Physics for University Students. Course of Modern Physics: textbook 9786010445956

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AL-FARABI KAZAKH NATIONAL UNIVERSITY

S.K. Kunakov

PHYSICS FOR UNIVERSITY STUDENTS COURSE OF MODERN PHYSICS Textbook First edition

Almaty «Qazaq University» 2020

UDC 53.01 (075.8) LBC 22.31я73 K-92 Recommended for publication by the decision of the Academic Council of the Faculty of Physics and Technology, Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol №2 dated 15.01.2020); Educational organizations by the educational-methodical association of the Republican textbook for university students studying in the field of training «Natural Sciences» (Protocol №2 dated 24.05.2019)

Reviewers: Doctor of physico-mathematical sciences, professor S.E. Kumekov Doctor of physico-mathematical sciences, professor V.D. Djunushaliev Candidate of physico-mathematical sciences, professor A.A. Komarov

Kunakov S.K. K-92 Physics for University Students. Course of Modern Physics: textbook / S.K. Kunakov. – 1st ed. – Almaty: Qazaq University, 2020. – 98 p. ISBN 978-601-04-4595-6

The proposed book “Physics for University Students” is devoted for gifted students of technical specialties as the general physics course stipulated by the program of the Ministry of Education and Science of the Republic of Kazakhstan. It should be noted that numerous editions of general physics courses are burdened with superabundant details and therefore this book provides a very short description of main achievments of contemporary physics and at the same time brings a fairly broad universality with an emphasis on descriptions of basic principles of fundamental laws of Nature. Published in authorial release.

UDC 53.01 (075.8) LBC 22.31я73 ISBN 978-601-04-4595-6

© Kunakov S.K., 2020 © Al-Farabi Kazakh National University, 2020

Preface Dear student! In this book I want to show you the beauty of the Universe and Nature in your surroundings by mathematic language known as physical phenoma or physics. We also were trying to keep clarity and brevity in the book content. At the same time this book is a great challenge to you. Presented definitions of main physical magnitudes are coupled by detailed comments and explanations, but it was done in the most high level mathematical form. At the end of each chapter you may find interesting examples and pictures which will help you to understand the origin and behavior of physical objects in the gravitional and electromagnetic fields. Much more attention paid to 4-vector analysis as the main language of description physical magnitudes and without which it is impossible to understand the variety of physical transfomations. We also explained thermodynamic laws after description of electromagnetic fields to reveal the deep unity of heat and quantum mechanical motion of elementary particles. In the first chapter we introduce you with main priciples in physics like least action priciple, the significant numbers lagrangian functions as a key of what and how physical objects should move and how to deal with them. We also present very brief recipe how to extract this equations of motion from these functions and demonstrated their significance under some system of references transformations. I also presented law of conservation of energy and momentum by detailed calculation of the elastic collision of two rigid particles. The third chapter describes the three types of forces like gravitational, electric and magnetic forces. Theory of relativity and nuclear physics reveals profound internal connection. Differences and similarities of this forces are thoroughly studied. The motion of free charged particles in electric and magnetic fields are described in detailed way from lagrangian functions. In the forth chapter the relation between electric and magnetic field are described and its mathematical expression of their relation known as Maxwell’s equations are presented. The forth chapter gives explanation why atoms emit energy, due to what interior atomic phenomena it takes place. Radiative emission and its connection with heat transfer also discussed. In the fifth chapter the electromagnetic radiation of heated objects studied, especially from the point of atomic structure of matter view. The sixth chapter describes the thermal phenomena and introduces the main thermodynamical laws and processes. At the end of each chapter tips and tricks in applied problems are thoroughly studied with some problems not burdened by tedious numerical calculations. Sandybek Kunakov Al-Farabi Kazakh National University First edition, August 2019

3

Contents

1

Lagrangian and Hamiltonian.........................................................9

1.1

Least Action Principle. Energy...........................................................................9

1.1.1 Kinetic energy....................................................................................................................9 1.2

Potential energy-U............................................................................................ 10

1.2.1 Fermat’s prinsiple...........................................................................................................11 1.3

Least Action Principle...................................................................................................12

1.4

Einstein rules....................................................................................................................14

1.4.1 Levi-Civita symbol and cross product vector, tensor.......................................14 1.5

Hamiltonian......................................................................................................................16

2

Space and Time. Lorentz Transformations.................................19

2.1

Lagrangian is your understanding and creativity level........................... 23

2.1.1 Free particle.....................................................................................................................24 2.2

Observable magnitudes and uncertainty principle.................................. 27

2.2.1 Creation and annihilation operators......................................................................29 2.3

Another face of Green’s function.................................................................. 33

2.4

Quantum mechanical transformations........................................................ 33

2.5

Lorentz transfomation..................................................................................... 34 5

6

Physics for University Students

3

Binary collisions.Scaterring matrix.............................................37

3.1

Rutherford scattering....................................................................................... 37

3.1.1 Cross section of binary collisions............................................................................37 3.2

S-matrix............................................................................................................... 41

3.2.1 Wick’s Time Odering symbol.....................................................................................42 3.3

S-matrix............................................................................................................... 46

4

Gravitation fields...........................................................................47

4.1

Gravitational forces result from the properties of spacetime itself..... 47

4.1.1 Newton’s Gravity Law...................................................................................................48 4.2

Cristoffel symbols and metrics...................................................................... 50

4.2.1 Einstein tensor.................................................................................................................52 4.3

Einstein field equations................................................................................... 56

4.4

Black holes and Schwarzschild metric......................................................... 57

4.5

Standard model and Higgs bosons.............................................................. 58

5

Electromagnetic fields..................................................................61

5.1

Lagrangian of charged moving particle................................................................61

5.1.1 Non interacting free particle ....................................................................................61 5.2

Gharged free moving particle...................................................................................63

5.2.1 Gauge invariance...........................................................................................................63 5.3

Electromagnetic waves................................................................................................66

6

Thermodynamics and Statistical Physics....................................69

6.1

Internal Energy.................................................................................................. 69

6.1.1 Avogadro number Ideal Gas State Equation.......................................................69 6.2

First Law of Thermodynamics ....................................................................... 70

6.3

Statistical Physics.............................................................................................. 76

6.3.1 Black body radiation.....................................................................................................77 6.4

Ideal Bose Gas.................................................................................................... 81

6.5

Ideal Fermi Gas.................................................................................................. 85

7

Hydrodynamics.............................................................................87

7.1

Boltzmann equation......................................................................................... 87

Contents 7.1.1 Function of energy distribution................................................................................87 7.1.2 Main assumptions.........................................................................................................87 7.2

Sound................................................................................................................... 90

7.3

Shock waves....................................................................................................... 91

7.4

Combustion........................................................................................................ 93

8

Complementary materials............................................................95

8.1

Appendix 1.Vector and tensor calculas....................................................... 95

8.2

Appendix 2 Residue.......................................................................................... 96

8.2.1 Residue..............................................................................................................................96 8.3

Useful tables....................................................................................................... 97

8.3.1 Some fundamental constants...................................................................................97 8.3.2 Physical data often used.............................................................................................97

7

1.2 Potential energy-U

9

1. Lagrangian and Hamiltonian

1. Lagrangian and Hamiltonian

1.1 1.1.1

Least Action Principle. Energy Kinetic energy Definition 1.1.1 — Work.  

W=



C

Fd s

multiplication = unit force * ­unit of length. = unit o f fsign orce1 ∗munit o f of length Work is measured in Joules.1 joule = 1kg ∗ 1 sm2 1m Energy measured also in Joules. This physical magnitude in some cases is measured in electron volts . Energyin in physics physics is kinetic energy (ev). 1ev = 1.6 ∗ 10−19 J {Joules → J}Energy is presented presentedinintwo twotypes: types:kinetic energy and potential energy Later , it will be shown that mass of any physical object if it is not equal to zero also represents Later, energy and is connected with energy in the following way:E = mc2 , c = 3.108 ms

→ − − → − −r → (x, y, z) or → −r = x→ i +y j +z k The position of the object presented by vector → Velocity and acceleration vectors

→ −v (t, x, y, z) =

−r (t,x,y,z) d→ dt

→ − a (t, x, y, z) = → − a (t, x, y, z) =

=

− − dy(t) → − dz(t) → dx(t) → dt i + dt j + dt k

−v (t,x,y,z) → − → − dv (t) → − d→ = dvdtx (t) i + dty j + dvdtz (t) k dt 2 → − − d 2 y(t) → − d 2 x(t) → −r i + dt 2 j + d dtz(t) k = ∂t2 → 2 dt 2

9

∗

−r = → −r = ∂t → ·

−v = → −v = ∂t →

10

Chapter 1. Lagrangian and Hamiltonian

Chapter 1. Lagrangian and Hamiltonian

10

We also introduce momentum and angular momentum vectors → → −  → −  −i j k   → − → − − → → − → − → −p = m→ − v and L = → r × −p =  x y z  = Lx i + Ly j + Lz k mv mv mv  x y z Lx = y ∗ mvz − z ∗ mvy , Ly = z ∗ mvx − x ∗ mvz , Lz = x ∗ mvy − y ∗ mvx

Work(W) and Kinetic energy(T)

W=



C

→ − → F d −s =



−p 2 d→ −s = mv d→ 2 C dt

This form of energy is called kinetic energy 

Example 1.1 Electron’s mass is equal

me = 9.1 ∗ 10−31 kg 

Electron’s tangential velocity around proton is equal ve = 106

m s

Kinetic energy is equal 2 me v2e kg 12 m 10 2 = 9.110−19 joules = 5.7ev = 9.110−31kg 2 s

Definition 1.1.2 — Newton’s law.

m

−r d2→ → − − = F (t, → r) 2 dt

From this differential equation you may know the position of the object at initial time and at its in theintermediate intermediate poinst points which which is final state.More over you are aware about what’s hapening inthe very unfriendly approuch to Heisenberg uncertainty principle which states that you are not able to get information about object’s coordinate and its velocity at once.

1.2

Potential energy-U Definition 1.2.1 — Potential energy equals to the work ,which might be done by the object taken with the minus sign. Let’s evaluate the work needed to tranfer some test charge

1.2 Potential energy-U

11

1.2 Potential energy-U

11

from a given point till infinity and what its potential energy will be equal to at this case. − 2→ → − − m ddt 2r = F (t, → r )=



1 e W = r∞ 4πε 2 dr = 0 r C2 −12 ε0 = 8.8510 Nm2 2

− 1 e2 → 4πε0 |r|3 r 1 e2 4πε0 r

Definition 1.2.2 — Electrons charge is the least charge in Nature.

e = 1.610−19 C 1,602 1C → 6.251018 electrons

6,2420

(3) (4)

Definition 1.2.3 Definition 1.2.3 — —Interaction Interactionisisaaresult resultofofbosons bosonsexchange.Boson exchange. Bosonisisaaspinless spinlessparticle particle due to which any interaction takes place.In case of electromagnetic interaction due to which any interaction takes place. In case of electromagnetic interactionwe wecall call photons,gravitional them photons, gravitionalinteraction interactiondue dueto toHiggs Higgsbosons,etc.. bosons, etc. Classical Classicalcontinuous continuousfield field theories are also in serious contradiction with quantum field field theories theories

Interpretation: Particle or wave.If wave we looking at the nature from one or if particle from another point of view-heads or tails. 

Example 1.2 Have a look at the picture



Definition 1.2.4 — Lagrangian is a fundumental function in physics which regulates how and in what way the physical object should move.

L = T (kinetic energy) −U(potential energy) 

Example 1.3 Snell’s law



1.2.1

Fermat’s prinsiple Definition 1.2.5 -Least Time. Time. Why Whythe thebeam beam light changes its direcDefinition 1.2.5 — — Fermat’s principle -Least of of light changes its direction? tion? Certainly from the picture youdeduce may deduce thattotime to the cover the length glassis slub lessif Certainly from the picture you may that time cover length of glassofslub less is then thenbeam if thewill beam will propagate thetrajectory straight trajectory as a continuation of itstrajectory. preveiuosIt the propagate along thealong straight as a continuation of its preveiuos is connected is with the fact with that light propagates inpropagates glass a littleinbit slower thanbitinslower the air,the although trajectory.It connected the fact that light glass a little in the itair,although seems that itbeam is choosing shorter some way. However this However principle this doesprinciple not present fornot an seems that beamsome is choosing shorter way. does observer thean key which reveals particular way physical objects present for observer the keythe which reveals theofparticular way of motion. physical objects motion.

