a rectifier. which converts the sinusoidal\y varying (alternating) current supplied to homes into the direct current needed by most eleclronic circuitry. Actually building a device that does this automatically, known as a requires some ingenuity. The primitive solution used evacuated glass tubes housing bulky electrodes and heaters. The advent of doped semiconducbrought a revolution. Asymmetric current flow can be achieved through a

diode, tors

simple physica l asymmetry : charge carriers of different sign in different regions. We produce a diode by joining an n-type semiconductor to a p-type �emiconductor. The area of contact is known as a p-n junction. As shown in figure lOA 1 , if we apply a potential difference with the p-type at higher potential. known as a forward-biased condition, free holes in the p-type region and free electrons in the n-type flow toward the junction. When they meet there, as we discuss below. they annihilate, or recombine. Current flows

continuously as electrons are added at the low-potential side and holes are added (valence electrons removed) at the high potential side. On the other

hand. if the n-type is at the higher potential-a reverse-biased condition­

both holes and electrons move away from the junction. A region devoid of free charge carriers quickly fonns, and current stops almost instantly. Energy-level diagrams help clarify the behaviors . Figure IOA2(a) shows an

unbiased condition. If we simply join a p-type and an n-type, without an exter­ nal potential difference, electrons "high" in the n-type conduction band cross to the p-type side, raising all electron energies there by repulsion. while corre­

spondingly lowering them in the n-type. Few electrons need actually cross

before an equilibrium is reached in which the highest occupied levels-the

figure 10.41 The motion of the charge carriers in forward and reverse biased p-n junctions.

Hole" added

Electron-hule r&ombin"ll\)O

If e e ® e ,'� 0- 0 e '-0

...

..

'

.

'

..

EI':"lron� JJ!.kd

.....

...

n-type

p-type

Electron noYo

«.tl

"---'lN1fIr-I 1 \�-­ +

Forward bia)'

Free !.:harge depleted

Holes removed

.

'

+

e e e

,.......,

p-type

1I'i \emper'J.\ure increases? Are these factors also present In II semiconductor. and if so. how can its conductivit), vat)' with temperature in the opposite sense" Il. Based only on me desire to limit m;nonlY carriers. why would silicon be preferable to gennanium as a fabric for doped semiconductors? 16. Why docs the small current flowing through a "versc­ biased diode depend much more strongly on tempera­ ture than on the applied (reverse) voltage? I7· In II concise yet fairly comprehensive way. explain why doped semiconductors are so pervasive in modem technology. 18. It h often said thaI Ihe transistor is the basic element of amplification, yet it supplies no energy of its own. Exactly what is its role in amplification? 19. The "floating magnet trick" is shown in Figure 10.50. If the disk on the bottom were a permanent magnet, rather than a superconductor, the trick wouldn't work. The superconductor does produce lin external field very similar to that of a permanent magnet. What other char­ acteristic is necessary to explain the effect? (Him: What happens when you hold two ordinary magnets so that !hey repel, and then you release one of them?) ZO. The isotope effect says that the critical temperature for superconductivity decreases as the moss of the positive ions increases. Can you argue why it should decrease? 21. What is a Cooper pair, and what role does it play in superconductivity? 22. Describe the similarities and differences between Type-! and Type-II superconductors. 23. In a buckyball, three of the bonds around each hexagon are so-called double bonds. They result from adjacent atoms sharing a state that does nOf panicipate in the Sp2 bonding. Which state is it, and is this extra bond a u-bond or a 1T-bond? Explain. 24. What are some of the properties of fullerenes that make them potentially so useful?

Exercises Section 10.1 25. Fonnulate an argument explaining why the even wave functions in Figure 10.1 should be lower in energy than their odd partners. 26. The accompanying diagrams represent the three lowest energy wave functions for three "atoms." As in all truly molecular states we consider. these states are shared among the atoms. At such large atomic separation. however, the energies are practically equal, so an

electron would be Just al< happ), occupying any combi­ nation. ta) Idenllt)· alg, but Il! can be :! I , then the allowed photon energie!: obey equation ( 1 0-6).

.n. Vibration-rotation speClra are richI For the CO mole­

cule (data are given in E",erci�e 42), roughly how many \'013tional levels would there be between the ground vibrational state and the first excited vibrational state?

�2. The carbon monoltide molecule CO has an effective spring con'itant of 1860 N/m and a bond length of

0.1 13

" one-dimcnsionlll ealcu� \.).\ion is in�tru..:t;\'e, C'(lnSlo.ler an infinite line of point charges alternating betwt'C'n +" and

... the lc) For thc'>C energies, by what per\:entage dot atomiC ..eparalion fluctuate" classical vibrational frequem:y (d) Cl1kulate the_ p- and lhe rotational frequency W w"t- =

10\'01\'«1 h'r a 3D lani�e. Nt

per

be helptul: In(1

Section

,

" I

2 band and the oo\t('m of the II

:III

3 band fur a

W8\'e number have \"" tl)' d,lkrcnt energie).

48. A, we ,ee in Figure, 10 :B. in a one·dimen�ional !.:ry�tal 01 finite well�. top-ol·the·t'land ....Ialc\ do\e!y re\Cmbk inliOltC well state'i, In la!.:t. the I'amou\ partldc-in.a.bOll. energy formula give� u fair \'alue for the energle, of

these ,tates and thus the energies (If the band, to '"hlch they belong,

(8) II for rt to thaI formula you u,e the

number of anllnodc\ to the ",hole W3\e

functlllO. '"hat would you U';C for the box length L? (b) If. in'tead. the n in the fannula ....ere . laken to refer to bdnd n. could you �lIl1 u�e the fomlula? It \0. what would )'1..11,1 u\c foc L?

in the lower-energy electronic ...tate? Explain. What

vibrational energy levels.?

-L{- \)"'11

wells. Explain \\-h� the� 1\\0 \Iah.'\ 01 rou�hl)' equal

(c) Explain why the energ;e� 10 a band do or do not

about the rotational levels?

Ie�'el!." (c) And what effect doe� it have on the

-

one-dimensIOnal "cry�tar' con",lIng 01 !-e\en finite

the same. farther apart, or c\m.er together than lho�

(b) What effect does it has'e on the rotational energy

"{)

10.5

the II

tional levels 10 the excited electronic state to be spaced

on the bond length and force con...tant" Explain.

+

41. Make rough ....kelehe' 01 thl' '"a\e IUnctlon, at the 1\IP uf

�3. From the qualitative shapes. of the interatomic potential energies in Figure 10.2 1 , would you expect the vlbra­

atom (a) What effect. if any. does the replacement have

ion. For \lmr\u':lty. ",,,,urn... that 3 po�ltive charge 1..

at the origin Th... fulloy,.m� puwl'r �nes e'pansion W il l

CO might absorb in vibration-rotation transitions.

from H,! 10 that a deuterium atom replaces a hydrogen

with a uniform

Wh)? (bl Calculate the ele..:tm\tatk potenltal ene'll)'

nm. Determine four wavelengths of light that

"-'. A noted in Example 10.2, the HD molecule differs

�(',

....paung bet....«n ad.la..:ent loppoo.ite) charges of Q, la) The cle..:tn\\tatir.: f"'tenltu\ tnefl), pt'r ion i.. the same illr a g!\cn Pl"ltl\'e I(ln a:o; lilt u " \'en Ilegati\'e ion.

49.

depend on the sile of the cry�tal 3., a y,.hole In Figure 1O.24.lhe n

I bilnd end\ at k

'"

4rrlL,

while in Figure 10,27 It end\ at rrla. Are t.he�e ..:ompill1· ble? If �o, how?

50, Assuming an interatomIC 'paCIng of 0.15 nrn. ot'luto a rough \'alue for the width (in eV) 01 the n one-dimensional u)\tal.

=

:! hand to 3

51. The density of copper 1\ 8.9 X 10-' kg/ml. it.. Fermi

energy i� 7,0 eV, and It h..� one condur.:tion el...ctroo per

Section

10.4

atom. At liquid nitrogen temperature

45. 1\Yo-dimensional lattices with three- or four-sided sym­ metries are possible. but there iCaller plot (c) Repem pan (b). but chOOSing

B In lhe " Impuri ly" alom. pUlling irs bollom 31 -2 unih. Cd) Discu!.s how Ihe impurilies added in pan�

- 0, I for

(h) and fc) cotre\pond to atom, I,I,ho\e ...alence differs from thaI of lhe intrin�ic atom!., (e) If each inlrinsic

110m comes with two eleelron\. and Ihe impurilies

come wilh one and three, R:specII\'ely. which slat�s would be filled in part:. (b) and (c)? Remember Ihal

there are IwO �pin Slate,. (n Discu,.. the overal l result

of adding Ihe impuritie�,

N u c l e a r Phys i cs

11

Chapter Outline 1 1. 1

@

I \ .2 1 1 .3

I tA

Basic Structure Binding Nuclear Models Nuclear Magnetic Resonance and MRI

1 1 .5 Radioactivity 1 1 .6 The Radioactive Decay Law 1 1 .7 Nuclear Reuctions

. n the preceding chapters, we tended to study ever-larger objects with photons and electrons. one-electron moved to We began � tron atoms, then to molecules and solids. In some sense, n the multielec

I

n a[o l.n" ',nue the trend would lead to other disciplines-chemi�try. engineering, to co and �o on. We now head the olh�r way. .

contam nuclei, studying the "atom" usually means Although all atoms of its orbiting electrons. The nuclei are assumed to behavior on the fIX'using nuclei . But are not always inert. For one thing. they can s n tiaU y inert

b< e se

' and such nuclear reactions release spontaneous1Y or otherwtse, ra f menl, grrnous amounts of energy compared to chemical reactions. In nuclear 0 ttmg e1ectrons t hat are 0f I'title concern. It IS true that they ,"h �ics, it is the orb" ,

P } bound to the nuclei, but compared with the energies within the nucleus,

�ctron �

bindi ng energies are usually negligible. The nucleus is the focus. hard problem t? solve. While much has been learned, it is nfortunalely, cs contmues to be an active area of research. auClear physi , ' cannot be overstated. h ex.plalOs the tance 0f nucIear phYSlcs impor The

�

and nuclear weapons; nuclear magnetic resonance perotion of nuclear reactors produce images of the body's interior; and radioactivily­ routinely used to

�I �

ergelic particles emanating from nuclei-is both a useful diagnostic tool, via n

dioactive tracing and dating. and a mounting disposal problem. as we con­ artificially produce more radioactive materia1s. Moreover, tinue to unearth and

the study of nuclear physics is an essential step in our quest to unravel the fun­ damental structure of the physical universe.

11,1

Basic Structure

All nuclei consist of protons and neutrons, known collectively as nucleons. Masses are given in Table I I . \ . Introducing important symbols, Figure 1 1 . 1

depicts a helium atom-two electrons orbiting a nucleus of two protons and twO neutrons. We use the symbol N for the number of neutrons and Z, the

475

-

-

-

-

�

-

-

-

-

-

-

-

-

- -

�-

- �-�-

.., , , , ... -

I 6126J17 )( '0 .17 ...,

"

•

••

I 6'49273 x 10 11 ..,

o

If •

•

-

9.1(9)( 10 u k,

•

....

hr.

.....

-

� '=>

J.(J07276 u

I 00tIM.'I u

� "'AA X /II � u

lor rbt number of prolons. The ..ymbol A. caJled the IRI5s

II &he lOCal number of nucleons.

At we found in C'bIpIer 8. an element's chemical behavior depend, onl}

a. die IIUIIIber of eJecuons miring il.\ nudeu.�. equal 10 Z. But nucle, of .he .... demenI rarely beha� alike jf they have different numbe" of neulmn,.

ftar ...-.xe. while tbt nucleus of ordinary hydrogen 1\ "" mply a prolon. about 0.01$" 01 hydroaen IIOmS in naluft' have nuclei con ..ie:nd on e orientation of the nucleon not �urpnsmgly, there IS no simple formula for this force, .. ins, Perhaps indeed has yet to be fully characterized. Nonetheless, we can explain quite a bit from what we do know about it. ore

�

::a,.,y :h.ich

Sta bility : A Theoretical

Model

engaging in an extremely strong attraction, we might lmag­ With all nucleons could bring together any combination and have a stable nucleus. vast majority would be unstable, disintegrating in a time span the owever. [allging from an instant to an eon. A few would be stable. The following discussion of nuclear stabiliry may seem rather s.peculative, forces in the nucleus have defied formulation of a compre­ but the complex htnsive theory. so we must resort [0 a model-well-informed but simplified guesswork. [n a later section, we discuss well-established models thai refine we one we begin here, bUI it is good to gain a quahtative understanding of the \arious factors fust. As we do. we won't speculate whether any particular combination of nucleons should actually be stable; we will simply argue ....hether a given factor should tend to make the combination more stable or le�5 stable. A nucleus is more stable when its constiruems are bound in a state of lower energy, requiring a greater expenditure of energy to extract a repre­ q:ntali\'e nucleon, Energy is all-imponant.

'ne that we

�

Two-Nucleon Nuclei

The simplest possible combinations of nucleons are two protons (p-p), two

oeulIOns (n-n), and a proton-neutron pair (p-n), Whkh should be most stable? . Let us look at the factors The short·range strong attraction shouJd create some. thing like a narrow but deep potential energy well. and if the force doesn't dcare" whether the particles are protons or neutrons. the weU should be the \.llIlC for all three combinations. This condition is no guarantee that any would hold together-wells can be too shallow to have any bound state.!. (see Exer­ cise 5.38)-but at least they should all be the same. On the other hand, the p-p combination ""ouJd mclude Coulomb repUlsion between the proton!>. which v.ould rai� the energ), and this argues that it should be somev.hat less stable than the other IWO. .. are ValId. but the ltuth is that onJy the p-n. known � a These argument dtUteron. form' a bound nudeu'i (that of the stable deuterium alom dio;cussed earlier). 1be n-n and p-p don't stick together at aJl. Why" The internucleon aruaclion i, .,pin-dependent. and II i... stronger ""hen spins are aligned. We will :e 1.5 Ie.. 'loigOlficant In larger nuclei. but here it call.. . • pendcn e find thaI milo d thai Ii crucial in all nuclei: the exclu�ion principle. factor r e h t o an play into

Fig.". 1 1 .4 The basic elements of the internuclean (strong force) poIe11tia1 energy-a strong. mort-range anractioD with a repulsive hard core. Sudeons Iolrongly atUaCI

lpo!cntia] energy drops). but only when close e/1Ough to "'toUch.�

ur,}

f -'7'!!'...-----" H-+ . rn -U

.

•

•••

:,0

.............

�p!

When roo dole, die.. han:! can::s repel.

,

,.... f1.l

The dnlrcroo'UtUlJ'Oll

Mtd profua buund In. well rnullIIJI

" -+�-r

from IIIftt w.:tt� poIftIuaJ cntlJ)"

Proton�

WId llC'utrons

art" I-pln

i f(mlion�, '0 they cannOI occupy

the same

�.)\I("m. The' lo�(�1 energ) po\�ible for Iwo nucleons in a "dl �Id haH' bolh in Ihe ground �piliioll �Iate with their ..pins aligned (the !OWe In lhc

�

mentation of slronger allm:l.on I, 001 ,ut:h a stolle is forbidden 10 the p-p and thC' eu-Iu�i(ln prinl,:iple. h Ihi' indeed why Ihe p-n forms:, but the P-p

I)-I) by

and n n do n,,(') E'p'-'nmental endence \lenfie.. thaI the deuteron i.� ani)' ... no exciled bound Slates, jusl the one �ly bc.-.und: 11\ potenllal well ha �mund slate, .\IO(f'O\er. the neulmn olnd proron ,pins art' aligned (its total SPin -

I� II. If til< p-n cannot �md any other way-with '>p,ns opposite or widJ one nudeon in a higher 'patial \tJte-lhen the other two should IIor bind at alL The enC'Iog) in the deuleron i� depir.:led in Figure 1 1 .5. (The repulsi\le nucleon

core, are i�nored.) Th'o nudeom occupy the lone bound slate in a well that ari\e\ from their mutual altrJclion.

fIsIu,. 11.6 RwumS tntfll} ret

Arbitrary Nucleon Number

nuclcvn JuC' II' Ihe ,'rpng mh.'mud("(ln

We no� com.ider nuclei compri�ing any number of nucleons. addressing in rum the efted� of the Ihree main faCIO,...,: the strong force, Coulomb repUlsion,

"lIr:atlion poly fill: �mJ"C'sl nudt.

t1a\e' lew Ix>nu. (Itt nudl'on, In 1.l11!C"

ilnd the e:(clu,ion prinCIple.

nucld. many lIurk.-ons J� !umlYnutd

Strong Intemucleon AUraction A two-nue/eon nucleus has only one bond. but a three·nucleon nudeu� ha.� three. Thus, the fonner has half a bond per

nlJ/1ron and the lalfer one bond per nucleon, A four·nucleon nucleus would ha\e m bond" or one·and.a-half bond� per nuc/eon. so a representative nudeon would be harder to extract than in a three·nucleon nucleus. This mcrea\ing [tend n i bond� per nucleon does not conlinue indefinitely. however. Nudeon� attract, bul they also have a hard core thai causes the nue/eus to

l .,

grow like a collection of hard spheres stuck logether. The intemucleon aUrae· lion i\ \0 short range that a given nucleon attraclS only Ihose immediately surrounding it. �o each surrounded nucleon will have Ihe same maximum number of bonds. (We ,ay that the force salurates.) Of course, nucleons al the surface are not completely surrounded. but as the nuclear sphere grows, the number .11 the surface is a diminishing fraction of the lotal. The ratio is essen­

F/vu1"9 11.7 [pu/mnb rrpul\ion faj'e, prI" "n eneq!ll,·'.

lially area/volume =

41Tr 2/� 1I"r3 oc

IJr. Thus, the average number of bonds

per nuc/eon should inilially increase fairly rapidly as the nucleus grows, then gradually approach a con�tanl, as more nucleons become surrounded. Figure

Cllulomh

J ,

I

1.6 show� the trend.

Rather than bonds per nucleon. however. we now speak

of binding energy per nucleon, Binding energy is the energy rnal would be required lO pull all nucleons apart. and binding energy per nucleon is the frac·

E,

E,

-I---,;�·l

-t--�:

Scurrun

T

;--

lion required for a representative nucleon.