12

Chapter 1. Lagrangian and Hamiltonian

Chapter 1. Lagrangian and Hamiltonian

12 R

1.3

Pierre Fermat was born on August 17, 1601 in the Gascon town of Beaumont-de-Lomagne (Beaumont-de-Lomagne, France). His father, Dominic Farm, was a wealthy leather trader, the second city consul. In the family, except Pierre, there was another son and two daughters. The farm received a law degree - first in Toulouse (1620-1625), and then in Bordeaux and Orleans (1625-1631). In 1631, after successfully completing his studies, Fermat bought the post of royal adviser to parliament (in other words, a member of the highest court) in Toulouse. In the same year he married a distant relative of his mother, Louise de Long. They had five children. Monument Farm in Beaumont de Lomagne. Rapid career growth allowed Farm to become a member of the Edict Chamber in the city of Castres (1648). It is to this position that he is obliged to add to his name a sign of nobility - de particles; from this time he becomes Pierre de Fermat. The quiet, measured life of a provincial lawyer left Farm time for self-education and mathematical research. In 1636, he wrote a treatise "Introduction to the theory of flat and spatial spaces", where, regardless of "Geometry" Descartes (published a year later) outlined the analytical geometry. In 1637 he formulated his “Great Theorem”. In 1640, he promulgated the less famous, but much more fundamental, Small Fermat theorem. He conducted an active correspondence (through Marin Mersenn) with the great mathematicians of that period. With his correspondence with Pascal begins the formation of ideas of the theory of probability. In 1637, the conflict between Farm and Descartes began. The farm destroyed the Cartesian "Diopterist", Descartes did not remain in debt, gave a devastating review of the work of the Farm on analysis and hinted that some of the results of Farm were a plagiarism from the Cartesian "Geometry". Descartes did not understand the Farm method for holding tangents (Fermat’s statement in the article was indeed brief and careless) and suggested that the author find the tangent to the curve, later called the Cartesian Leaf, as a challenge. The farm was not slow to give two correct solutions - one according to the article by Fermat, the other - based on the ideas of “Geometry” by Descartes, and it became obvious that the method of Farm was simpler and more convenient. The mediator in the dispute was made by Gerard Desargues - he recognized that the Fermat method is universal and correct in its essence, but is not clearly and incompletely stated. Descartes apologized to his rival, but until the end of his life, Fermat was hostile.

Least Action Principle Definition 1.3.1 — Lagrangian is a fundumental function in physics which regulates how and in what way the physical object should move.

1.3 Least Principle L =Action T (kinetic energy) −U(potential energy)

13

m ddt 2x = − dVdx(x) δ δ x(t) (T (x) −V (x)) = 0 L = T (kinetic energy) −V (potential energy)  S =0 S(action) = 0τ L dt, δδx(t)   ·  δS δ L δ x(u) δ L δ x(u) δ x(t) = du δ x(u) δ x(t) + · δ x(t) = δ x(u)    δL δL d = du δ x(u) δ (u − t) + · dt δ (u − t) = δ txf(u)   L = δδx(t) + δ (u − t) δ· L − duδ (u − t) dtd δ· L = δ x(u) ti δ x(u) d δL δL = δ x(t) − dt · , δ x(t) δL d δL − = 0 → Euler − Lagrange equation dt · δ x(t) δ x(t) 2

Definition 1.3.2 — Action is the following functional from L (variable here is a function-Lagrangian function).

S=

 t2 t1

L(t, q, dq dt )dt

where Lagrangian is a fundumental function in physics which regulates how and in what way the physical object should move

x(t) δδx(t)

x(u)δδx(t) x(t) δδx(u)

x(t) δδx(t)

··

(u) δδxx(u)    δδLL δδLL dd (u−−t)t)++ · · dtdtδδ(u du δδx(u) (u−−t)t) == == du x(u)δδ(u (u) δδxx(u)  t ft f



δLL + δδ(u (u−−t)t) δ· δ·LL duδ(u (u−−t)t)dtddtd δ· δ·LL == ==δδδx(t) −− duδ x(t)+ (u) titi (u) δδxx(u) δδxx(u) δLL dd δδLL − , ==δδδx(t) − , x(t) dtdt · · (t) δxx(t) δPrinciple 1.3 Least Action δδLL dd δδLL −dtdt · · ==00→ →Euler Euler−−Lagrange Lagrangeequation equation x(t)− δδx(t) (t) δδxx(t)

13

Definition1.3.2 1.3.2— —Action Actionisisthe thefollowing followingfunctional functionalfrom fromLL(variable (variablehere hereisisaafunction-Lafunction-LaDefinition grangianfunction). function). grangian 

dq L(t,q,q,dq )dt SS== tt1t2t12L(t, dtdt)dt

where Lagrangian is aisis fundumental function in physics which regulates howhow andand in what wayway the where Lagrangian fundumental function physics which regulates how and what way where Lagrangian aafundumental function ininphysics which regulates ininwhat physical object should move thephysical physicalobject objectshould shouldmove move the

(kineticenergy) energy)−U(potential −U(potentialenergy) energy) LL==TT(kinetic m(dqdq))22 dq m( dq −U(q) L(t,q,q, ))== dtdt −U(q) L(t, 22 dtdt Definition 1.3.3 1.3.3 — — Least Least Action Action Principle Principle means means that that the the variation variation of of this this functional functional isis Definition equalto tozero zeroand andisisestablishing establishingthe theway wayaccording accordingto towhich whichNature Natureobliges obligesany anyphysiphysiequal calobjects objectsto tomove. move. cal 

··

L(t,q,q,q) q)dt dt==00 δδSS== tt1t2t12L(t, t2t2 ∂∂LL ∂∂LL · · + · ·δδq}δ q}δqdt qdt==00 t1t1{{∂∂qqδδqq+ ∂∂qq

Fact1.3.1 1.3.1 The TheLeast LeastAction ActionPrinciple Principlepresents presentsfor forasasititwas wasexpected expectedthe theequation equationofofphysical physicalobject object Fact

motion motion 

··

q}δqqdt dt==00 δδSS== tt1t2t12{{∂∂∂∂LqLqδδqq++dtddtd ∂∂L· L·δδq}δ ∂∂qq ·· ∂∂LL +dtddtd ∂∂L· L·δδqq==00 ∂∂qqδδqq+ ∂∂qq Forbetter betterunderstanding understandingforthcoming forthcomingmaterial material let’s introduce vectors which have four compoFor forthcoming material let’s introduce vectors which have four compoFor better understanding let’s introduce vectors which have four component. nent.For example x=(t,x,y,z) or more stricktly nent.For example x=(t,x,y,z) or more stricktly For example x = (t, x, y, z) or more strictly 14

xxµµ==(x(x00, ,xx11, ,xx22, ,xx33))

Chapter 1. Lagrangian and Hamiltonian

• The energy -momentum four-vector p = (E, p1 , p2 , p3 ) • The current density four vector J = (ρ, J 1 , J 2 , J 3 ) • The vector potential four vector A = (ϕ, A1 , A2 , A3 ) • The Nabla Nubla operator operator also also might might be bepresent presentas assudo sudofour fourvector vector ∂µ ≡ 

∂ ∂ ∂ ∂ ∂ ∂ = ( , ∇) = ( , , , ) µ ∂x ∂t ∂t ∂ x ∂ y ∂ z

Example 1.4 — Waves on a string.  L =  dxL, L − Lagrangian density

S = dtL = dtdxL

The displacement from equilibrium position in this case is a function of time and x -coordinate along which the string is streched .So ,then the kinetic energy of the string is equal Kinetic energy of spring is as follows 

J = (ρ, J 1 , J 2 , J 3 ) • The vector potential four vector

14



A = (ϕ, A1 , A2 , A3 ) • The Nubla operator also might be present as sudo four vector

Chapter 1. Lagrangian and Hamiltonian

∂ ∂ ∂ ∂ ∂ ∂ ∂µ ≡ µ = ( , ∇) = ( , , , ) ∂t ∂ x ∂ y ∂ z ∂t ∂x

Example 1.4 — Waves on a string.  L =  dxL, L − Lagrangian density

S = dtL = dtdxL

The displacementfrom fromequilibrium equilibriumposition positionininthis thiscase caseisisaafunction functionof of time time and and xx -coordinate -coordinate The displacement the string is streched .Sothe ,then the kinetic of theisstring equal Kinetic along along whichwhich the string is streched. So, then kinetic energy energy of the string equalisKinetic energy energy of spring is as follows of spring is as follows T = 12 ρ = ml

l

∂ψ 2 0 ρ[ ∂t ] d x

Potential energy equals U=

1 2

 l 0

τ(

∂ψ 2 ) dx ∂x

Lagrangian is equal: 1 ∂ψ 2 1 ∂ψ 2 L = ρ( ) + τ( ) 2 ∂t 2 ∂x Taking operation of varience over the action we get the motion equation δS δψ = 0       d  ∂L  − dtd  ∂ L  − dx  ∂ψ  ∂ψ ∂( ) ∂( ) ∂t ∂x ∂ 2ψ ∂ 2ψ = 0 + τ ∂ x2 − ρ ∂t 2 δS δψ

=

∂L ∂ψ

This is wave equation 2 ∂ 2ψ − ρτ ∂∂ xψ2 = 0 ∂t 2

1.4 1.4.1



Einstein rules Levi -Civita symbol and cross product vector,tensor 1.4 Einstein rules Definition 1.4.1 — Levi-Civita tensor. A tensor Levi-Civita looks like this



 0, i f any two labels are the same εi jk =  1, i f i, j, k is an even permutation o f 1, 2, 3 −1, i f i, j, k is an odd permutation o f 1, 2, 3

R

• Repeated indices are imlicitly summed over. • Each index can appear at most twice in any term • Each term must contain identical non-repeated indices

Definition 1.4.2 — Cross poduct of two vecrors presentation with Levi-Civita symbol.

   e 1 e2 e3    → − → − A × B = A1 A2 A3  = εi jk A j Bk ei B1 B2 B3 

Definition 1.4.3

15

0, i f any two labels are the same  εi jk = 1, i f i, j, k is an even permutation o f 1, 2, 3 −1, i f i, j, k is an odd permutation o f 1, 2, 3 R

1.4 Einstein rules indices are imlicitly summed over. • Repeated

• Each index can appear at most twice in any term • Each term must contain identical non-repeated indices

15

Definition 1.4.2 — Cross poduct of two vecrors vectors presentation with Levi-Civita symbol.

   e e2 e3   → − → −  1 A × B = A1 A2 A3  = εi jk A j Bk ei B1 B2 B3 

Definition 1.4.3

εi jk εimn= δ jm δkn −δ jn δkm 0, j = m δ jm = 1 j=m Some useful examples presented below 

Example 1.5

→ − → − ∇T = ei ∂∂ xTi , ∇ · A = ∂∂Axii , ∇ × A = εi jk ei ∂∂Ax kj −r = 3, ∇ × → −r = 0 ∇→  

Example 1.6

→ − → − ∇ · ( A × B ) = ei ∂∂xi (ε jkl Ak Bl )e j = ∂∂xi (ε jkl Ak Bl ) = = εikl ( ∂∂Axik Bl + Ak ∂∂Bxil ) = εikl ∂∂Axik Bl + εikl Ak ∂∂Bxil = = εlik ∂∂Axik Bl − Ak εkil ∂∂Bxil = → − → − → − → − − → → − → − → − = (∇ × A ) · B − A · (∇ × B ) = B·(∇ × A ) − A · (∇ × B ) 

Fact 1.4.1 — Four -vector version of the Euler-Lagrange Equation.