I:

E,

Coulomb Repulsion All pairs of protons in the nucleus repel. In effect, this positi\e potential energy shift� all proton energies closer 10 rhe top o f the finite well in 'II. hich all nucleons are bound. as depicted ID Figure I I .7 for the case of helium-4. In such small nuclei. all nucleons are

"

touchi ng:· \0 there is slill a

,(rong net attraction between all pail"!. But in large nuclei, paiJ"\ of protons can be too far apart to attract via the strong force, and these pairs add to the nucleu, an uncompensated nel repulsion. Thus, while under control in small

Figur. 11.8 Binding energy per nucleon due to both the �trong inlemudeon attraction and Coulomb repulsion. t.ur�e nudei hone proltln\ thai do nol

Small nude. ha\c:

allriltl.'� \

Ie..... NInth pc:r nudeun

remove lrom lin

2

A

Coulomb repulsion should be an increasingly destabilizing factor in The energy needed to extract an "average" nucleon becomes pro­ ones. larger Combining this with our previous arguments. we should smaller. gressively expect the binding energy per nucleon to vary as tn Figure 1 t .8. Accordingly. �omewhere between the extremes, there should be a mass number more !ltable than all others. for which the nucleons are the most lightly bound possible. nuclei,

The Exclusion Principle To this poim. it would seem that the beM way to produce a stable nucleus is to build it of essentially all neutrons, for only pro­ tons repel. This argument overlooks the fact that protons and neutrons tn the same nucleus must obey the exclusion principle, each independently. Putting Coulomb repulsion temporarily aside. the most important consequence of the e'(c\usion principle is that for a given A, the state of lowest energy would have equal numbers of protons and neutrons. Why? Suppose that Z does equal N. as illustrated on the left in Figure 1 1 .9. The lowest energy state consistent with the exclusion principle would have aU the Figur. 1 1 .9 I gnoring Coulomb repulsion. the exclusion princi­ ple argues that for a given number of nucleons. energy should have N = Z. Slanmg wIth equal

•

numben. of nCUlron� and

f: E,

' higher energy level_ _ _ _

\

-

•

E,

E,

_

./

E)

�7

Neutron

d conveni ng one to

an

the other force� a nucleon to

pruto"",

E,

__

the lowest

Ez

E,

Proton

.

--=- : - - - -

- .�--=..-..:. :=.=..;..- =� --

..... ft. to In .... nucki. .heo Coulomb�ltJOCI becomes l.I,,"ti·

e.I. dw JowaI tnftJY IAouJd

�N>Z

Anllmn cnerJ} Ic\-e!, filled. c&:h �ilh an 0ppo"He-�p,"s pair. up 10 some maxi_ mum cnergy_ Comlde-nng only !he inlemudeon altraction, protons �ould occup) 110 lde-nllf.:aJ ""cll and would fill le\eI� 10 the same maximum energy. NtM. keeping A that a dtcreas(' b)' J in Ihe prolan number would mean an 1f)(:Tea.\e b} 1 In the neutron number. Becau�e all lower energy levels are filled,

ncubun wQUld ha\e 10 occupy an energy levcl higher than the previ00\1) occupied prolOn leH!I. PJld me 10lal energy would increase. The argumenl applic\ )1.1\1 � well If we chan!!e a neutron 10 a proton. Thus. Ihe slate Z :::: N is

lhc

.. -+.....;.-j-

.. -+.....;.-j-

.. -+.....;.-j-

.. -+.....,-1-

�

of 10....('\1 energy (If elmer top level initially had only One neutron or prOlan, we might ha\'C' to changc a I.C':cond particle before the energy increased.) UIU_\ now faclor in Coulomb repulsion. In �mall nuclei, where essentially

all nucleons lauch, all prolon repulsions are overwhelmed by the strong attrac_ tion. so Z :!' N should be me more �table Slate. In lurge nuclei, uncompensated Coulomb repul�ion, may shif! me prolon energies upward enough to align wim

different neutron 1C'\'els. as depicted in Figure 1 1 .10. The lowest energy slate­ with me lap.. oflhe proton and neutron levels roughly equal-should then ha\'e

more neutrons than protons. A balance is struck between having eXira neutrons \toe as nonrepuhl\e "glue" and forcing them into higher-energy states by the

e).du�ion principle. O\craJl. our modcl �uggests mat a represenlative nucleon should be most ti!!htly bound-in a 10we�1 energy stale-in a nucleus of intermediate mass number, but il is nOI a function of A alone. The bindlOg energy per nucleon

should ha\-e a rtla/h'e maximum at Z a N for small A and al N > Z for large A.

Stability: The Experimenta l Truth The \alidilY of an)' theory rests on ils agreement with experimental evidence_ Obtaining evidence to support our model of binding might seem 10 inVOlve pulling a nucleus apart while measuring forces. bUI it is much simpler than that. All we need to know are nuclear masses. If we were to pull a stationary nucleus apart into separate stationary nucleons, we would have to expend our own energy. which would increase the system's inlernal energy and therefore It� ma�s. The mass of the parts must be greater than that of the whole bound nucleus. Figure 1 1 , 1 / bears this oul for our simple!!t nucleus.

deuteron mass = 2.013553 u proton mass

,.... t t.11 A dtUlC'ron ""C'IJlh) IC'" lhIn the sum of liS parh

+ neutron mass =

mas� of pans - mass of whole

=

1.007276 u + 1 .008665 u = 2.015941 u 2.015941 u - 2.013553 u = 0.002388 u

We o..ee lhal we must do work to pull the deuteron apart. For an arbitrary nucleus. lhe energy required to pull all the nucleons apart-Ihe total binding 2 OIJ�.H u

energy BE-is lhe difference between the final and Ihe initial energ ies which ,

Proton

Nrutroa

1.007276 1.1

1.008665 1.1

o

mc IIOnra­

•

dIoKtiw: IMIIS'iah move about or how they slick to one another. radioactive IIOIDI lie

iDIrodIlCed, either as a loose mixture or by being bonded 10 specific

IIIDIcaIJ.r 1fOUPI. 1be subsequenl behavior of the materia] is then easily fol. lowed by the lelhale decay of the radioactive tracer. In one of many biologi.

cal

research

applications• • slerOid molecule is lagged (covalently bondtd) WIth tritium. • /J entitlei'. and then mixed with a protein. The .H�rojd·s abilil), to bind to the protein is clearly indicated by how much trilium ends up stuck to the proIein. Let us now look II another characteristic of radioactive decay that I allO much e..ploited in ICteDlific investigation.

1 1 .6 The Radioactive Decay Law

All forms of radiOKtive decay fundamentally change the parent nucleus, and the nuckus may decay in a particular way only once. A good analogy is a light. bulb. which can bum out only once, after which 11

IS fundamentally different

from a wading lighlbulb. Furthennore. decay!>. are governed by probabililles. It is irnpo5jible 10 know exactly when specific nudel In a sample will decay. One

nucleus may decay righl away and another after

very long time. In a large sample. however. there should be a predictable a\'eroge time. characteristic of n

the panicular decay: it may be long for the a decay of isotope X and shon for the /3. decay of isotope Y. The lightbulb analogy is again helpful. One neVer knows when

8

IighlbuJb will bum out. but in a huge office building with thou­

sand", of IightbuJb X. thert would be a predictable number burning out per day. A rtliable average lifetime would be apparent. In a similar building with thou­ sands of lightbulb Y-a competitor's lower-quaJity unit-a predictable average lifetime would aJso be apparent. though it might be much shorter. No mailer whar value the lifetime might be. if we increase the size of the sample (building). the number decaying (burning OUI) per unit lime should increase proportionally. In OIher word�. the change per unit time i n the number

of nuclei present should be directly proponional to the number presen!.

dN ex N dl

-

We make this an equation by using the symbol A for the proponionality con­ stant.

(Note: In this seclion. N stands for Ihe number of nuclei. rather than the

number of neurrons in a given nucleus.) Becau�e the number of radioactive nuclei that are present is constantly decreasing �uires a minus sign,

dN dl

=

-AN

(dNldl


uits in an e\plosion . releasing

5-6 orders of magnitude more energy per kilogram

txplosives based on chemical reactions.

lhan

lEllr'� III

Ih Iluckar IIge, bl1:'\\'ill� lh� "" kUI ...

IIpa" WI\) populllrly

,

",k=d II'> u� "�phllJl\I! Ih�

nlom." cv�n Ihough onhnM)' ionitllll(ll\ ,phi, un alOIn. tarly h"nln ",(';Iron_ w�rc \lUh \tnn�.J "aloml� hamh......

1l1I.J lh'" nilffie Mud.

Lller

wcapoll' m\(llvlnll lu'Illn of hydrogcn "",lllf'l'. w('rt

lermed

bomh�."

·hy.Jrogcn homb,_' LJu\h lin: '[lude,I
dnd . few years are found in nature

Ifl

small abundunfo."eS Ih31

do not chan¥e at J.1l mer man)', many )can, How is Ihis po�\lble" (Him_ Natural uranium and Ihonu m have l't'/)' long half-Ji\e\.)

9, Why dlle' fi,,'lon of he:a,y nudci tcnd 10 produl:e li'ce neutron,')

10. I n bolh

( I I-IS)

and

fusion, which ekploit the vel')' tight bindin, of inrenncdiale­

O-D rcac.:lj�'n" ln clju;Ulun ( 1 1·/8). 1....0

deuleron, fu\C 10 produce tw.I} particles, .. nuckus of

A = 3 and It free nu..:k"tlfl

M.lU ds nf!utrons. the tol31 binding

37,

n

t

a,

""

are

section 1 1 .4

All target nuclei used in MRI have an odd number of protons or neutrons or both. What does this suggest about nuclear spms? (Not�: 80th the proton and the neutron have gyromagnetic ratios.) 35. In electron spin resonance (Section 8. t). incoming elec· tromagnetic radiation of the proper (resonant) frequency causes the electron's magnetic moment to go from its lower-energy. or "relaxed," orientation, aligned wilh the ex.temal field. 10 Its hlgher-energy antialigned state. MRJ IS analogous. A quantity commonly discussed in MRJ IS lhe ratio of lhe frequency of lhe incoming radia· tion to the external magnetic field. Calculate this ratio for hydrogen. You Will need to adapt equation (8·7) to !.he proton. Note that the proton gyromagnetic rauo, gp' 155.6. 36. MRI relies on only a tmy majority of the nuclear mag· netic moments aligning with the eXlernal field. Can· sider lhe common target nucleus hydrogen. The difference between thl! aligned and antiahgned states of a dipole a magneuc field IS 2JA..B. Equation (8·7) can be used to find JA.. for the proton, provided that the correct mass and gyrOmagnetic rallo (gl? 5.6) mserted USing the Boltzmann dlstnbuuon, show that for a 1.0 T field and a re�onable temperature, the

)4.