∂ L(φ , ∂µ φ ) ∂ L(φ , ∂µ φ ) δS = − ∂µ ( )=0 δφ ∂φ ∂µ φ Suppose 16

L=

Then

2 1 1 ∂µ φ − m2 φ 2 2 2

Chapter 1. Lagrangian and Hamiltonian

∂L ∂φ

= −m2 φ ∂L = ∂ µφ ∂ (∂µ φ ) δS 2 µ δ φ = −m φ − ∂µ ∂ φ

(∂ 2 + m2 )φ = 0 We shall encounter many times with this equation untill we will get its profound sense and meaning This equation also might be derived in some diferent way 

L =  dxL, L − Lagrangian density S = dtL = dtdxL S = d 4 xL(φ , ∂ φ )

∂L ∂φ

= −m2 φ ∂L = ∂ µφ ∂ (∂µ φ ) δS 2 µ δ φ = −m φ − ∂µ ∂ φ

16

2

Chapter 1. Lagrangian and Hamiltonian

2

(∂ + m )φ = 0 We shall encounter withequation this equation until get its profound and We shall encounter manymany times times with this untill we willwe getwill its profound sense andsense meaning meaning This equation also beinderived in some way diferent way This equation also might be might derived some diferent 

L =  dxL, L − Lagrangian density S = dtL = dtdxL S = d 4 xL(φ ,∂µ φ ) 

− ∂µ ∂ (∂∂ µLφ ) = 0 2  L = 21 ∂µ φ − 12 m2 φ 2 ∂L ∂L 2 µ ∂ φ = −m φ , ∂ (∂µ φ ) = ∂ φ  δS 2 µ 2 2 δ φ = −m φ − ∂µ ∂ φ = 0 , ∂ + m φ = 0 δS δφ

1.5

=

∂L ∂φ

Hamiltonian

Definition Lagrangian is a function in physics which regulates how Definition1.5.1 1.5.1—— Lagrangian is fundumental a fundamental function in physics which regulates and what way the object should move.move. If Lagrangian doesn’t change with time how in and in what wayphysical the physical object should If Lagrangian doesn’t change with

then for energy is conserved and called Hamiltonian. time the thenexpression the expression for energy is conserved and called Hamiltonian. ∂L · ∂ ·· ∂ qi qi + · qi qi  ∂ dL = d ∂ L q· + ∂ q·· · · i i dt dt  ∂ qi  ∂ qi ∂ L q· − L = 0 d i · dt ∂ qi dL dt

=

pi = H=

R

∂ L dH · , dt = ∂ qi · p i qi − L

d = dt



∂ L q· ∂ qi i



0

Joseph Joseph Louis Lagrange, Italian. Giuseppe Lodovico Lagrangia; January 25, JosephLouis LouisLagrange Lagrange(Fr. (Fr. Joseph Louis Lagrange, Italian. Giuseppe Lodovico Lagrangia; 1736, Turin April Turin 10, 1813, Paris) French mathematician, astronomer andastronomer mechanic of January 25, ‒1736, - April 10, ‒1813, Paris) - French mathematician, andItalian mechanic of Italian origin. Along with Euler - the largest century. origin. Along with Euler ‒ the largest mathematician of themathematician XVIII century. of Hethe wasXVIII especially famous He was especially famous forin histhe field exceptional mastery in the field of generalization and synthesis for his exceptional mastery of generalization and synthesis of accumulated scientific of accumulated scientific material. material. The author author of of the the classic classictreatise treatise“Analytical "AnalyticalMechanics”, Mechanics",ininwhich whichheheestablished establishedthe thefunda-mental fundaThe mental "principle of possible displacements" and completed the mathematization of mechanics “principle of possible displacements” and completed the mathematization of mechanics. He made . He made a huge contribution to mathematical analysis, number theory, probability theory a huge contribution to mathematical analysis, number theory, probability theory and numerical methods, he created the calculus of variations. Monument to Lagrange in Turin Lagrange’s father ‒ half-French, semi-Italian ‒ served in the Italian city of Turin as military treasurer of the Sardinian kingdom. Lagrange was born January 25, 1736 in Turin, in a wealthy family. However, his father, taking risky speculations, lost both his personal fortune and his wife’s fortune. Due to the financial difficulties of the family, he was forced to start an independent life early. First, Lagrange became interested in philology. His father wanted his son to become a lawyer, and therefore appointed him to the University of Turin. But Lagrange accidentally got a treatise on mathematical optics in his hands, and he began to study mathematical literature with enthusiasm. In 1755, Lagrange sent Euler his work on isoperimetric properties, which later became the basis of the calculus of variations. In this work, he solved a number of tasks that Euler himself could not overcome. Euler included Lagrange’s praise in his work and (together with d’Alambert) recommended the young scientist as a foreign member of the Berlin Academy of Sciences (elected October 1756). In 1755, Lagrange was appointed teacher

and numerical he createdand thetherefore calculus appointed of variations. Monument to Lagrange in wanted his son tomethods, become a lawyer, him to the University of Turin. inTurin a wealthy family. However, his father, taking risky speculations, lost both hisofpersonal Lagrange’s father - got half-French, semi-Italian - served in in thehisItalian city Turin as But Lagrange accidentally a treatise on mathematical optics hands, and he began fortune his wife’s fortune. Due kingdom. to the financial difficulties of the family, wasinforced militaryand treasurer of the Sardinian Lagrange was born January 25,he 1736 Turin, to mathematical literature enthusiasm. In 1755, Lagrange Euler hisHis work on toinstudy start an independent life early.with First, Lagrange became interested insent philology. father a wealthy family. However, his father, taking risky speculations, lost both his personal isoperimetric properties, which later became the basis of the calculus of variations. In this wanted his son become a lawyer, and therefore appointed himoftothe thefamily, University of Turin. fortune histoawife’s fortune. Duethat to the financial he was forced work, he and solved number of tasks Euler himselfdifficulties could not in overcome. Euler included But Lagrange accidentally got a treatise on mathematical optics his hands, and he began to start an independent life early. First, Lagrange became interested in philology. His father 17 Lagrange’s praise in his work and (together with d’Alambert) recommended the young scientist 1.5 Hamiltonian towanted study mathematical literature with enthusiasm. Inappointed 1755, Lagrange sent Euler his work on his son to become aBerlin lawyer, and therefore him to the University of1755, Turin. as a foreign member of the Academy of Sciences (elected October 1756). In isoperimetric properties, which later became the basis of the calculus ofhands, variations. Inbegan this But Lagrange accidentally got a treatise on mathematical optics in his and he Lagrange was appointed teacher of mathematics at of thevariations. Royal Artillery School to in Turin, where work, he solved a number tasks that Euler himself couldLagrange not Monument overcome. Euler included and numerical methods, heofcreated the calculus Lagrange 1.5 Hamiltonian toused, study mathematical literature with enthusiasm. In 1755, sent Euler his work in on 17 he despite his youth, the glory of an excellent teacher. Lagrange organized a scientific Lagrange’s praise infather his work and (together with the d’Alambert) recommended young scientist Turin Lagrange’s - which half-French, semi-Italian - served incalculus the Italian city of Turin as isoperimetric properties, later became basis of the ofthe variations. In this society there, from which the Turin Academy of Sciences subsequently grew, publishes works aswork, a foreign member of Sardinian the of Berlin Academy ofhimself Sciences (elected October In 1755, military treasurer ofnumber the kingdom. Lagrange was born January 25,1756). 1736 Turin, he solved tasks that (1759). Euler could not Eulerin included on andacalculus of variations Here, for the firstovercome. time, applies analysis Lagrange was appointed teacher of(together mathematics at the Royal Artillery School inyoung Turin, where in amechanics wealthy family. However, his father, taking risky speculations, lostheboth his personal Lagrange’s praise in his work and with d’Alambert) recommended the scientist to the theory of wife’s probability, develops the of oscillations and acoustics. 1762: the first he despite his Royal youth, the glory of antheory excellent teacher. Lagrange organized a scientific and his fortune. DueSchool to the financial difficulties of the family, he was forced asused, a foreign member of the Berlin Academy of Sciences October In 1755, offortune mathematics atthe the Artillery in Turin, where he(elected used, despite his1756). youth, the glory of description of general solution of a variational problem. It was not clearly substantiated society which the Turinof Academy of Sciences subsequently publishes to start there, an independent life early. First, Lagrange became interested ingrew, philology. His works father Lagrange wasfrom appointed teacher mathematics at the Royal Artillery School in Turin, where of and met with harsh criticism. Euler in 1766 gave a rigorous justification of the variational anwanted excellent teacher. Lagrange organized a scientific society there, from which the Turin Academy on calculus of variations (1759). Here, for thehim firsttotime, he appliesa of analysis hisdespite sonand tohis become lawyer, therefore appointed the organized University Turin. hemechanics used, youth,apublishes the gloryand of an excellent teacher. Lagrange scientific methods and of subsequently supported Lagrange in oscillations every possible In 1764, French Sciences subsequently grew, works on mechanics and calculus variations (1759). Here, tosociety the theory probability, develops the theory of and acoustics. 1762: first But Lagrange accidentally got a treatise on mathematical optics inway. hisofgrew, hands, andthe hethe began there, from which the Turin Academy of Sciences subsequently publishes works of Sciences announced a competition for the best work on the problem ofhis the motion forAcademy the first time, he applies analysis to the theory of probability, develops the theory of oscillations description of the general solution of a variational problem. It was not clearly substantiated to study mathematical literature with enthusiasm. In 1755, Lagrange sent Euler work on onthe mechanics and calculus of variations (1759). Here, for the Moon first time, he applies Points), analysis of moon. Lagrange presented a work on the the (see Lagrange and met with1762: harsh criticism. Euler in 1766 gave aoscillations rigorous justification of problem. the variational isoperimetric properties, which later became thelibration basis ofofthe calculus of variations. In this and the first description of the general solution of aand variational ItParis was toacoustics. the theory of probability, develops the theory of acoustics. 1762: the firstnot which was awarded the first prize. In 1766, Lagrange received the second prize of the methods subsequently supported Lagrange in every possible way. In 1764, theincluded French work, he and solved a number of tasks that Euler himself could not overcome. Euler description of the general solution of a variational problem. It was not clearly substantiated clearly substantiated and met with harsh criticism. Euler in 1766 gave a rigorous justification of the Academy for his in research onand thea(together theory ofwith thefor motion of work the satellites ofthe Jupiter, and until Academy of Sciences announced competition thea best on the problem of the motion Lagrange’s praise his work d’Alambert) recommended young scientist and met with harsh criticism. Euler in 1766 gave rigorous justification of the variational variational methods and subsequently supported Lagrange every possible way.atInthe 1764, the French 1778 he was awarded three Joseph LouisinLagrange In 1766, invitation ofmethods the moon. Lagrange presented aprizes. work on the ofpossible the Moon (seeInLagrange Points), as a foreign member of the more Berlin Academy oflibration Sciences (elected October 1756).the In 1755, and subsequently supported Lagrange in every way. 1764, French of Prussian King Frederick II, Lagrange moved to Berlin (also on the recommendation of of Academy of Sciences announced a competition for the best work on the problem of the motion which was awarded the first prize. In 1766, Lagrange received the second prize of the Paris Lagrange was appointed teacher of mathematics at the Royal Artillery School in Turin, where Academy of and Sciences announced afirst competition for the best work on the problem of the motion D’Alembert Euler). Here he headed the physics and mathematics department of the moon. Lagrange presented a work on the libration of the Moon (see Lagrange Points), which Academy for his research on the theory of the motion of the satellites of Jupiter, and until he despite his youth, the glory of an on excellent teacher. Lagrange a scientific ofused, the moon. Lagrange presented a work the libration of the Moon organized (seeInLagrange Points),was the Academy of Sciences, and later became president of the Academy. her "Memoirs" 1778 hethe first wasawarded awarded three prizes. Joseph Louis Lagrange 1766, at the invitation society there, from which themore Turin Academy of Sciences subsequently grew, publishes works awarded prize. In 1766, Lagrange received the second prize ofsecond the Paris Academy for his which was the first prize. In Lagrange received theIn prize of the Paris published many outstanding works. He1766, married (1767) in his mother-in-law cousin, Vittoria of Prussian King Frederick II, Lagrange moved to Berlin (also on the recommendation ofthree on mechanics and calculus of variations (1759). Here, for the first time, he applies analysis research on the theory the motion of the satellites of(1766–1787) Jupiter, andsatellites until 1778 he was awarded Academy his research on theThe theory of period the motion of the Jupiter, and until Konti, but infor 1783 hisofwife died. Berlin was the of most fruitful infirst the D’Alembert and Euler). Here he first headed the physics and mathematics department of to the theory of probability, develops the theory of oscillations and acoustics. 1762: the 1778 he was awarded more Louis Lagrange InKing 1766, at theincluding invitation more prizes. Joseph Louishethree Lagrange Inprizes. 1766, atJoseph thework invitation of Prussian Frederick II, Lagrange life of Lagrange. carried out on algebra and number the Academy of Sciences, and later became president of the Intheory, hersubstantiated "Memoirs"of description ofKing theHere general solution ofimportant a variational problem. It Academy. was not clearly of Prussian Frederick II, Lagrange moved to Berlin (also on the recommendation strictly several statements and Wilson’s 1767: Here Lagrange publishes moved toproved Berlin (also onFermat’s the recommendation of D’Alembert Euler). he first headed the published many outstanding works. He married (1767) intheorem. hisand mother-in-law Vittoria and met with harsh criticism. Euler in 1766 gave a physics rigorous justification ofcousin, the variational D’Alembert and Euler). Here heequations” first headed the and mathematics department of the memoir “On solving numerical and then a number of additions to it. Later, physics and mathematics department of the Academy of Sciences, and later became president of the Konti, butand in 1783 his wife died. The Berlin period (1766–1787) was theInmost fruitful inAbel the methods subsequently supported Lagrange in every possible way. 1764, the French theGalois Academy of Sciences, and this later became president of the Academy. In her "Memoirs" and took inspiration from brilliant work. For the first time in mathematics, a finite life of Lagrange. Here he carried important work algebra and number theory, Academy. Inofmany her “Memoirs” published many outstanding works. He (1767) inincluding his motherAcademy Sciences announced aout competition for theon best on married the problem of the motion published outstanding works. He married (1767) inwork hisequations mother-in-law cousin, Vittoria group of substitutions appears. Lagrange suggested that not above the 4th degree strictly proved several Fermat’s statements and Wilson’s theorem. 1767: Lagrange publishes of thecousin, moon. Lagrange presented aThe work on the libration ofall the Moon (see Lagrange Points), in-law Vittoria Konti, but in 1783 his wife died. The Berlin period (1766–1787) was Konti, but in 1783 his wife died. Berlin period (1766–1787) was the most fruitful in thethe are solvable in radicals. Strong proof of thisLagrange factthen and areceived concrete examples of such equations the memoir “On solving numerical equations” and number of additions to it. of Later, Abel which was awarded theofhe first prize. InHere 1766, the second prize the Paris life of Lagrange. Here carried out important work on algebra and number theory, including most fruitful in the life Lagrange. he carried out important work on algebra and number were givenfor by his Abel in 1824–1826, and Galois found theof general conditions for solvability in and Galois took inspiration from brilliant work. For the firstsatellites time in mathematics, a finite Academy research on thethis theory of the motion the ofLagrange Jupiter, and until strictly proved several Fermat’s statements and Wilson’s theorem. 1767: publishes theory, including strictly proved several Fermat’s statements and Wilson’s theorem. 1767: Lagrange 1830–32. 1772: Elected foreign member of the Paris Academy of Sciences. group of was substitutions appears. Lagrange suggested that not all equations above degree 1778 he awarded three more prizes. Joseph 1766, attothe the invitation the memoir “On solving numerical equations” andLouis then a Lagrange number ofIn additions it. 4th Later, Abel publishes the memoir “On solving numerical equations” and then aexamples number ofofadditions to it. Later, are solvable in radicals. Strong proof of this fact concrete such equations ofand Prussian Frederick II, Lagrange moved toand Berlin of Galois King took inspiration from this brilliant work. For the(also first on timethe in recommendation mathematics, a finite Abel and Galois took inspiration from this brilliant work. For the first time in mathematics, a finite were given by Abel in 1824–1826, and Galois found the general conditions for solvability in D’Alembert and Euler). Here he first headed the physics and mathematics department of group of substitutions appears. Lagrange suggested that not all equations above the 4th degree 1830–32. 1772: foreign member thepresident Paris Academy of examples Sciences. the ofElected Sciences, and later became of theequations Academy. Inofher group of substitutions appears. Lagrange suggested that not all above the"Memoirs" 4th degree are areAcademy solvable in radicals. Strong proof ofofthis fact and concrete such equations  Example 1.7 published many outstanding works. Hefact married (1767)the inexamples his mother-in-law cousin, Vittoria solvable in radicals. Strong proof of this and found concrete of such equations were given were given by Abel in 1824–1826, and Galois general conditions for solvability in Konti, but in 1783Elected his wife died. member The Berlin period (1766–1787) was the most fruitful in the 1830–32. 1772: foreign of the Paris Academy of Sciences. by Abel in 1824–1826, and Galois found the general conditions for solvability in 1830–32. 1772: life of Lagrange. Here he carried out important work on algebra and number theory, including Elected the Paris Academyand of Sciences. strictlyforeign provedmember several of Fermat’s statements Wilson’s theorem. 1767: Lagrange publishes  Example 1.7 2 the memoir solving numerical equations” and then a number of additions to it. Later, Abel 1 v mc “On √ Land =− = inspiration ,β = c this brilliant work. For the first time in mathematics, a finite γ , γtook Galois from 1−β 2  Example 1.7   12 group of substitutions appears.2 Lagrange suggested that not all equations above the 4th degree ∂L ∂ 2 (1 − V pare = = γmv −mc in radicals. Strong proof of this fact and concrete examples of such equations i =solvable i 2 ∂ vmc cv i 2 ∂ vi 1in 1824–1826,  2found  conditions for solvability in  the general √ by Abel and Galois Lwere = −given , γ = , β = 2 γ 2 +cmc = γmc2 v + 1 − v2 2 2 =ofγmc E1830–32. ≡H = pi vi −Elected L1−β = γmv 2 2 1772: foreign member of the Paris Academy Sciences. c c  γ 1 2