10

-

are

Q

arc

=

... " �_� . .. 0.

CIIIIJY would Ktually dlCmur. bplatn wtaM .t wrong e.

�

with the naIVC' atpmcnt. il lht reooll lpced of tht dluJhtef nudeu� when ,.,Ho 0 dcaY''t CTrcal aU mobon Il!o nontd'II\1i�bc.}

_ " .6 51. Given IOJbally 100 a of plutoOJum.239. how much time mUlt pus for the amou.ntto drop 10 1 g? 51. 1l1e IMial decay rate of a wnple of a eertal.n radioac· tlve ilOlopc II 2.00 X 10" � I After haJf an hour. !he decay rate 11 6.42 x 1010,'1 Detennine the halr·life

of the itOlope.

51. Tbc half·hfe TII2 1$ not the a�ngc lifetime Tof 3

radloacll� nudeu!> We find the a"'erage lifetime by multlpJyma t by the probability per umt IInlC P(t} that the nucleu\ will " Iwe" thai long.lhen inlegrating over all IItnC. (a) Show that P(t) \hould � gi\'en by At-AI (Hint. What mu�t be the lotal probability?) (b) Show that T - Tlr../ln 2. 53. Eighty centurie� atter its dealh, what will be the: decay rate: of I g of carbon from the thigh bone of an animal" S4. A fOiU,il specimen ha.\ a carbon· 14 decay rate of 3.0 s' 1 . (a) How many carbon· 14 nuciel are pre�nt? (b) lrthi� number 1\ rn the num�r that must have been present when the animal died. how old i� the fouil') 55. A fossil \pecimen has a 1 4C decay rate of 5.0 s I Ca) How many carbon-14 nuclei are prt!otnt? (b) If the specimen I� 20.000 yean; old. how many carbon·14 nuclei were present when the animal died? (c) How much kinetit: energy (in MeV) is released in each /3 decay, and what is the tOlal amount released in all fJ decays 5in� the animal died') 56. A bone of an animal contains tli mol of carbon when it diei. (a) How many carbon-14 atoms would be left after 200,000 yr') (b) h carbon-14 dating useful to predici the age of such an old bone? Eltplain. 57. Pow.sium-40 has a half-life of 1,26 X 109 yr. decaying 10 calcium-40 and argon-40 in a ratio of 8.54 to I If a rock sample conllllnod no argon when it formed a solid but now contains one argon-40 atom for every pot8ssium-40 atom. how old is the rock'! 58. (a) Oetennine the total amount of energy released in the complete decay of I mg of lritium. (b) According to the law of radiOllctive decay. how much time would this release of energy span? (c) 10 a practical sense. how much time will it span? 59. Gi"'en initially 40 mg of radium-226 (one of the decay product!. of uranium-238), detennine (a) the amounl that will be left after 500 yr, (b) the number of a parti­ cles the radium will have emined during this time. and ecl me amount of kinetic energy thaI will have been

released (d) Find the decay nne of the radium at the end of the 500 yr. 60. Ten milligrams of pure polonium-210 is placed in 500 of water. If no heat is allowed to escape 10 the Sur_ g roundings, how much will the Icmperature rise in I hr? S.ction 1',7 6 1. Determine Q for the reaction

62. Calculate the net amount of energy released in the: deuterium-tritium reaction I 2,0 + J, T - "2He + on 63. In an assembly of fissionable material. the larger the SUr.

fnce area pcr fissioning nucleus (i.e., per unit volume)' more hkely is the escape of valuable neutrons. (a) What is the surface-to-volume ratio of a sphere of radius ro? (b) What is the surface-to-volume ratio of a cube of the same volume? (e) What is the surface_to­ mlume ratio of a sphere of twice the volume? 64. If all the nuclei in a pure sample of uranium-235 were to fission, yielding about 200 MeV eaCh, what is the kinetic energy yield in joules per kilogram of fuel? 65. (a) To release 100 MW of power. approximately how many uranium fiSsions must occur every second? (b) How many kilograms of U·235 would have to fis. sion in I yr? 66, 1\yo deuterons can fuse 10 form different products. Although nol the mosl probable oUicome. one possi. bility is hclium-4 plus a gamma particle (see Concep. tual Question 10). Calculme the net energy released in this process. Consider equal numbers of deuterium and tritium 67. nuclei fusing 10 fonn helium-4 nuciei, as given in equa. tion (1 1-18). (a) What is {he yield in joules of kinetic energy liberated per kilogram of fuel? (b) How does this compare with a typical yield of 106 Jlkg for chemical fuels? 68. For the carbon cycle 10 become eSlablished, helium-4 nuclei must fuse 10 fonn beryllium·a. Calculate Q for this reaction. 69. Ignoring annihilalion energies of the positrons, how much total kinetic energy is released in the silt·step carbon cycle? (There is a quick way to answer this. and a much slower way.) 70. (a) How much energy can be extracted by deuterium fusion from a gallon of sea water? Assume that an average D-O fusion yield is about 2 MeV per atom. the

I!un:"

(b) A modem supertanker can hold 9 X 107 gallons. How many "water tankers" would be needed to supply the energy need:. of greater Los Angeles, consuming electricity at a rate of about 20 GW, for 1 yr? Assume that only 20% of the available energy ac(ually becomes electrical energy. 71· A fusion reaClion used to produce neutron beams ('lee Progress and Applications) is 3 2 I iD + I D - 2 He + on

,

M\uming that Ihe kinetic energy before the fusion is negligible compared with the energy released, calculate the neutron kinetic energy after the fusion.

Com prehensive Exercises 72. The binding energy per nucleon in helium�3 is 2.57

MeV/nucleon. Assuming a nucleon separation of2.5 fm, detennine (a) the gravitational potential energy per nucleOli and (b) the electrostatic energy per proton between the protons. (c) What is the approximate value oflhe intemucleon potential energy per nucleon? (d) Do these results agree qualitatively with Table 1 1.2? 73. For the lightest of nuclei, binding energy per nucleon is not a very reliable gauge of stability. There is no nucleon binding at all for a single prOlon or neutron,

111

)·el one' is "table (so far as We' knOW) and the other is not. (a) Helium-_' and hydrogen-3 (tritium) differ only in the switch of a nucleon. Which has the higher bind­ ing energy ptr nucleon? (b) Helium�3 is stable, while tritium, in fael, da:ays into helium�3. Does this 5Ome­ how violate law,,? 74_ You occupy a (me-dimensional world in which beads­ of mas" 1110 when isolaled-attract each other if and only if in contact. Were the �ads to interact solely by thi� uttr"olclion, it would take energy H to break the con­ tac!. Con�equently, we could extract this much energy by stick.ing two together. However, they also share a repul�ivc forcc, no mattcr what their separation, for which the potcntial energy is U(r) = O.85Halr, where a is a bc"d'«;ay Mode, and Con'lervation Rules in the Standard Model 1 2 6 Panty. aw¥e ConjugatIOn. and Time R�eBa1 1 2 7 Unified ThC'une.. and Coo.mology

r"f"hc \oColt�h 10 lind the fundamtntal building bloch of nalUrt tw 1 been gwnS nn fur lung tunt_ Ancient ptu� po!olUlaled •

..- die: "'�lfld compn£

'"

h

parucle of energy ,lE could appear and disappear without "erifiably up�et­ so long as it existed for no longer than ,lr :::::: h/ �E. ,n \ g energy conservatton the temporary problem with momentum conservation.) By covers iS also a force whose mediating particle has mass must be a shortargument, ge force. If a massive particle must be created, there IS a lower limit on the

; �� :

ount by which the total energy could deviate: the mass/internal energy of This, in turn, implies that the maximum time the particle could lnal particle. Even at the speed of light, the particle could travel at ive is 6.1 �

hlme2 .

\Ilrv

most

ax � cat :::::

h

-

me

h 1 range ::::: -­ em

(12-1)

If lhe mediating particle has mass, the range of the force is lirruted. Conversely, a force whose mediating particle is massless could have infi­ nite range. With no mass, there would be no lower limit on the energy of the mediating particle. Its strictly kinetic energy could be arbitrarily small, so the

time and range could be arbitrarily large. Consider the electromagnetic force. Although it does fall off with distance, it nevertheless reaches infinitely far, so

it would have to be conveyed by massless particles. Indeed, electromagnetic

forces are conveyed by the exchange of massless photons. An electrostatic

repulsion between electrons. for instance, is conveyed when one electron emits a virtual photon that is absorbed by the second electron.

We will return to equation ( 1 2-1) when we discuss specific fundamental

forces in Section 12.3. Here. we merely note that it has been used in several

instances to predict fairly accurately the mediating particle mass from knowl­

edge of the force's range

before the mediating particle was first detected. This

ISone of the most convincing arguments for treating forces as the exchange of particles.