2

∂ Lmc , γ∂ = √ 1 2 , β = = ∂− pLi = (1 − Vc2 vc = γmvi γ ∂ vi −mc vi = 1−β 2 18 1. Lagrangian and Hamiltonian  Chapter    2 1 2 v2 + 1 − v2 2 + mc E ≡ H∂= = γmc2 L pi vi∂− L = γmv V 2γ 2= γmc 2 c2 pi = ∂ vi = ∂ vi −mc (1 − c2 = γmvi c2 Example Example1.7 1.8   18  2 Chapter   2 2 1. Lagrangian and Hamiltonian E ≡ H = pi vi − L = γmv2 + mcγ = γmc2 cv2 + 1 − vc2 = γmc2 E 2 − p2 = m2 c4  Example 1.8 c = 2.9981010 cm  s , → −v22E → −v 2 → − → − mc 1 v v m 2 2 4 2 2 4 LEp= = ,,γEm =(1 βm = cc , p =  − pγc2 = c√− c2 2),=   v2 cm1−β 10   1  c = 2.99810  s , 1− 2 2 → − → −vc2 ∂ Lv E ∂ 2 V 22 → − → − v m 2 4 pip==∂ vi → =, E i → → −→ )(1 =−m c2c → , p==γmv ∂ vi (1−mc − −→ −c2 → → − −  c−2 −  − − 3 β( 2β F = ddtp , → a = ddtv , F =2 mγmc a2 + mγ  2 vv2 +a )1 − v2 = γmc2 E ≡ H = pi vi→ − L = γmv + = γmc 1− c2 − → −v γ c2 2 1 c , β = γ = √→ 2 − − → − → −− → −c→ → − − 1−β − F = ddtp , → a = ddtv , F = mγ → a + mγ 3 β ( β → a) → − → −v 1  γ = √ 2, β = c 2

1−β





R R

2e + 4p → 4He + 2n + Ekin , Ekin = 29.3Mev m p → 938Mev, e → 0.5Mev, m m = 0.8% 235U + n → f f + f f + 200Mev, m = 0.910−3 2e + 4p → 4Hel + 2nh+ Ekin , Ekin =m29.3Mev m CH +938Mev, 2O2 → CO 2H2 O, m = 0.8% 10−10 2+ m p 4→ e→ 0.5Mev, m = 235U

m

−3 + n → f fl + f fh + 200Mev, m m = 0.910 m −10 CH4 + 2O2 → CO2 + 2H2 O, m = 10

2. Space and Time. Lorentz Transformations

2. Space and Time.Lorentz Transformations

Definition 2.0.2 Time

One can measure time and treat it as a geometrical dimension, such as length, and perform mathematical operations on it. It is a scalar quantity and, like length, mass, and charge, is usually listed in most physics books as a fundamental quantity. Time can be combined mathematically with other fundamental quantities to derive other concepts such as motion, energy and fields. Time is largely defined by its measurement in physics. What exactly time "is" and how it works is still largely undefined, except in relation to the other fundamental quantities. Currently, the standard time interval (called conventional second, or simply second) is defined as 9 192 631 770 oscillations of a hyperfine transition in the 133 cesium atom. Definition 2.0.3 Length In the physical sciences and engineering, when one speaks of "units of

length", the word "length" is synonymous with distance. In the International System of Units(SI), the basic unit of length is the meter and is now defined in terms of the speed of light . The centimeter and the kilometer, derived from the meter, are also commonly used units. Units used to denote distances in the vastness of space, as in astronomy, are much longer than those typically used on Earth and include the astronomical unit, the light year.

R

Proper time or time interval is the time interval maesured by an observer located within the inertial system where the observer makes time interval measurements and being at the state of rest relative to this inertial system of reference.The proper length is also defined and treated in the same way.

Definition 2.0.4 Mass

Mass is the internal property of the matter and shows its resistance to change its speed. Mass, in physics, the quantity of matter in a body regardless of its volume or of any forces acting on it. The term should not be confused with weight, which is the measure of the force of gravity acting on a body. Under ordinary conditions the mass of a body can be considered to be constant; its

19

20

Chapter 2. Space and Time. Lorenth Transformations Chapter 2. Space and Time.Lorentz Transformations

20

weight, however, is not constant, since the force of gravity varies from place to place. Because the numerical value for the mass of a body is the same anywhere in the world, it is used as a basis of reference for many physical measurements, such as density and heat capacity. The SI unit of mass is kilograms. Definition 2.0.5 Einstein’s postulates

R

• The laws of physics are the same in every inertial system of reference • The speed of light has the same value for all observers , independent of their motion or of the motion of the light source

Definition 2.0.6 Galilean transformation of coordinates 

x = x − vt  y =y  z =z  t =t  ux = ux − v Definition 2.0.7 Lorentz Transformations Time is the subject of inertial system’s velocity moving

relative to another inertial system Lorents transformation then presented as follows: 

x = 

x−vt   v2  1− c2

y =y  z =z 

t =

= γ(x − vt)

v

t− x 2  c 2 1− v2 c

= γ(t − cv2 x)

Definition 2.0.8 Lorentz velocity transformations 

dx =

dx−vdt    v2  1− c2

= γ(x − vt)

v



dt = 

ux =



dt− d x 2 c 2 1− v2 

dx  dt

c

=

= γ(t − cv2 x)

dx−vdt

v = dt− dx c2

u−v 1−

vu c2

Example 2.1 According to a stationary observer ,a moving clock runs slower than an identical

stationary clock by a factor of  v2 −1 γ = 1− 2 c

21

Chapter 2. Space and Time. Lorenth Transformations

t =



t (proper time)   2  v  1− c2

21



= γt (proper time)



R

If an observer at rest with respect to an object measures its length to be L*(proper length),an observer moving with a relative speed v with respect to the object will find it to be shorter than its proper length by the factor gamma

R  2  2  2  2 |x|2 = x · x = x0 − x1 − x2 − x3 → − → − a · b = a0 b0 − a1 b1 − a2 b2 − a3 b3 → − → − a·b  = gµν aµ bν ,  1 0 0 0 0 −1 0 0  gµν =  0 0 −1 0  0 0 0 −1 Matrix form of Lorentz transformations

   t γ −β γ x  −β γ γ   =  y   0 0  0 0 z



0 0 1 0

  0 t x 0   0 y 1 z

Example 2.2

−p ) · (E, → −p ) = E 2 − → −p · → −p = m2 , c = 1 p · p = pµ pµ = (E, → −p = 0, then E = mc2 if → 

22

Chapter 2. Space and Time. Lorenth Transformations Chapter 2. Space and Time.Lorentz Transformations

22 

Example 2.3

h = 6.62610−34 j · s me = 9.110−31 kg, c = 3 ∗ 108 ms me V 2 E0 = me c2 , β = Vc 2 = K, 2 K = √me c 2 − me c2 , K = √ E0 2 − E0 1−β  1−β 2 E02 E 1 − β 2 = (K+E0 )2 , β = 1 − 2 0

9.1 ∗ 10−31 ∗ 9 ∗ 1016

(E0 +K)

E0 = = 8.1 ∗ 10−16 J 6 K = 1 Mev = 1. ∗ 10 1.610−19 = 1.6 ∗ 10−13 Ve = 2.3 ∗ 106 ms , p = meVe = 9.110−31 2.3106 = 2.0910−24 kgm/s kg m/s −34 −10 m = 0.3nm λ = 6.62610 = 3 ∗ 10 2.0910−24



For any any four four vector vector let’s let’s introduce introduce Kronecker Kroneker symbol: For symbol:

gαβ

R



 1 if α = β = 0 = −1 i f α = β = 0 0 otherwise

⇒ Einstein Einsteinrules is rules • • • • •

Repeated indices are implicitly summed over Each index can appear at least twice in any term Each term must contain identical non-repeated indices Raising or lowering index(1,2,3) changes sign Raising or lowering index(0) does not change the sign

2.1 Lagrangian is your understanding and creativity level 2.1 Lagrangian is your understanding and creativity level

23

23

R θ

θ

sh(θ ) = e −e , ch(θ ) = e +e , (ch(θ ))2 − (sh(θ ))2 = 1 2 2   2   2        2   x = x ch(θ ) + ct sh(θ )  x = x ch(θ ) + ct sh(θ ) + 2x ct ch(θ )sh(θ )    2   2   →     2 ct = x sh(θ ) + ct ch(θ ) (ct) = x sh(θ ) + ct ch(θ ) + 2x ct ch(θ )sh(θ )   2   2 = inv, th(θ ) = ctx = Vc c2t 2 − x2 = c2 t − x V c , ch(θ ) =  1 , sh(θ ) =     V2  V2   1− 1− c2 c2  V  t+ x    2 x = x +Vt , y = y , z = z , t =  c −θ

−θ

2

1− V2 c

2.1

2

1− V2 c

Lagrangian is your understanding and creativity level Definition 2.1.1 — Lagrangians and Hamiltonians.