1 2.2 Antiparticles

Before delving further into fundamental interactions between particles, we must understand antiparticles. For each kind of particle, there is an antiparticle

that shares

essentially all the properties of the particle except that it is of

III

_ 0." tI

" "n _ t .., � .. ... i_ _

ClfIPlCiIe dtIqe. Section 14 introduced the positron-me antiparticle of � eJoc:croa.. 1be poUlrOII bas the same mass and spin as an eleclTOn btu is ofpas_ ime cblrJe. An el«Cron and posib'Oll can be crealed in me process of pair production. When they meet. pair annihilation may follow. in which they diuppear and Iheit mass enerv is convened 10 pholon energy. Many OOltr

anap.tides have been found. such as the anliprolon (negalh'ely charged) and

antinturron funcharpd). II mighl seem mal an uncharged neutron has no prop­ erty to disliJlguisb il from irs anlipattide. However. the antineutron's distinct

idenbfy is coofumed by the fael that it does annihilale with the neutron

�

whereas rwo neuuons do nor annihilate. Acrually. as we will see in Sectio

12.3. the neutron is not fundamental-nor is the proton-and its internal struc­

hft disbnguishes particle from antiparticle. (Some uncharged particles. such as the .0. lack such distin,uishing sU'Ucture and are their own antiparticles.)

The convenliona.! symbol for an antiparticle is (he same as for the particle but with an overbar. The antiproton is thus p and the antineutron ii. An alternn_ live convenbon is often used for charged particles. The positron is usually rep. rtsenled ef- rather than e -, and the antiproton is somelimes wrillen p-. Similarly. the JJ. f- and JJ.

-

are anlipartic/es

of one anOlher.

The existence of antiparticles is an experimental faci. bUI there is a theo­

retical basis: relativistic quantum mechanics. Let us pursue jusl enough of this advanced theory to gain some idea of how combining relativity with quantum mechanics might suggest the existence of antiparticles. The SchrOdinger equatIOn for a free particle 1ft one dimension is

A2 a2 2m ax>

---- "'(X, I)

=

Expressed in lenns of operators, where becomes

- jJ' "'(X, I) 2m I

ih

a aI

-

"'(x I)

(12·2)

'

p = -ih(ajax)

and

E = ih(a/at). it

"

= E "'(x, I)

As we know, this equalion is based on energy. In the absence of external potential energies. the kinetic energy of a particle. enerxY E. However. because for kinetic energy.

p2/2m

p212m,

equals its total

p212m is not the relativistically correct expression

= E cannot serve as the basis of a relativistic

replacement for the Schrooinger equation. A logical basis is the relativistically correct expression

(2-28):

(12·3) To obtain a relativistic matter wave equation, we might try inserting the appro­ priate operators and then have each tenn operate on a wave function:

c'jJ' ''' (X, I)

+

m'c' ''' (x, I)

=

£' '''(X,I)

11.2

( 1 2-4)

Klein-Gordon equation, this has been shown to yield correct �nown a� the the behavior of spinless massive particles at all speeds. Par­ about tiolls redic the related Dirac equation. In nonrelativistic quantum obey spin de� with spatial states that are solutions of the SchrOdinger meII)

;\ 2 )( 10

Me-V

I�t Me-V 'II MeV

� I Me\' ' I MeV 50 MeV

�\II1I.lJ",nR qllJrli �)t.'Ild the �tlange

J,I/I (c d

,' 10l)

II

11. 0

H7 lc\'

("")

"-IN.l

II

n.o

50 lcV

\

Ib antiparticle KO. A d,...cU'.slOn of this miKing may be found m higher-level . le�l\ on particle phy...ic... Yet anolher mlrtTl... ic property given m Table 12.2 i.. strangeness. Sen... , bI). II is the \lrange quark that endow... a particle with slrangenes"" By arbl

tr.tI)' sign chOice. po......e ..-,Ion of one strange quark gives a strangenes\ of

.

I,

of twO, a "trangenes.. - 2; o f an antistrange. a st.rangene!�uckarph)\ic\

P. low energy

12.6 At � Slanford LH�ar AccderaIOr Center(SLAC), eleclrom and positrons Irom II .H,m·long hnear 3Ixeicrator (C.listing Injector) are u�l!d for variou� e,\periments. "fhto BJ.BJt del�tor \tuche� CP \iol:lI!on in the decay of B me'>On� (Seclion 12.6). F!gln

Electron

PEP·Ll nng'

,,"""- 1.1."['\(..1 111 ,{rong anti elcctromagnetlc Inleractlon, kit no. In

....

caL line' (another c\ample 01 the '1,)'n1nlclr)

the eIC(;lro'Wcal

III

mhT.,('{llm),

fh( �tand.nJ nlnwnalllin IJ'"

01' da"ical phy,ic, appear 10 be \ule,

MI11lk"nlum. cn(r�). •\fl�ular momentum, ilnd chilrge are alway .. con�en'cd"

And all tnh:ra..:ll\)fl' "l.'l.'m to Cllll,cnc colm, Another quanlllY apparently con­

"i."'nC'd In all

CJ'l"

1\ blryon number 9UI),01l number

B

I"

+ I for baryons.

I t(lf antlhar)llO\_ Imd 0 1M Ilonbar),on" linlll falrl) f(\·(nll�. lfpton numMf Wih aho belie\'ed 10 be universally \'t.ln-.eneJ_ It 1\ l"\lnllnlln 10 define one number for each generation. The dl',:tf\ln icpllln numlxr Lt ,' � I for the electron and the eleclron neutrino, I hlr their Ilnllp.irtlde'. and 0 for all olher particle�" The muon leplon

numt'lud Arl'hl,.·,llh1n,l ,,\,,1 Q1Q(" t thaI C(lll'CC\illilln III lertolt I1\lmlx-r mOl) 11\11 l'oe a UI\I\\'"",I rule \#Ihd�". "Inhl ..t "II pml'e,"c'" ('on"'I:(\I,.' the Ihr\.'(" Il'rh1n "'tm�r' .adcrn, howe....er, a quiCk. tremendous e.�pan�ion occurred, called inOation. Thi� is the pre\'aihng e�planalion for a "problem" in todoJ's uO/verse. The univel"\C IS actually very homogenous. �uggesllflg an equilibrium. But its parts are much too far apart to have communi­ cated, Jln/�ss they were Once in closer contact before undergoing a drastic expan­ �ion. We aren't sure whal caused innation. bUI a nareup rn Ihe co�mological constant is one possibility. Somewhere in this early period. sJXlIltancou'l symme­ try breaking would occur. FirM. gravity would ...ever ItS relanlln\hip with the

genealogy 1 8 The

_ · 1 2. ll ' fig

of p3nll.:Je�·)

,trong-electroweak force. "lriking out on it.. own at an carly agc. rand unified next rebel would be the 'Itrong. �plitting off from the eleclroweak. Cooling ....ould . continue. but '>0 long a� paJ1ic1e energic'\ remained above about would \till be plcnty of energy to en,ure an abundance of 1 TeV (loJ GeV). there ... of fundamental fennions: the least ma�"'I\'e tier (u, d. e. " e)' re gencmtion all lh e the middle (s. c, l./... I'J.!)' and the heavie...t (b. t. T. " t)' along with their antiparticle.... called the quark-lepton soup. thiS hoi mixture 1\ \ometimc\ I By perhap" 1 0 � ... average energies would have dropped to approxi­ mately I TeY. whereupon the electromagnetic and weak force... would part company. thu .. beginnmg the four-force, asymmetric behavior we ob\erve today. Thi\ 1\ roughly at the Imllt of experimentally attainable energie,. At around 1 - 1 0 I-l' .lntl I 0. 1 GeV. the heavier fermion\ \vould have become

�e

.

lOcreasingly \caree. lor the} are hable to decay to lighter particle\. and avail· able energle\ v.ould hI>! \O\ufficlent to renew the !'upply. Below thi, limit. the univer,e would (.:011'1" mn,tly of nucleons and light lepton\. After \everal minutes. average enc.:rgy v.ould have dropped to -I MeV. Thi\ 1\ roughly the binding energy per nudeon 10 the atomic nucleu'\. \0 nuclei would form. At around a million y�:lfS lnLl. a temperature of 3000 K. a, noted above. atom.., would form. Heawilly bodie, arrive on the !'cene much later

.. a.,., t2

... I " sd ... _ . ' ..... .. ... ... · , · ..

Condusion

AI presenr.

we' hive I fairly ple.t.�ing view of fundamental panicles and mler.

acboDs: Silo qlW'b. six leprons. and

three torees. But many que.�tions remam:

JUSI how does the uymmerry in the e'ectroweak interaction come about'> Will the Hills be found? Dots il tll:pJain mass' 1 . Wi" a theory encompassing all forces be established. and whar will it lell us about the universe? What really

happened right after !he- Big Bang? Was a uniled (orce shared by panic'e� III(ft fundamental than quatb and 'eplons'l Are there other fundamental forces'? &citing discoveries lie ahead.