    · · ·· d ∂ dL d ∂ L ∂ L = dt = dt ∂ q· qi + ∂ q· q = dt ∂ q· qi i i i   · · ∂ L dH d pi = · q − L = 0 q − L p = 0, H = p i i i i dt ∂ qi dt   · · · δ qi δ H = pi δ qi + δ pi qi − ∂ L δ qi − ∂ L · ∂ qi  ∂qi · ∂ L ∂H δ q δ H = ∂∂ H δ H = δ pi qi − qi δ qi + ∂ pi δ pi ∂ qi ∂ H = − p· ∂ H = q· , i i ∂ pi ∂ qi ∂L · ∂ ·· ∂ qi qi + · qi ∂ qi

dL dt

Definition 2.1.2 — Poisson bracket. ∂A ∂B ∂A ∂B ∂ qi ∂ pi − ∂ pi ∂ qi · ∂F ∂F ∂F · ∂F ∂t + ∂ qi qi + ∂ pi pi = ∂t {F, H}PB , i f ∂∂tF = 0

{A, B}PB = dF dt dF dt

= =

+ {F, H}PB

Definition 2.1.3 — Operator expectation value.

 d F dt



Example 2.4

=

1     F, H ih

∂q ∂p q j , pk PB = ∂ q jj ∂∂ ppki − ∂ qij ∂∂ qpki = δi j δik − 0 ∗ 0 = δ jk

h qj , pk PB = i 2π δ jk



Chapter 2. Space and Time.Lorentz Transformations

24 2.1.1

Free particle

Definition 2.1.4 — Lagrangian for free moving particle.

   b − proper time S = ττ12 −mc2  dτ = −mc a ds, τ  Chapter 2. Space and Time. Lorenth Transformations → − 2 1 → −p = ∂ L = ∂ −mc2  (1Chapter −vTime.Lorentz Transformations mv − v ) 2 =2. Space =and γm→

24

24

2.1.1

−v ∂→

−v ∂→

c2

−p → −v − L = γmv2 + mc = Free particle E =H =→ γ 2

2

1

(1− v2 ) 2 c 2 2 γmc2 [ vc2 + (1 − vc2 )]

Definition 2.1.4 — Lagrangian for free moving particle.

= γmc2

Next: Free charged field.  in electromagnetic    particle − proper time S = ττ12 −mc2  dτ = −mc ab ds, τ  Definition 2.1.5 — Free charged field.  particle in electromagnetic −v → −p = ∂ L = ∂ → −v m→ 2 (1 − v2 ) 12 = −mc = γm → − → − c2 ∂ v ∂ v v2 1 ) 2    −→ −v(1− c2 µ = τ2 −mc2 2+ q→ 2 S = −mcds −→ qA dx A − qϕ dt 2 −vµ− L = γmv τ1 2 γmc E = H = −p → + γ = γmc2 [ vc2 + (1 − vc2 )] = γmc2 → − → −p = ∂ L = γm→ −v + q A −v ∂→   0 E 1 field. E2 E3 Next: Free charged particle in electromagnetic −E 1 0 in electromagnetic −B3 B2   particle , Definition field. Aν −— ∂νFree Aµ, Fcharged Fµν = ∂µ2.1.5 µν =  2 3 −E B 0 −B1   3 2 −B→ 2    −mc − → −v B−1 qϕ 0dt µ = τ2 −E S = −mcds − qA dx + q A   µ τ γ 1 3 1 2 −E −E → − → −p = ∂ L 0= → −v + q−E 1 3 2 γm A → −     0 −B B ∂vE   , 2= 2 EB32 − (E/c)2 , L = 1 d 3 xFµν F µν F µν =  F1µν F µν 1 0 E E 4  E 2 B3  0 −B −E 1 3 −B2 1 0 −B3 B2  E B 0 , Fµν = ∂µ Aν − ∂ν Aµ, Fµν =   2 B3 0 −B1  E i = −F 0i = F i0 , Bi = −ε i jk F jk−E 3 2 −E B1 0   −B 1 2 3 0 −E −E −E E 1    0 −B3 B2  , R F µν =  Fµν F µν = 2 B2 − (E/c)2 , L = 14 d 3 xFµν F µν 1  E 2 B3  0 −B → −r )2 = ε 1 ∂ x = 0 3 ×−B i jk k E(∇ B i ∂xj 0 i 0i i0 i i jk E = −F = F , B = −ε F jk Definition 2.1.6 — Massless particle.Wave Equation.

2  L = 12 ∂ µ φ ∂µ φ = 12 ∂µ φ = 12 (∂0 φ )2 − 12 ∇φ ∇φ −r∂)L= ε ∂µ x = 0 ∂ L (∇ × → i jk k ∂ φ = 0, ∂ (∂µi φ ) = ∂∂ xφ  j  ∂ 2φ ∂L ∂L µ 2 ∂ φ − ∂µ ∂ (∂µ φ ) = 0 → ∂µ ∂ φ = 0 → ∂t 2 − ∇ φ = 0 Definition particle.Wave Equation. Definition2.1.6 2.1.6——Massless Massless particle. Wave Equation. φ (x,t) = ∑ p a p e−i(E pt−px) , E p = c |p| 2  L = 12 ∂ µ φ ∂µ φ = 12 ∂µ φ = 12 (∂0 φ )2 − 12 ∇φ ∇φ ∂L ∂L µ ∂ φ = 0, ∂ (∂µ φ ) = ∂ φ   2 ∂L ∂L = 0 → ∂µ ∂ µ φ = 0 → ∂∂tφ2 − ∇2 φ = 0 − ∂ µ ∂ ∂ φ ∂φ (µ ) φ (x,t) = ∑ p a p e−i(E pt−px) , E p = c |p| R

R

Any particle obeying to this law of motion is free and non-interacting

R

Any particle obeying to this law of motion is free and non-interacting

2.1 Lagrangian is your understanding and creativity level

25

2.1 Lagrangian is your understanding and creativity level

25

Definition 2.1.7 — Massive particle.Klein -Gordon equation.



2

− 12 m2 φ 2 , ∂∂Lφ = −m2 φ , ∂ ∂∂Lφ = ∂ µ φ (µ )    ∂L ∂L µ 2 ∂ φ − ∂µ ∂ (∂µ φ ) = 0 → ∂µ ∂ + m φ = 0, Klein − Gordon equation φ (x,t) = Ae−i(E pt−px) , E p2 = p2 c2 + m2 c4 , c = 1 → E p2 = p2 + m2 L =

1 2

∂µ φ 

These particles possess mass but do not interact 

Example 2.5 — Interacting particles.

∂L ∂φ

− ∂µ





∂L ∂ (∂ µ φ ) 2  ∂µ φ − 2  ∂µ φ −

=0

1 2 2 L = 12 2 m φ + J(x)φ (x), → External source 1 2 2 1 4 4 L = 12 2 m φ − 4! λφ , φ − theory  2 2 For two scalar f ields → L = 12 ∂µ φ1 − 12 m2 φ12 + 12 ∂µ φ2 − 12 m2 φ22 − g(φ12 + φ22 )  2 For complex scalar f ield L = ∂ µ ψ f ∂µ ψ − m2 ψ f ψ − g ψ f ψ , ψ = √12 [φ1 + iφ2 ], ψ f =

√1 [φ1 − iφ2 ] 2 

Definition 2.1.8 — Simple Harmonic Oscillator. The Scrodinger equation for harmonic oscil-

lator equals: 

h ∂2 − 2m ∂ x2

h=

h 2π ,

+

1 2 2 Kx



√1 2n n!



mω πh

 14

Hn (ξ )e− ψ = Eψ → ψn (ξ ) =     1 K ξ = mω h x, En = n + 2 hω , ω = m

ξ2 2

The Hamiltonian is equal: 

   14 2 h ∂2 1 mω 2 ψ = Eψ → ψ (ξ ) = √ 1 − ξ2 + Kx H (ξ )e − 2m n n 2 2 ∂x 2n n! πh    h mω 1 K h = 2π , ξ= h x, En = n + 2 hω , ω = m Definition 2.1.9 — Hamiltonian of harmonic oscillator. 2

 = p + 1 mω 2 x2 , K = mω 2, , p = −h ∂ , x = x H 2m 2 ∂x   2  = 1 mω 2 x− i p x+ i p = 1 mω 2 x+ p + iω { } H 2 mω mω 2 2m 2 x, p { x, p} ≡ xp − px = ih

Theorem 2.1.1 Theorem 2.1.1 Moving Moving clock clock goes goes more more slowly slowly then thanthat thatatatrest rest

Proof. Speed Speedofoflight lightisismaximum maximumspeed sped of any interaction interaction Proof.



26

Chapter 2. Space and Time. Lorenth Transformations Chapter 2. Space and Time.Lorentz Transformations

26 

Example 2.6 — Tensor algebra.

→ → − → − → → −  → − → − → − − → − − → − ∇×( A × B ) = B ·∇ A − B ·∇· A + ∇· B A − A ·∇ B →  − → − = εi jk ∂∂xi A × B el = +εikl ∂∂xi εklm Al Bm ei = k   = εki j εklm ∂∂ Ax jl Bm + ∂∂Bxmj Al ei =   = εki j εklm ∂∂ Ax jl Bm + ∂∂Bxmj Al e =    = δi j δ jm − δim δ jl ∂∂ Ax jl Bm + ∂∂Bxmj Al ei =   ∂Aj ∂ Bi − B − A = B j ∂∂ Ax ji + Ai ∂∂ Bi i j xj ∂xj ∂ x j  ei =  ∂ B ∂ A = B j ∂∂ Ax ji + ∂ x jj Ai − ∂ x jj Bi − A j ∂∂ Bx ji ei = → − → → −  → − → − → − → − − → − = B ·∇ A − B ·∇· A + ∇· B A − A ·∇ B 

Example 2.7 — One more example of tensor representation.

→ − → → − → − − → −  → − ∇·( A × B ) = B · ∇× A − A ·∇× B   → − → − ∇ · ( A × B ) = ei ∂∂xi ε jkl Ak Bl e j = εikl ∂∂Axik Bl + εikl Ak ∂∂Bxil =  → − → −  − → → − = εikl ∂∂Axik Bl − Ak εkil ∂∂Bxil = ∇ × A · B − A · ∇ × B

R

2.2



Hendrik (often spelled Henndrick) Anton Lorenz (niderl. Hendrik Antoon Lorentz; July 18, 1853, Arnhem, Netherlands - February 4, 1928, Harlem, Netherlands) - Dutch theoretical physicist, Nobel Prize winner in physics (1902, together with Peter Zeeman) and other awards, a member of the Royal Netherlands Academy of Sciences (1881), a number of foreign academies of science and scientific societies. Lorenz is best known for his work in the field of electrodynamics and optics. Combining the concept of a continuous electromagnetic field with the concept of discrete electric charges that make up the substance, he created the classical electronic theory and applied it to solve many particular problems: he obtained the expression for the force acting on a moving charge from the electromagnetic field (Lorentz force), the formula connecting the refractive index of a substance with its density (Lorentz-Lorentz formula) developed the theory of light dispersion, explained a number of magneto-optical phenomena (in particular, the effect Zeeman) and some properties of metals. On the basis of electronic theory, the scientist developed the electrodynamics of moving media, including hypothesized to reduce bodies in the direction of their movement (Fitzgerald-Lorentz reduction), introduced the concept of "local time", obtained relativistic expression for the dependence of mass on speed, derived relations between coordinates and time in moving inertial reference frames (Lorentz transformation). The works of Lorentz contributed to the formation and development of the ideas of the special theory of relativity and quantum physics. In addition, he obtained a number of significant results in thermodynamics and the kinetic theory of gases, the general theory of relativity, and the theory of thermal radiation.

Observable magnitudes and uncertainty principle



2.2

(in particular, the effect Zeeman) and some properties of metals. On the basis of electronic theory, the scientist developed the electrodynamics of moving media, including hypothesized to reduce bodies in the direction of their movement (Fitzgerald-Lorentz reduction), introduced the concept of "local time", obtained relativistic expression for the dependence of mass on speed, derived relations between coordinates and time in moving inertial reference frames (Lorentz transformation). The works of Lorentz contributed to the formation and development of the ideas of the special theory of relativity and quantum physics. In addition, he obtained a number of significant results in thermodynamics and the kinetic theory of gases, the general Observable magnitudes and of uncertainty principle theory of relativity, and the theory thermal radiation.

2.2 Observable magnitudes and uncertainty 2.2 Observable magnitudes and uncertaintyprinciple principle

27

27

Definition 2.2.1 — Observable physical magnitude is a physical magnitude which can be measured. A crucial difference between classical quantities and quantum mechanical ob-

servables is that the latter may not be simultaneously measurable, a property referred to as complementarity principle. This is mathematically expressed by non-commutativity of the corresponding operators, to the effect that the commutator p = −h ∂∂x , x = x then { x, p} ≡ xp − px = ih       {A, B} = AB − BA = 0, Physical magnitudes A and B can not be measured at one and the same time

Definition 2.2.2 — Hermit operatos in physics. Let C ∈ Rn be convex, f : C → R is strictly

convex on f if x, y ∈ C ×C. ∀α ∈ (0, 1), f (αx + (1 − α)y) f (αx) + f ((1 − α)y)

Definition 2.2.3 — Adjoint operator. The symbol

f → dagger most commonly used to denote the adjoint operator. Interpretation: Albert Einstein did not like the principle of uncertainty, and he challenged Niels Bohr and Werner Heisenberg with a famous thought experiment (See Bor-Einstein’s debates for more information): fill the box with radioactive material that emits radiation in a random way. The box has an open shutter, which immediately after filling closes with a clock at a certain point in time, allowing a small amount of radiation to escape. Thus, the time is already known. We still want to accurately measure the conjugate energy variable. Einstein suggested doing this by weighing the box before and after. Equivalence between mass and energy by the special theory of relativity will allow you to accurately determine how much energy is left in the box. Bohr objected as follows: if the energy leaves, then the lighter box will move a little on the scales. This will change the position

28

Chapter 2. Space and Time. Lorenth Transformations Chapter 2. Space and Time.Lorentz Transformations

28

of the clock. Thus, the clock deviates from our fixed reference frame, and according to the special theory of relativity, their measurement of time will differ from ours, leading to some inevitable error value. Detailed analysis shows that inaccuracy is correctly given by the Heisenberg relation. Within the limits of the widely but not universally accepted Copenhagen interpretation of quantum mechanics, the uncertainty principle is adopted at the elementary level. The physical universe does not exist in a deterministic form, but rather as a set of probabilities, or possibilities. For example, a picture (probability distribution) produced by millions of photons diffracted through a slit can be computed using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation believes that this cannot be predicted at all by any method. It was was this this interpretation interpretationthat thatEinstein Einsteinquestioned questioned when wrote to Max Born: not when he he wrote to Max Born: “God“God doesdoes not play dice”. Niels.Bohr, waswho one of theone authors of authors the Copenhagen interpretation, replied: “Einstein, do play dice” Nielswho Bohr, was of the of the Copenhagen interpretation, replied: not tell God to do”. “Einstein, dowhat not tell God what to do” . Einstein was convinced that this interpretation was wrong. His reasoning was based on the fact that all already known probability distributions were the result of deterministic events. The distribution of a coin toss or a rolling bone can be described by a probability distribution (50One more joke: The unusual nature of Heisenberg’s uncertainty principle and its memorable name made it the source of a number of jokes. It is claimed that the popular inscription on the walls of the physical faculty of university campuses is: “Heisenberg may have been here.” In another joke about the principle of uncertainty, a quantum physics specialist was stoped by a policeman on the highway and was questioned: “Do you know how fast you were driving, sir?” To which the physicist replied: "No, but I know exactly where I am!" 

Example 2.8 — Return back to harmonic oscillator.

   2  = p + 1 mω 2 x2 = 1 mω 2 x− i p x+ i p , H 2m 2 2 mω  mω    mω   i i {  − mω  x, p} ≡ xp − px = ih → a = mω x + p , af = p 2h mω 2h x     i  mω  h h 1 {  } {   } a, af = mω x , p + p , x + =1 − = 2h mω mω     2h mω mω h a + af , p = −i hmω a − af x = 2mω 2    = hω aaf + 1 H 2

Useful to turn this into an unconstrained problem.  f (x) i f x ∈ C f¯(x) = ∞ elsewhere



Definition 2.2.4 — Hermitian operator. The expectation value of any physical magnitude is

 which is given by the following: allianced by the correspondent operator A A =



 ψ ∗ (r)Aψ(r)dr

Any physical observable magnitude is represented by such expectation value,and al of them

2.2 Observable magnitudes and uncertainty principle

2.2 Observable magnitudes and uncertainty principle

29

29

should be real, so

2.2.1

A = A∗ ∗  ∗    Aψ(r) ψ (r)Aψ(r)dr = ψ(r)dr ∗  ∗   1 (r) ψ2 (r)dr  2 (r)dr = Aψ ψ1 (r)Aψ

Creation and annihilation operators

Definition 2.2.5 — Creation operator. The creation operator af creates a single particle which

has the momentum value . 1  f (x) = √ ∑ afp e−ipx ψ v p

This field operator creates a particle at position x, while afp creates a particle in a state with three momentum p. Definition 2.2.6 — Annihilation operator. The annihilation operator destroys single particle

which has the momentum value. 1  ψ(x) = √ ∑ ap e−ipx v p

This field operator annihilates a particle at position x, while a p annihilates a particle in a state with three momentum p

R

      i i + mω − mω n = af a, af = mω p , a = mω p x = x, p = −ih ∂∂x 2h x 2h x      h  a + af , p = −i hmω a + af x = 2mω 2  f { x, p} = ih, a, a = 1

Definition 2.2.7 — Creation and annihilation operators normalization. a| n >=

1>

af |n >=

√ n + 1 |n + 1 >

    1   H 0 >= hω n + 2

af |0 >= |1 >,  2 √ (af )  f a 1 >= 2 |2 > ⇒ |2 >= √2 |0 > , 3  √ (af ) af 2 >= 3 |3 > ⇒ |3 >= √3∗2 |0 > n  √ (af ) af n >= n + 1 |n + 1 > ⇒ |n >= √n! |0 >

√ n | n−

30

Chapter 2. Space and Time. Lorenth Transformations Chapter 2. Space and Time.Lorentz Transformations

30

R

German physicist Werner-Karl Heisenberg was born in Duisburg to the family of August Heisenberg, a professor of the ancient Greek language at the University of Munich, and born Annie Veklein. Heisenberg’s childhood years were spent in Duisburg, where he studied at the Maximilian Gymnasium. In 1920, he entered the University of Munich, where he studied physics under the guidance of the famous Arnold Sommerfeld. Heisenberg was an outstanding student and in 1923 defended his doctoral dissertation. She was devoted to some aspects of quantum theory. The following year he spent at the University of Göttingen as an assistant with Max Born, and then, having received a Rockefeller Foundation scholarship, went to Niels Bohr in Copenhagen, where he stayed until 1927, except for the lengthy visits to Göttingen. The greatest interest in Heisenberg was caused by unsolved problems of the structure of the atom and the increasing inconsistency of the model proposed by Bohr to the experimental and theoretical data. In 1925, during a short rest after an attack of hay fever, Heisenberg, in a rush of inspiration, saw a completely new approach that allows applying quantum theory to the resolution of all difficulties in the Bohr model. A few weeks later he laid out his ideas in an article. Max Planck laid the foundation of quantum theory in 1900. He explained the relationship between body temperature and the radiation emitted by him, putting forward the hypothesis that energy is emitted in small discrete portions. The energy of each such portion, or quantum, as Albert Einstein proposed to call it, is proportional to the frequency of the radiation. The concept of a quantum of energy was radically new, since in the last century it was proved that radiation, for example, light, propagates in the form of continuous waves. In 1905, Einstein used quanta to explain the photoelectric effect — the emission of electrons by a metal surface illuminated by ultraviolet light. More intense radiation leads to an increase in the number of electrons emitted by the surface, but not their energy. Einstein made the assumption that every quantum (of light or any other radiant energy), later called a photon, transfers energy to one electron. Some energy is expended on the release of an electron, and the rest goes into kinetic energy, i.e. manifested as electron velocity. The flux of a more intense radiation incident on the surface of the metal contains a greater number of photons, which release a greater number of electrons, but the energy of each photon remains fixed, which sets the speed limit of the electrons. Around 1913, Bohr proposed his own model of an atom: electrons orbit around a dense central nucleus in orbits of different radii. Using quantum theory, he showed that an atom excited by the burning of a substance or by an electric discharge emits energy at certain characteristic frequencies. According to Bohr, only well-defined electron orbits were allowed. When an electron "jumps" from one orbit to another, with less energy, its excess is converted into a quantum of emitted radiation at a frequency determined, according to Planck’s theory, by the difference in energy between the levels. At first, the Bohr model was a great success, but it soon became necessary to introduce corrections to eliminate the discrepancies between theory and experimental data. Many scientists pointed out that, despite its seeming simplicity, it cannot serve as the basis for a consistent approach to solving many problems of quantum physics. Heisenberg in the laboratory A brilliant idea that came to Heisenberg’s mind was to consider quantum events as phenomena at a completely different level than in classical physics. He approached them as phenomena that did not allow an accurate visual representation, for example, with the help of a picture of electrons circulating in orbits. Instead of visual images, Heisenberg proposed an abstract, purely mathematical representation based on the use of “principally observable” quantities, such as the frequencies of spectral lines. The equations derived by Heisenberg included tables of observable quantities: frequencies, spatial coordinates, and pulses. He pointed out the rules for performing various mathematical operations on these tables. Born recognized in the Heisenberg tables the matrices well known to mathematicians and showed that operations on them can be carried out according to the rules of matrix algebra - a well-developed area of mathematics, but little known at the time to physicists. Born, his student, Pascual Jordan and Heisenberg, developed this concept into matrix mechanics and created a method to apply quantum theory to the study of the structure of the atom. A

2.2 Observable magnitudes and uncertainty principle

2.2 Observable magnitudes and uncertainty principle

31

31

few months later, Erwin Schrödinger proposed another formulation of quantum mechanics describing these phenomena in the language of wave concepts. Schrödinger’s approach originates in the works of Louis de Broglie, who expressed the hypothesis of the so-called matter waves: just as light, traditionally considered as waves, can have Definition 2.2.8 — Replace state labels by operators.

 nk f 1 ak |0 >   (nk !)2      2  21000... >= √1 af √1 af2 |0 > 1  2! 1!

|n1 n2 ... >= ∏k

1

Particles in Universe are divided into two types:bosons and fermions: There are two types of particles in Universe: Bose particles and Fermi particles Two identical bosons swap brings the same state Exchange of two fermions changes the sign previous state 

Example 2.9 — A particle in a box.

For simplicity h = 1, p = −i ∂∂x , ψ(x) =

√1 eipx L ψ(x + L), eipx

pψ(x) = −i ∂∂ψx = pψ(x), ψ(x) = = eip(x+L) , eipL = 1, 2πm pL = 2πm, pm = L , p|p1 p2 >= (p1 + p2 ) |p1 p2 >

Definition 2.2.9 — Particles exchange. We have three energy states and we start to move

particle from one state to another: |110 > this state with particles in states 1 and 2 → |101 > swap particle f rom 2 to 3 → |011 > swap particle f rom 1 to 2 → |110 > swap particle f rom 3 to 1 The full sequence of events can be described by the following string of operators: af1 a3 af2 a1 af3 a2 |110 >= ± |110 >

Definition 2.2.10 — Green’s function. The amplitude that a particle starts at point y at a time ty and ends up at point x at time tx is a propagator Theorem 2.2.1

 Lx(t) = f (t)  LG(t, u) = δ (t − u)

G(t,u) is the Green’s function



32

Chapter 2. Space and Time. Lorenth Transformations Chapter 2. Space and Time.Lorentz Transformations

32

Proof. 2

m dtd 2 x(t) + Kx(t) = f (t),  f (t) = 0∞ du f (u)δ (t − u) 2

m dtd 2 + K G(t, u) = δ (t − u),

 = m d 22 + K L dt



x(t) = 0∞ duG(t, u) f (u)     u) f (u) = duδ (t − u) f (u) = f (t) Lx(t) = duLG(t, 



Example 2.10 — Green’s function for spherical wave.