Producing I au.rtc-GIuon ,..."..

nllJ)' be ih own antlp.utJde. and if 11 1\, the accounlmg fads. MOl"C(wtr. the ncUlnno produced in the "first" decay of a

��lIy Iht world\

mo6I·pow(f"(ul kcdetaror o' ions is lht RelltlH�lic He.a\·)·

Ion Colhdet IRHIClIlI Brooihavcn

National uboralory It

!muhe ,lold iOM tOllelher at II rrcmendou\

100 (i(V per nudeon. Sut;h • (cn"ll:lVu\ Impa.:t of man)' nudrons i\ ot8SS 1u capture. SF §lands for spontaneous fiSSIOn. s for elecuun st.aIld EC t

\-1

.. .... ",, . f ..... ", .. -

z

27.976927 28.976495

28 29

29.97377

3D

IS 16

3/

p

PhMphOrus

J2 J2

s

Sulfl.lf

JJ 34

"

36 17

CI

Clllorine

"

J7 36

Argon

III

"

40

20

Scandium

So

22

Ti!anium

Ti

24

25 26

27

Cllromium

Manganese

Iron

Cobalt

Mo

F,

Co

0.02 75.77

34.968852

24.23

36.965903 35.967545

0.3]7

37.962732

0.06]

39.962384

99.600

0.01 17

41

40.961825

6.7302

"

v

Vanadium

34.969031 35.96708

9].2581

40

34.99523 39.962591

41

40.962278

42

41.958618

96.941 0.647

43

42.958766

0.135

44

43.95548

2.086

45.953689

0.004

48

47.952533

0.18 7

"

44.95591

100

46

45.952629

8.0

47

46.951764

7.3

"

47,947947

73.8

49

48.947871

5.5

49.944792

5.4

50 23

0.75 4.21

33.967866

38.963707

46

21

3 1.97207 32.97146

39.963999

39

C,

Calcium

14.28 d (fJ-) 95,02

'"

K

pO!lI5Sium

19

30.973762

31.973907

50

49.947161

0.250

51

50,943962

99.750

50

49.946046

"

51.940509

4.345 8].79

53

52.940651

9.50

54

53.938882

2.365

55

54.938047

100

54

53.939612

5.9

56

55.934939

91.72

57

56.935396

2.1

"

57.933277

0.28

"

58.933 198

60

59.93381 9

100

1.26 Gyr (P-. r r , Eq 50 ms

(fJ+)

0.103 Myr (EC)

Atlp·ndta l PIopIdia ol 1L11II'

8Jeldent

t

� �

57.935346 59_930788 60.93 1058 61 .928346 63.927968

63 65

62.939598 64.927793

69.\7

64 66 67 68 70

63.929145 65.926304 66.927 129 67.924846 69.925325

48.6 27.9 4.1 18.8 0.6

'"

G,

69 71

68.92558 70.9247

Gallium

)1

60.108 39.892

G,

70 71

69.92425 7 1 .922079 12.923463

21.24

Germanium

)2

p.rsenic

Jl

Selenium

}I

Bromine

)5

Cu

Zn

Zinc

J1 18

)9

40

Rubidium

Strontium

Yttrium Zirconium

68.077 26.223 1.140 3.634 0.926

30.83

73.921177

76

75.921401

7.44

A,

75

74.921594

S,

74

73.922475

0.89

76

75.919212

77

76.919912

9.36 7.63

78

77.917308

23.77

80

79.91625

49.61

81

81.916698

8.74

79

78.918336

50.69

81

80.916289

49.61

78

77.9204

80

79.91638

81

81 .913482

1 1 .6

83

82.914135

11.5

84

83.9 1 1 507

57.0

86

85.910616

17.3

91

9 1.926270

Rb

85

84.9 11794

72.17

87

86.909187

27.83

S,

84

83.91343

0.56

86

85.909267

9.86

87

86.908884

7.00

88

87.905619

82.58

94

93.915367

Y

89

88.905849

Z,

90

89.904703

51.45

91

90.905644

1 1 .22

92

91 .905039

17.15

B,

-......

(Ilo-\l1l1Tl

l e lill n 1elln r.

�

Symbol

A

Atomic Mus

'" Nalural AbuDCb.oc:e.

S.

2.3

263 . 1 1 82

8h

262

262.1231

H,

265

265. 1300

Mt

266

266.1318

......ur. lDecay

MocI-.)t

O.M , \Sf. Q)

O.tO '\Q} 2 m\{al "'3.4 m\ \0)

AP pe ndix

J:

Pro b a b i l ity, M e a n , Sta n d a rd Deviati o n , a n d N u m b e rs of Ways

I lighl the main ideas arising. whenever we speak of probabililie� we h'gh , I fiere most equatIon numbers have an n, b. c, or d. The ad physics. Note that 10 [11

�m

not 10 sneak in more equations, but to emphasize the fact that. .... ,.a50n 1S ' Ih'mgs. we realty deaI Wit ' h ingly endIess ways 0f reexpressmg , Ihe seem

despite t.s. few basic concep ' " ' . . onIY a The first concept is �robablltty Itself A quantity Q IS the focus. �nd. 11 k . various possIble values-Q t _ Q2' Q3' and so on-each wIth liS 1a e on . P2' Pl, and so on. For example, the quantity Q might be the an bility-P 1 � Prob3 . ' · '· . of a pet house cat. AI an arbIl rary InSlant, It may be at a Window, Q I ; iocal1on . ' ' a probabl'I'lIy. e sleeping location. Q2 : or Its f00d d'ISh . Q3' each wIth its fa\'oril P, = 0.065. And there may be some P., = 0.780, 0.040, P = ce _ s.an L ' For ,m where the probability is 0 unless an eXlernal agent intervenes, such locaU'ons ' ' ht have a theory Ihat predicts W mIg the probabl'I"Illes. or he pet washtub. e observation, experimenlal but in either case, one thing i s on may rest i n: The sum of all the probabilities-the total probability of finding the a

�t ce:

cat somewhere in the house-must be I .

L , p,

�

I

(j· 1 aJ

Often probabilitic� are based on a collection of data. such as N exams. where the quantity Q is the Score. or N repetitions of an air-quality experi­ ment. where Q is a fluctuating pollulant leveJ. If value Q; turns up Ni limes out of a tolal of N. then the probability is simply

p,

=

N, N

which filS perfectly.

N, I I L, P. � L , - � - L N � - N � N N " N

On the other hand. Ni is often a theoretical number of ways of obtaining a

particular value. For example, suppose we have an office with four identical

cubicles and four workers. We define Q as the number of workers in cubicle I . As Figure 1.1 shows. of all the possible ways of distributing the workers. one

subset has three worker� i n cubicle I , which we designate Q3' and can be done

J·1

FIgun J.t ,·ubldes.

WilYi Q( arranging

r� �r . r

w()(k�r'i a. b. c. and d in four

Cubide J

J

� [�;-�

(j

::�

,-

(j "" .., , (jj) :(jj)

- - -,- - - -

-

, (j :@� -------

, (jj)

:(jj)� (j : -,-- . -

,

12 ways. Another subse[ has all four [here, designated

Q4' and this can be

done only one way. Wha[ are the probabilities? We need a bit more to go on. The mo.�t common addi[ional assumption is thai any way of arranging specific obJecb In specific loca[ions is equally likely. If [his were [0 apply to the work· ers (0 dubious assumptjon for real coworkers), then all 1 3 ways shown in Figure J.I and all the other ways not shown are equally likely. If we were to do an experiment. each way should tum up as often as any other. Therefore,

Q),

In which fhree workers (independent of their identities) occupy cubicle I , $hould [urn up 12 times as oflen as

Q4' simply because there are 1 2 times as

many ways to do it. II happens Ihat the ,otal number of ways is 256 (thai is,

44), so !.he probabilities

are

121256 and 1/256. Again,

Pi

=

NJN. where N is

now a number of ways.

Mean No matler how probability arises. !.he concept of a mean, or average, is always the same. Suppose a quantity can take on only two values,

Q , = - 60 and Q = 2

+60. and that PI is 213, so !.hat P2 is 1/3. With twice the probability, it is logical

thaI we should give

QJ twice the weighting. A mean of -20 would indeed

be half as far from -60 as from +60 and would follow from the general prescription: To find the mean

Q, multiply each value by its probability and add.

(l-2a)

u.t' u� P,

•

NJN. thi,

a....\ume...

_Q=

a common fonn thai may be familiar�

N" �Q,N, Q .... . ' N = N

'"

(l·2b)

value Qj

limes the number of tllne!Jwa)'\ nu' �)... that the mean is a possible obtained, summed over all values. then dl\llded by the total \\ tit)' dle quan times/ways. number of

Qulle often It is relatIVely easy to obtain something thai is. pmfHJrrionai to

�abililY, but actually expres..mg the proportionality constant i� difficult or �sy. In these cases. II IS common to write the mean differently. Suppose P' i� proportional to the actual probability P: P, = A P,' ,

"bere A is a constant. The total probability must be 1 . so

�'Pr =

I =>

L,A P; = A L, P.' = I � A =

I

� p' ""

,

Thus

(1-2c)

Again. in this form,

P'

need only be proportional to P and the proportionality

Q.

constant disappears. the prob­ Finally, if we wish to average something that is ajunction of ability of obtainingj{Qi) is the same as obtaining QI' and the probabilities stlU add to I , so the mean is

KQ)

=

k,j(Q, )P,

(1-3,)

Standard Deviation Our nex.t concept is a way of quantifying how much the values of Q deviate from the mean. Many recipes are pOSSible, but the most common is the e standard d viation. How do we come to choose it? A given value deviates from the mean by deviation: Qj - Q Some values of

Q i will be above the mean, some below. The deviation is a

function of Qj' which we can average via (J-3a). The result of the average IS logical, but not very helpful.

U

_I.. J

....1.., . . ..... .s..IInf � ... Nambm ol W ...P

I1IQII ofdo'......

1:.( e,

- O)P, = 1:. e, p. - Q ,£, P, = Q - Q I

= 0

.

So Ikt don', '\'cragt the dC'\'ration' Ho"'ever. if we a\'erage i rs squOfl'. rhe sum ,an fant no ntg.O\'C' ".Jut.. and 'he farther tha! 0, vaJues stray from the

mttn. lht larger lht a\tnge �houJd be.

deviation. 1:,(Q; Q)lp, To Yield ..omelhing that ha,� the same dimemiom as Q. we rake [he square mean oflhe ..quare of the

RJOt. Siloing

u� !he root-mean-square deviation.

_

known as the standard devia-

1IC)n, Of the many symbols u.�ed for this important concepl. we choose a della.

vr,(e, - Q)'p, 0-4.) only if Pi i zero whenever Q, - Q is nonzero, Note thai tlllS can be ;\1;1lhemalically expressed. Q, Q � P, = 0 In ocher words. the value IIQ ·

..

:no

0,

:I-

=

Q I� Ihe only one ever obcained, Combined with ilie fael ilia[ 6Q does

'pread a., de\'iacion,� increase. we see Ihal slandard deviation is a very logical definition, 51andard deviation i� often not presented or calculated in form 0-4a), but

1O\leJd in a closely related fonn. whose derivation is a good exercise in sorting

oul the \'anous quanlities we have discussed thus far.

�Q = vr,(Q, - Q )'p, = vr,( Qf - 2Q, Q + Q')p, = V"i, Q�P, - 2Q Li Ql, + Q'2 L/ P,

We have used the fact that

Q is not a function of summnlion index i, but sim­

ply a number thai can be broughl outside a summation, In the second term

Q.

imide the radical. we now recognize the definition of Q, and in the third, a unit lolal probability. In the firs!. we have the mean of Ihe square of

IIQ = VQ' - 2QQ + QI

or

IIQ = VQ' - Q'

(J-4b)

While obscuring the fact that the radical's argument is necessarily nonnegative. !hie; fonn makes a simple point: To ca1culate standard deviation. we need only find the mean of the square.

Q2. and the square of the mean. Q2,

JUSI as equation (1-2b) follows from (J-2a), an alternative fonn for stan­ dard deviation follows from (J-4a).

-

( J 4c )

� Differe

nt Route

deviation can hi! el(p�,�d In a seemingly different form and ..tandard t a. matter of redefinitlon_ If the sum.. are nm over Pl l....!Oible .. rtall) ju ..

�

�5

Q

y but Instead over a.n "tnals"-aU mdi"idual mstam:c!f. of of the quantit we will uc;,c J rather than l-then a sum over I With an N In hlch - 'ut'-- for .... I all '''' " the same as a sum over j alone. For example, If in a �ne!io of SIX is lbe sum - 1..5. and e'( riments we obtain only three values-Q \ = \,2. al� pe t .8. 1.5, 1.2. 1.2. \.8. the following are cqui"alent: 1.2. ce �uen

Q, ­

Q�

�8-ln

Q=

�:"'I Qj N, N

=

Q= With this

=

-'-'----'-----""-;-'----==-=6 1 .2 · 3 + L5 · l + 1 .8 ' 2

1 .2

+

1 .8 + 1.5 + 1 .2 + 1.2 + 1.8

6

redefmition. a\l N, would disappear from sums. and (J·2b) and (J.4c)

\\ould appear as

/l. Q =

Q=

�'Zj (

)( QJ - Q)'

....

N

We avoid this route because it tends to obscure the role of probability and clut­ ters the otherwise seamless transition to continuous quantities.

Continuo us Quantities The number of workers in cubicles is a discrete quantity. its values being nonnegative integers. Except on average, we don't obtain 1 .3 restricted to �'orkers in a cubicle. Some quantities are inherently continuous, meaning that

from one value to the nex.t is an infinitesimal change. An example would be

the locations of a swinging pendulum. In such a case, a sum naturally becomes the probabilities of being at particular point locations must an integral. and

beCome infinitesimal; otherwise. summing over the infmity of locations could not yield a unit total probability. The basic formulas translate in a straightfor­ ward way.

'Z , P, = \ Q = L; Q, P , KQ) = 'Z;!(Q,)P,

AQ " V}; (Q, - Qj'P, ,

--->

--. ---> --->

j dP(Q) = \ Q = j Q dP( Q) KQ) = jJ( Q)dP( Q) /l.Q = Vj(Q - Q)'dP(Q)

Note that equation (JAb) 1� unchanged, as its fonn is independent of whether

a sum or integral is involved.

" - , ,1. '' . ,. .,,1 a J

.. 7 7•5 .... .... " .. ... .. . _ . ... _ .. _

FiuUy. I« «JIIIIDuous varillb� iI i� u:o.ually mo� t:onH'ment 10 lkaJ noc

..uti , dilf'crtati&I probebilil)' bur with a prolNlbUUy de-Mi.y. a probabllil), �r _

Q. __ u follow

,

U(Q) D(QI - -- ,,, .... dP(Q) - D((J)dQ dQ

With thi� delimllon. formullL\

0·1a), {J-1al. (J-Ja). IlI1d (1-4a) become

/D(Q)dQ - 1 0 - /QD(Q)dQ iW) - /M) D((J)dQ � Q - "rrrQ - OrD(Q)dQ

N,'IIe thai In quantum rnt\:h4nit:,. the quanfJIy

(J- I b)

(J-2dl (J-3b)

(J-4d1

Q might be location x and the

prohabiliry drn..il)' comt, from me wa ..,C' function,

D(x) = 'I/I(x)i2.

EXAMPLE J . 1

I

,

I

I I/'

VatU(' CJ, �lImbn of Tunn. .\

I

J

,

,

.

.

"

•

4

7 1

Yo'hat 0Itt W mean and \land.lnJ de\iatlon" SOLlJTION The \Ioa)' in whi,h the data are gncn make� _

Q-

(J-2b) quickest for the mean.

" 1 + 2 ,4 + 3 ' 9 + 4 ' 1 2 + 5 , 9 + 6 ' 4 + 7 " . � _ . • -

40

= 4

hlr the: \,andard dC\'ia!lon, we can u\c (J-k) directly.

.lQ

-

1 0,].1)

/.

2 ,

/(1- '1' 1 " ( 2--- ')"

1-------

\

( J -

+ ( 7 - 4)' 1

+

'0

=

1.30

An...'lIher way i'i equation 0-4b). Thj� requires caku/aling the mean of the square. for whll'h equation (J-Jal i, appropriate. the " funclion" being ju�t the square. Equation

Thus,

i� ex�s�d in

N, IN puIS �J{Q,)N,

term, of probability. but P, •

JlQ)

�

N

II in the form

NoW usin, (J-4b)

�Q

=

\/17.7

- (4)' - \.30

We see that 30 of the 40 "alues are withm t standard �iation of the mean, Standud df:Yiation usually cover-, the majority of \la1UC!i.

EXA MPLE J . 2 "'"""' 1be probability density-probabihty per unit ht:ight- for finding a gi\len object at 1 tw:iJht )' is given by

and applies to all values of)' from 0 to +00, Find the mean and standard de.\liation of " in terms of the constant The following Integral will be useful-

b.

fooo Y"�-b"dy

""

SOLU1 c'...

Apparently our resuh should not depend on naturally using y in place of Q_

m!jb'".. t

A, We can use (l·lb) to detennine it,

Thus,

{Nott: In quantum mechanics, the process of ensuring unit probability i!i called nor· malization.} Now using (1·2d),

The result is sensible, If

b increases, the exponential faUs off faster. and we would

expect the average height to be smaller. Moreover, its dimensions are correct. The

argument of the exponential must be dimensionless. so the dimensions of b must be

one over length.

For the standard deviation. we could use (J.4d) directly, but let uS instead use

(J-4b) after finding the mean of the square via (1·3b). As in Example 1 . 1 . the func·

tion here is just the square,

y' = fo

-

00

y' be-b'dy =

b fo

00

y'e-b'dy =

Now inserting in (J-4b)

t.y =

I

\/(211)')

b(2!/b')

(I/bl' =

= 2/b'

l ib

Not only does this have the correct dimensions of length. but it also happens to equal

the mean. We conclude that most values obtained should be in the fange of heights between zero to twice the mean.

...... J� "'-)"101 JWl.. two cl .. c.- I. two ,.,t". 1If*ft.

FH ', r·n;

-r

;' .. ' f, ,

,

; .. f

/' I

r,

1

-

Factorials-Numbers of Ways

1

Suppose we h.t,-e Nd.ffel'C'nl can. and N md.,iduaJ parking splices. How many d,fft'ft'nl .. a)\ L"iU1 we arrange' Ihe' c� m Ihe space�? Any of the N cars could

be in � tim: 'paL't', and an) Ilf th� I'C'maJnlng N - I could be m the second Thu�. tilhng Ju�t the' tiN IWO ,p"ccs. Ihere are N(N - I ) ways. In the nd c. and �rec �pace�. these six possible Ways \pc.:.aJ ca.'iC' of three J.:aJ"\. O. b. .. _ are �hOwn in Figure 1.2. Conllnumg, any 01 the remaining N - 2 cars might be 'pao.'C'.

In the thiN 'pace, �o to thi� point we would have N(N - I )(N - 2) ways. Of cotJ.r>C. in lhe thfU-car case. N - 2 i� I. and no more ways are added. as there

i_' onl)' Oil(: choJ('c for the la\1 car. For arbitrnry N. the number of ways would 2. then N - J. and so on. until again there is only one car lefl So the 10lal number of ways is N(N - I HN - 2) . . . 3·2· 1 . which is the delinJlion of Nf and j� referred 10 a� "N factorial." Nov.- let us de,ignale a group of NI parking �paces as region i. as depicted be multiplied by N

,..".. J,J A �'''Il .." hln ..hJ�·h

�,"anpffiC'nlll' lotoJn.:h " da:l�

IrrrkW'allI

in Figurt J3. By the aOO\'e arguments, there are Ni! ways of rearranging cars among lhe-.e ..paces alone. without affecting caTS elsewhere. What happens if

�e om.. declare thaI it doc�o't maner where the N, cars are in this region. and we will con�ider it as anI)" one way of arranging all N cars? For (lll)' previous

,in,gle way of parking all car; in all individual �paces. there II'ould hUl'e been N, ' different ....ays . (including Ihal single way) thaI would leave the cars not in

region i exactly where ,hey were before. and we have now declared these N '

�f

fomlerly different ways as jusl one way. Thus. the previous lotal number ....ap. . N'. j, simply divided by N/. the number of rearrangements within region j thlll change nothing out�ide. Repeating the process. we choose

another group of parking spaces from the N - N; not in region i, calling this

ne� group of N �paces region j. and again declaring that rearrangemeOls

l

"ilhin il are irre e\'ant. By the same arguments. we must divide the existing

number of ways by

N ! to obtain the new number of ways. If in the end we N spaces into M regions. the number of ways W to park

1

have broken the line 0

cat\ in !!tpaces. where rearrangement within any given region is not considered J different WIl)·. is

(l-5) Consider the limits. If each space were a "region:' there would be N regions. each with one cat-SO that NI '"" I for all i-and W would be simply N! (divided by I ! 10 the power N). as we expect. At the other extreme, if there were just one region encompassing all

�nsible.

N spaces,

IV would be

1,

which is also

We have used cars and parking spaces as the framework, but the arguments

are general. Formula

among

M

(J-5) gives the number of ways of arranging N objeclS

separnte calegories (regions, boxes. energy levels. etc.), where

N

�

panicles are in calegory I-rearrangements within being irrelevant-N2 i category 2. and so on.

N-

al case i!> just two categories, one \\lth rnmon speci ost cO m objects. in which case we ha\'e II e 'flI ),lith \' other

)C'11\ 10g

the

tl

obJe4.:h.

"::N-"-W � _II!(N -! ,,--,

fI)\

ial coefficient and is so common lhllt It hu,

vn as the binom IS kllO\ r to l. fac . 0wn special symbo ""' 1"1� ItS • en

1> .... a IptZ'. ... by which \he fun'

(K-4)

Solution of a first-order equation yields one arbitrary constant, essentially a constant of integration, which. in the above result, is A. It c�n take on any . techmque of sOlving value while still solving the e� uation. The m�thodlcal this basic differential equation IS to rearrange It to dflf = b dx, then integrate both sides.

Second-Order Linear

d'�;)

bf{x)

{A �

=>

f(x)

�

( VIbIx) + B cos ( VIbIx) 0' Ae+iv1il• + Be-iv1il• b Ae+v'bx + Be-v'bx or A sinh ( v'bx ) + B cosh ( Vbx) b sin

Ax + B

(K-5) < 0 > 0

b = O

A second-order equation yields two arbitrary constants (of integration). Note that the sign of the constant b is crucial, for different signs lead to functions whose behaviors are entirely different. If b is negative, the functionJis oscilla_ tory; if b is positive.! is exponential. growing or decaying or some combina_ tion thereof; and if b is O,fis a straight iine.

Useful Integrals Below is a short list of integrals most often needed in the text.

J . (L ) dx = "2 L = ""4 J . ,(mTX) J . (--L ) sm2

x sm -

mT X

x

-

x'

-

d.x

T x, sm2 n7 X dx

=

( -) ( ) (-- \,

2"-,, sin - X L 4mT

L

Lx 4mT

2mrx sin -L -

2mTX Lx' - -- sin 6 4mT L J

x'

LOS

(2mTX) -L

Appendix K

Some Important Math

K-S

An sw e rs to Sel e cted Exercises

Chaple. 2

99. -48 1"

17. 0.1'''"

ll. 43.75 m 23. ]:ller, 0.8 m 15. 60 m. 2.67 x 10-7 ,

lOS.

n. \'!c = 0.78 1 . 1 .04 x 1 0 -1 s, 24.375 m

19. 24 m. I'le = 0.6 31. 0.0067 s behind )5. 9.8 ps earlier

41. 1549 m

.0. (a) - l OO ns: (b) 1 4 1 ns; (c) I OO ns, zero 45. Bob is 60 yr, Anna is 52 yr

47. (a) 32 y" (b) 32 y,

(a) jumps ahead 1 2 8 days; (b) 250 os; (e) behind by same amounts

53. (a) toward. I'le =: 0.25 ; (b) 687 om; (c) 549 nm

55. yes

:/ T : ,= 3k r y m A, a . 1 5 0m 7 2 5 . c

59. O.385c. c_ ", , " (b) (0, c) c 63. (3) 2

II;:

65. (.) (-0.8c. 0.6c): (b) (O. c)

1016 J. 6 X 1 0 1 6 1 , 1 . 5 X 1017 J 71. 9 10-17 kg X 73. 2.5 . 75. (a) 2.19 X 1 0- 26 kg mls; (b) 3.64 X 10-22 kg . mfs; low 40% low. 10-7% X 3 (e) X

77. c/V'i

79. 1.83 X lOS kg/day 8I . 4.71 X 109 kgls

o

83. -c 2

85. 25.1 MV 87. (3) O.759c; (b) 2.07c; (e) O.948c 89. ue = 0.9997 I c.uc = - 1 .62 X 10-Jc. 20.6 MeV, [7.1 keY

91. "'2 = 6.43 kg. ml

I

0

0

0

0 - �,/C

I I I.

J7. c/V'i

V

0 0 -V/l'

0

0

0

0

I

(.) 4mo' and 5"'0"': (b) 0"'uc and

lIS. (.) A \/8/3 and 5A \/8/ 12: (b) bolh

39. it must; lOP passes through (v/e2)Lo cos 80 earlier

49.

I

=

1.43 kg, 1 .93 X 1 017 J

93. Ca) 1.29moc2; (b) 0.795c; (e) 2.57mo' 3.29mo' expo B

2 0"'0'" VSqA/24".o'

Chaple. 3

17. 3.12 eV. 399nm

19. 82.4 nm

21. 6.42

23. 1.22 29. 8.15

X 1031 phorons per sec X 1()6 mls X 107 mls

31. 130.5°,23.9° 35. 0.00659 nm

41. (b) opposite, 1 . 1 8 X 10 14 m 43. 9.1 X 10-27 kg 45, 60°, 5.73 X 10-3 degrees. visible light

47. 7.38

X 10' 10 m. 600

49, 5 X 10-6 Pa, 6.37

51. je.,E'A/h

x

IO� N

53. 2.91 X 10-12 m

55, imoCl, lmo

Chapter 4 11. 40 13. 0.333 nm 15. 728 mls 17. 2.43 X 10-12 m 19. (a) 6.29 nm; (b)D.147 nm

21. (a) 1.46 X IO-I o m to 1.32 X 10-0 m;

(b) 6.23 X 10-9 m 10 2.43 X 10- ro m 23. 2.2 X 10- 10 m 25. cannot be treated classically 27. Ca) 0.14 mm; (b) 1600; (c) 400 29. (a) s/m: (b) 10 s- 1/2v'b; (c) 5 s'I/2 v'b. (25 S - I)b, 25 s - I

AN-1

AIWI

_ .. _ _

71. 1.61 X 1 0 22 kI an.. ' )S x 1 0 11 J. 1.67 )( 10'l7 kI ) 6 )( 10 li mit

.L G. 4I. #7.

10' "'" 10 " )( 23 (11'1 ' 3 )( 10 . J .. (.)0.01 67'. Cb) 1.0:5 )( 10 J I dtpn - 1.83 )( 10 '\) rtd 51. r < 10 ' m

6.l &r

51 (b)

"

�V'A'.'2M'

5'7. .. (4 ....,)A.. 2 2 x 10' mi. 59. c .) O lI4nm. O ' nm 11. 0 1 '. 6J. 9' nm 65. 67.

...

I

2.

Ct

Cbapc.r S

53.

I

I X 10' �

lJ. (a) 2t1m. (b) 2Lln 31. 'a) .boul 9'1 as large 33. (.) 0.080 and 0.16; (b) 5.8 x 1O-17nnd7.1 X 10-1 4.. (cn,4 x 10 1M and 1.8 x 1O- 1l1 •

'" v ph.ouot

0,0-'3 s

partid�

- B.n«' - 1»

+ c, '11, u.... nh U.

�( �: y �

(b) R => k?L2/(4 + .l:1Ll). T "'" 4/(4 + klL2)

Chapter 7

- lAo A �inkL + B cos kL - F�L B 'I n U.l ... o(FrrL �� QL) _

m ji(�:t.J��(mK)I ·.lh I

21.

O,� om

2J. 0.609. 0.196.0.609 25. 107 'TrZfa2/2mL2, at center 27. -mi'!2rr4iffi2nl 29. n, "'" 6. nr' '''' 3, 1 . 1 X 10-6 m 31. n "" 6. 13,2 cV. lOS K

33. 5.5 X 10-8 eV, 5 X 10-9

37. 150°. 125.)0, 106,8°. 90°, 73.2°, 54.7°, 30°

45. -0.85 cV, magnitude of angular momentum: 0, V 2h.,

V"i2JI, l...component of angular momentum:

-Ii, 0, +11. +211, +311

51. (a) yes; (b) no '3. 0.2 l 2

\lH "'It'll v ICf

_

59, lrm

'7. 0.238 59, ellipse

&J.

61. no 67. 13.5 nm

73. (a) plane wa,'(' (1). Dirac df'lta (I); (b) A =< 1I

,,11-;

Ie) B _ 11 75. tb) no� Ie) 1

3�

,

v2b :

X 10 :!O J 77. (al and (bl do