 = ∇2 ∇2V (x) = − ρ(x) L ε0 ε0 ∇2V (x) = −δ (3) (x − u), V (x) =   ∇2 4πε01|x−u| = −δ (x − u)

1 4πε0 |x−u|

G(x, u) = − 4πε01|x−u| 2  k2 L  =2 ∇ +  ∇ + k2 Gk (x, u) = δ (3) (x − u) ei|k||x−u| Gk (x − u) = − 4π|x−u|

 

Example 2.11 — Propagators in Quantum mechanics.  +

φ (x,tx )= dyG (x,tx , y,ty )φ (y,ty ) G tx > t y G+ = 0 t x < ty G+ = θ (t − t )G x y  0 tx > t y − G = G t x < ty G− = θ (ty − tx )G

 

Example 2.12 — Green’s function and time evolution operator.

       −iH(t  x −ty )  G+ (x,tx , y,ty ) = θ (tx − ty ) < x U(t y > x − ty ) y >= θ (tx − ty ) < x e           + −i H(t −t ) −i H(t −t ) x y x y G (x,tx , y,ty ) = θ (tx − ty ) < x e  y >= θ (tx − ty ) < x e  |n >< n| y >

+ ∗ −iEn (tx −ty ) G (x,tx , y,ty ) = θ (tx − ty ) ∑n φn (x)φ (y)e x − i ∂ G+ (x,tx , y,ty ) = −iδ (x − y)δ (tx − ty ) = −iδ (2) (x − y) H ∂tx    Hx − i ∂t∂ x θ (tx − ty ) ∑n φn (x)φ ∗ (y)e−iEn (tx −ty )

∂ ∂t θ (tx − ty ) = δ (tx − ty ) i ∂t∂ x G+ = iδ (tx − ty ) ∑n φn (x)φ ∗ (y)e−iEn (tx −ty ) + θ (tx − ty ) ∑n En φn (x)φ ∗ (y)e−iEn (tx −ty ) x δ (tx − ty ) < x(tx ) |y(ty ) >= θ (tx − ty ) ∑n En φn (x)φ ∗ (y)e−iEn (tx −ty ) x G+ = H H

 x − i H

∂ ∂tx



G+ (x,tx , y,ty ) = −iδ (tx − ty ) ∑ φn (x)φ ∗ (y)e−iEn (tx −ty ) = −iδ (tx − ty )δ (x − y) 

2.3 Another face of Green’s function

33

2.3 Another face of Green’s function 

Example 2.13 — Non-relativistic free particle.

= H

p2 2m

G+ (x,t

∞



φ (x) = Ep =

x , y,ty ) 2

√1 eipx L

⇒

p2 2m

= θ (t x − ty )L

− ax2 +bx −∞ dxe

2



 dp

2π b2a a e ,a

=



G+ (x,tx , y,ty ) = θ (tx − ty )

2.3

33

 dp 

∑n → L  ∑p → V

2π d3 p (2π)3 ∗ (x)φ (y)e−iE p (tx −ty )

2π φ p i(t −t ) = xm y , b

m 2πi(tx −ty ) e

= i(x − y)

= θ (tx − ty )L

 dp

2π φ p (x)φ

2

p (tx −ty ) ∗ (y)e−i 2m

im(x−y)2 2(tx −ty )



Another face of Green’s function ∗

∗

(x)φn (y) n (x)φn (y) G+ (x, y, E) = ∑n iφnE−E = Lim ∑n iφE−E n n +iε ε→0

1 1 1 1 1 1 1 (H − E)G = −1, G = E−H , G = E−H10 −V = E−H + E−H V E−H + E−H V E−H V E−H + ... 0 0 0 0 0 0 Go 1 G = G0 + G0V G0 + G0V G0V G0 + ... = 1−V G0 = G−1 −V 0

Fact 2.3.1 



iEt + iEt −i(E p −iε)t = G+ 0 (p, E) = dte G0 (p,t, 0) = dte θ (t)e 

G+ 0 (p, E, q, E ) =

2.4

i E−E p +iε δ (p − q)

∞ −iei(E−E p +iε)  E−E p +iε 0

=

i E−E p +iε

Quantum mechanical transformations Translation the position of a particle in space at point x by tre vector a might be given by  x = T(a)x = x + a

Rotation of three vector in space by angle θ makes its transformation in the following way:    1 cos θ − sin θ 0 x      x2  x = R(θ )x = sin θ cos θ 0 0 0 1 x3

The particle during its translation wouldn’t change the probability density and we may write:     f  (a)U(a) |ψ(x) > < ψ(x) ψ(x) >=< ψ(x + a) |ψ(x + a) >= < ψ(x) U  U  f (a) = 1 • U(a)

   is unitary U

 + b) (composition  U  f (b) = U(a • U(a) rule) (compsition rule)

 • U(0) = 1 (a zero translation does nothing)

34

Chapter 2. Space and Time. Lorenth Transformations

Chapter 2. Space and Time.Lorentz Transformations

34

ψ(x + δ a) = ψ(x) + dψ(x) dx δ a + ... ψ(x + δ a) = (1 + i pδ a)ψ(x), p = i d • ψ(x + a) = lim (1 + i pδ a)N ψ(x) = edxi pa ψ(x) N→∞  U(a) = e−i pa

ψ(t + δta) = ψ(t) + dψ(t) dt δta + ...  =id ψ(t + δta ) = (1 + iHδta )ψ(t), H dt • a  a )N ψ(t) = eiHt ψ(t + ta ) = lim (1 + iHδt ψ(t) N→∞

a  a ) = e−iHt U(t     p ⇒ U(a)  p = H, = ei pa = eiHta −i pa

2.5

Lorentz transfomation

Definition 2.5.1 — Lorentz transformation. Lorentz transformation are usually presented as

follows      γ −β γ 0 0 t t   x −β γ  x ν 0 0  =    ⇒ xµ = Λνµ xν , Λνµ ≡ ∂ xµν , γ =  1 , β = v  c ∂x y   0 0 1 0 y  v2  1− z 0 0 0 1 z c2  0 2  1 2  2 2  3 2 2 → − −p ) = E 2 − p2 = m2 |x| = x − x − x − x , ab = a0 b0 − a · b, pp = pµ pµ = (E, p ) · (E, → x-direction of Lorentz transformation is given by  1  γ −β 1 γ 1 0 0 −β 1 γ 1 γ1 0 0  Λ(β 1 ) =   0 0 1 0 0 0 0 1

This transformation connects the four coordinates of two inertial systems of reference moving with relative speed v = cβ 1   cos hφ 1 sin hφ 1 0 0  sin hφ 1 cos hφ 1 0 0  ⇒ U(φ  ) |p > = |K (φ )p > K(φ 1 ) =   0 0 1 0 0 0  0 1 1 ∂ D(φ i )  iKφ i D(φ ) = e , K = i ∂ φ i  i φ =0

So Lorentz transformation taken together both boosts and rotations presented like D(θ , φ ) = e−i(Jθ −Kφ )

2.5 Lorentz transfomation 2.5 Lorentz transfomation R

35

35

He [Lorenz] created the life to the smallest details as they create a precious work of art. The kindness, generosity and justice that never left him, along with a deep, intuitive understanding of people and environments, made him a leader wherever he worked. Everyone was happy to follow him, feeling that he seeks not to dominate people, but to serve them. - Einstein A. Speech at the grave of Lorenz

By the beginning of Lorentz’s scientific career, Maxwell’s electrodynamics was able to fully describe only the propagation of light waves in empty space, while the question of the interaction of light with matter was still waiting to be solved. Already in the first works of the Dutch scientist, some steps were taken to explain the optical properties of matter in the framework of the electromagnetic theory of light. Based on this theory (more precisely, on its interpretation in the spirit of long-range action proposed by Hermann Helmholtz, in his doctoral dissertation (1875) Lorentz solved the problem of light reflection and refraction at the interface of two transparent media. in which light is treated as a mechanical wave, propagating in a special luminiferous ether, faced with fundamental difficulties. The method for eliminating these difficulties was suggested by Helmholtz in 1870; The first proof was given by Lorentz, who showed that the processes of reflection and refraction of light are determined by four boundary conditions imposed on the electric and magnetic field vectors on the interface, and derived the well-known Fresnel formulas. crystals and metals. Thus, the work of Lorentz contained the foundations of modern electromagnetic optics. No less important, here the first signs appeared of the peculiarity of the creative method of Lorenz, which Paul Ehrenfest expressed in the following words: “a clear separation of the role played by the ether in each given case of optical or electromagnetic phenomena occurring in a piece of glass or metal on the one hand, and "weighty matter" - on the other. " The distinction between ether and matter contributed to the formation of ideas about the electromagnetic field as an independent form of matter, as opposed to the earlier interpretation of the field as a mechanical state of matter. Previous results concerned the general laws of the propagation of light. In order to make more specific conclusions about the optical properties of bodies, Lorenz turned to ideas about the molecular structure of matter. He published the first results of his analysis in 1879 in his work “On the relationship between the speed of light and density and medium composition”. Assuming that ether inside a substance has the same properties as in free space, and that in each molecule under the influence of an external electrical force is proportional to the electric moment, Lorenz has obtained the ratio between the refractive index and the density of matter. R

Hendrik (often spelled Henndrick) Anton Lorenz (niderl. Hendrik Antoon Lorentz; July 18, 1853, Arnhem, Netherlands - February 4, 1928, Harlem, Netherlands) - Dutch theoretical physicist, Nobel Prize winner in physics (1902, together with Peter Zeeman) and other awards, a member of the Royal Netherlands Academy of Sciences (1881), a number of foreign academies of science and scientific societies.

3. Binary collisions. Scaterring matrix

3. Binary collisions.Scaterring matrix

3.1 3.1.1

Rutherford scattering Cross section of binary collisions Definition 3.1.1 — Cross section. The effective cross section is a physical quantity characterizing the probability of a system of two interacting particles moving to a certain final state, a quantitative characteristic of the collision of particles of a flow incident on a target with particles of a target. It is widely used in atomic and nuclear physics in the study of the scattering of particle beams on targets. Effective cross section has the dimension of the area. This value can be visualized as a conditional sum of the cross sections of the particles that make up the target. When this target is irradiated with a uniform flow, the particles that make up the flow must fall into this cross section. Particles that “miss” will not participate in the interaction channel . An effective cross section is widely used in nuclear and neutron physics to express the probability of a particular nuclear reaction occurring when two particles collide. We reduce the problem of the motion of two bodies to the problem of the motion of one body affected by central field, central field,ininwhich whichitsitspotential potential energy depends only distance r at a fixed point; by energy depends only on on thethe distance r at a fixed point; such such type type of potential is field is called central field.The the physical is equal: of potential field called central field. The forceforce actingacteing on the on physical object object then isthen equal: −r ∂U(r) ∂U → → − F =− → = − ∂r r ∂ −r

The angular momentum is conserved. conserved.This This might might be shown considering the Lagrangian function and isotropic space space property. property. isotropic  t2

·

L(t,  q, q)dt, δ S =  0, ∂L ∂L δ L = ∑i ∂ ri δ ri + ∂ vi δ vi = 0, S=

t1

∂L ∂ vi

·

= pi , ∂∂ rLi = pi

·   ·  → − − − −r × → −p ) = 0 → −r  + p δ → → −v  = δ → = p δ r + p δ v p δ ϕ × ϕ × ϕ dtd ∑ (→ ∑i i ∑i i i i i i i i i → −r × → −p ) = 0 d ( ∑ i dt → − −r × → −p ) = const L = ∑i (→

37

38

Chapter 3. Binary collisions. Scaterring matrix Chapter 3. Binary collisions.Scaterring matrix

38

The Lagrangian then might be written in the following way:

 · · · r 2 + r2 ϕ 2 −U(r), pϕ = mr2 ϕ, mr2 ϕ = const = M   ·2 m · 2 mr M2 2 2 E = 2 r + r ϕ +U(r) = 2 + 2mr 2 +U(r),  · 2 r = dr = m2 (E −U(r)) − mM2 r2 , dt dr M t = 2 + const, dϕ = mr 2 dt, M2 L=

m 2

m (E−U(r))− m2 r2

ϕ=





M dr r2

+ const

2 M2 m (E−U(r))− m2 r2

p – momentum of the projective

summerly plane

Δp pi

θ π–θ 2

Δp

pi

π–θ 2 π–θ 2

Path of projective

b

φ after Beiser

θ

b Impact parameter Target nucleus

θ = |π − φ | φ= φ=

∞

rmin

∞

rmin

M dr r2

 2m(E−u(r))− M2

, M = mbv∞ , E =

r

mv2∞ 2

M b 2 dr r    2 2U  b    1− − r2 mv2∞

Definition 3.1.2 Cross section

dσ = dσ =

Number o f scatted particles = dN All particle number moving n  to target   ) 2πbdb = 2πb(θ )  db(θ dθ , b =  dθ

or rigid spheres spheresooff radius radius a. a. a cos θ2 ffor

3.1 Rutherford 3.1 Rutherford scatteringscattering 3.1 Rutherford scattering

39

Rutherford’s fomula Rutherford’s fomula

dr dr r2 θ0 = rmin  ,b 2 U = αr −Coulomb f ield   ∞2 2U r θ0 = brmin  , U = αr −Coulomb f ield  2 2U − 1−  b 2 2 ∞ −  r αmv1− 2 2  r αmv∞   mv2 b  π−φ α 2 2 θ0 = arc cos    ∞ 2  mv, 2∞ bb =mv2∞ tg2 θ0 ,αθ0 =2 2 π−φ α  θ0 = arc1+cosmv  2  , b = mv2 tg θ0 , θ0 = 2 2b ∞

b

∞

1+

α

∞

mv2∞ b = α2 ctg2 φ2 φ α 2 2 mv∞b = 2 ctg mv∞ φ 2    2 2 cos  22 cos φα  dΩ 2 α dσ = π = 2 2 φ α dΩ 2 mv= mv∞ = (sinαφ2 )4 dσ ∞ π ( sin 2 )2 φ φ 2 mv2∞ mv 2 ∞ ( sin ) (sin )4 2 2

b2

39

39

40

Chapter 3. Binary collisions. Scaterring matrix

Chapter 3. Binary collisions.Scaterring matrix

40

Elastic and Inelastic collisions



Example 3.1 Two particle collision

m1



dr1 dt

2

m2



dr2 dt

2

+ −U (|r1 − r2 |) L= −r + m → − → −r = → −r 2 − → −r , m2 → 1 2 1 1 2 r 2 = 0, → −r = m2 → −r , → −r = − m1 → −r , 1

m1 +m2 ·

2

m1 +m2

0

m1 +m2

→ − → −v +m → − → − m2 1 2 v2 L = µ 2r −U(r) , µ = mm11+m , V 0 = m1 m 2 1 +m2 → −v = m2 → − → −v = − m1 → − ⇒ be f ore cllision collision 10 20 m1 +m2 v , m1 +m2 v , before − → − → −v = m2 → −v + → −v + V −v = − m1 → V , → ⇒ a f ter collision 10

20

m1 +m2

0



Theorem 3.1.1 — Idealization brings model. Idealization in the ordinary sense is a concept

meaning an idea about something (or about someone) in a more perfect form than it actually is. In science, this term is used in a somewhat different sense: as one of the methods of cognition, namely, how far the abstraction has gone. Idealization in the creative activity of man contributes to going beyond ordinary thinking and deeper comprehension of reality.

3.2 S-matrix

41

3.2 S-matrix

41

See below the Rutherford experimental set with alpha particles:

3.2

S-matrix

Definition 3.2.1 — Green’s function. We remind you that Green’s function is defined as follows:

 Lx(t) = f (t)  LG(t, u) = δ(t − u) ∞,t = u  +∞ δ (t − u) = , −∞ f (x)δ (x − a)dx = f (a) 0,t = u Definition Definition3.2.2 3.2.2——Another Anotherface faceofofGreen’s Green’sfunction. function.IfIfyou youknow knowthe theamplitude amplitude which which belongs to the thetransformation to the the point point xx at at aa time time txtx you you transformation of of aa particle particle at at point point yy at at aa time time ttyy to of the the amplitude the the propagator propagator-x(t -‹x(txx))|y(t |y(tyy)›. call this amplitude ) The The propagators propagators are are the Green’s functions of equations of motion for a particle particle

iφn (x)φ ∗ (x)

n ∑ E − En − ε→0+ iε

G+ (x, y, E) = Lim

n

• We see that this definition contains singularities or poles on the real axis when E = En corresponding to eigenstates of wave function phin (x) • The residues at the poles are the wave functions Here we are able able to to declair declair that that ifif we we have have Green’s Gren’s functionof function ofaasystem systemwe we have have access access to to the the energies and wave functions of the system system as as well well as as to to their their energy energy eigenvalues eigenvalues Definition Definition3.2.3 3.2.3——Green’s Green’sfunction functionininpertubation pertubationtheory. theory.The Thesystem systemHamiltonian Hamiltonianmight might

be splitted into two parts: parts:H +V Symbolicly .Symboliclywe wemay maywrite write(H (H−−E)G E)G==−1−1 H= = H00 +V. 1 1 1 1 1 1 1 = E−H10 −V = E−H + E−H V E−H + E−H V E−H V E−H + ... G = E−H 0 0 0 0 0 0 1 G0 = E−H0 , G = G0V G0 + G0V G0V G0 + ...

G00is known as as the the free free propagator propagator and and G-full G-full propagator propagator G 

Example 3.2 



iEt G+ (p,t, 0) = dteiEt θ (t)e−i(E p −iε)t = G+ 0 (p, E) = dte 0 ∞ −iei(E−E p +iε)t  = E−E p +iε  = E−Eip +iε 0

42

Chapter 3. Binary collisions. Scaterring matrix

Chapter 3. Binary collisions.Scaterring matrix

42 42

Chapter 3. Binary collisions.Scaterring matrix 

42Definition 3.2.4 — Dyson equation.

Chapter 3. Binary collisions.Scaterring matrix

G0 1  = −1 1 −V G0 G0 −V Definition 3.2.4 — Dyson equation. G0 1 G = G0V G0 + G0V G0V G0 + ... = = −1 1 −V G0  G     0 1 −V G0  Particle Particle + G = G V G + G V G V G + ... = =  0 0 0 0 0 Ω G (x, y) = Ω  i (0, 1, G02, 3)G−1 −V at point yi (0, 1, 2, 3   at point 1x−V 0 created  annihilated     Particle Particle  GG++(x, (x0 − y0 ) Ω φ(x) φf (y) |Ω i (x,y) y)= =θ   created at point yi (0, 1, 2, 3 Ω  Ω annihilated (0, 1, 2, 3) at point x   Particle   Particle  0 0 Ω y) = Ω GG++(x, f (y) |Ωi i (0, 1, 2,as (x, y) = θ (x −annihilated y ) that Ω φ (x) φ  Example 3.3 Remember Heisenberg operator has a time dependence created at point ydefined 3 follows: at point x (0, 1, 2, 3)   0 − y0 ) Ω φ f (y) |Ω G+(x, y) = θ (x    (x) φ +iHt −Ht φ (t, x)3.3 = eRemember φ (x)e  Example that Heisenberg operator hasa time dependence defined as follows:   0 −y0 )  0 −iH(x  0 iHx (x)e f (y))e−iH(y ) |Ω |e G(x, y) = Ω φ φ   +i Ht − Ht  Example a time dependence defined as follows: φ(t,3.3 x) =Remember e φ(x)ethat Heisenberg operator has   0 −y0 )  0) −iH(x −i H(y  0 i Hx f  |e   0 y) = +iΩ − φHt (x)e Hy φG(x, (t, x)|Ω=>eisHtthe φ(x)e state Ω evolved tφa (y))e time y0 |Ω • e−i  0 −y0 )  0) 0 0 −i H(x   f (y)e−iHy |Ω >iHx f (y))e−iH(y state with aφparticle created • φG(x, |Ω at a time y0 at a positon y φ(x)e  0y) = Ω |e is the −i Hy 0 • eiHx  0  |Ω > is the state Ω evolved t a 0time y • e f φ (x)is  0the state evolved to time x −i(y)e  0 −iHy |Ω > is the state with a particle0created at a time y0 at a positon y • φ Hy on the left and ystarted on the right This propagator present the • This |Ω > is is terminated the state Ω evolved t a time e  0 string iHx φ −i 0 the state evolved to time x0 0 and get 0  that (x)is • e f Hy  amplitude we put a particle at position at a time at position |Ω > is the state with a particleycreated at ay time y0 atout • φ (y)e a positon y x at time x • This  0 string is terminated on the left and started on the right This propagator present the  • eiHx φ(x)is the state evolved to time x0 amplitude that we put a particle at position y at a time y0 and get out at position x at time x0 • This string is terminated on the left and started on the right This propagator present the  Wick’s Time Odering symbol amplitude that we put a particle at position y at a time y0 and get out at position x at time x0 Definition 3.2.4 — Dyson equation.

G = G0V G0 + G0V G0V G0 + ... =

3.2.1 3.2.1

Definition 3.2.5 — Feynman propagator.



Wick’s Time Odering  symbol Definition 3.2.5 — Feynman )φ(y0 ), x0 > y0 φ(x0propagator. 0 )φ (y0 ) = symbol 3.2.1 Wick’s Time T φ(xOdering 0  φ(ypropagator. )φ(x00), x00< y00 Definition 3.2.5 — 0  Feynman  )φ(y ), x > y 0 ) =  φ (x T φ(xy)0 )= φ(yΩ |T  G(x, φ (x)φf0(y)|Ω0 = 0 (y0 )φ (x0 ), x0 < y00    φ(x )φ (y ), x >0 y 0 φ  0 0 0)φ (yy ) =Ωφ(x)φff(y) f (y)φ(x) |Ω T φθ(x |Ω − + θ (y − x ) Ω| φ = (x   0 0 0 0 G(x, y) = Ω |T φ (x) φ(x|Ω), =x < y φ(yφ )(y)      0 − y0 ) Ω 0 − x0 ) Ω| φ (x) ff(y) f (y)φ(x) |Ω 0  0 |Ω φ (x) φ (y) + θ (y = θ (x |T |Ω G(x, y) = Ω φ φ = The Feynman propagator  consists of two  parts.The first part operates forx later thany .It creates a0 f (y) |Ω to+xθwhere 0 ) destroyed.The particle at=y θand theφparticle second part operates when y − y0 ) Ωφ(x) (y0 − xit’s Ω| φf (y)φ(x) |Ω (x0propagates 0 .It creates a The Feynman propagator consists of an twoantiparticle parts.The first part propagates operates forthe x0 system later thany later than x0 .The second part creates at x and to point y. Ω is particle at y and propagates the particle to x where it’s destroyed.The second part operates when ya0 the interacting ground state of theof system.If theThe first system doesn’t contain interaction we have so The Feynman propagator consists two for xx00 later than It creates creates The Feynman propagator consists of two parts. parts.The first part part operates operates forany later thanyy00..It a 0 later than x and .The second part creates antoantiparticle at x and propagates the system to pointwhen y. Ω is 0 called propagator: particle at the The second second part operates operates particlefree at yyFeynman and propagates propagates the particle particle to xx where where it’s it’s destroyed. destroyed.The part when yy0 the interacting ground state of the system.If the system doesn’t contain any interaction we have so 00   later The second second part part creates creates an an antiparticle antiparticle at at xx and and propagates propagates the the system system to to point point y. y. Ω is later than than xx ..The Ω is (x)φf (y) called free y) Feynman |0system. If the system doesn’t contain any interaction we have so (x, =ground 0| Tpropagator: φstate the interacting of the the interacting ground state of the system.If the system doesn’t contain any interaction we have so  propagator:  called Feynman called free free (x,Feynman y) = 0|propagator: T φ(x)φf (y) |0   (x, y) = 0| T φ(x)φf (y) |0

3.2 S-matrix

43

3.2 S-matrix

43

Definition 3.2.6 — Free Feynman propagator.

 f ipy +  −ipy ⇒   |0 |0 a > e > e b b p |0 >= 0 p p 3 1 (2π) 2 (2E p ) 2    d3 p f ipy , y ↔ x, p ↔ q |p > e φ (y) |0 >= 3 1 φf (y) |0 >=





d3 p

(2π) 2 (2E p ) 2 d3q

  −iqx 1 < q| e 3 (2π) 2 (2Eq ) 2  d 3 qd 3 p f −iqx+ipy δ (3) (q − p) = < 0| φ(x)φ (y) |0 > = 1e

< 0| φ(x) =

=





(2π)3 (2E p 2Eq ) 2

d3 p

(2π)3 (2E p )

(x, y) =



e−ip(x−y)

 0  θ (x − y0 )e−ip(x−y) + θ (y0 − x0 )eip(x−y) 3 (2π) (2E p ) d3 p

 ∞ dz e−iz(x0 −y0 ) −∞ 2π z+iε , 0 0 ∞  dzd 3 p e−i(E p +z)(x −y )+ip(x−y) [(x, y)](1) = i −∞ , ⇒ z = z + Ep 4 z+iε (2π) (2E p )  ∞ dz d 3 p e−iz (x0 −y0 )+ip(x−y)  0

θ (x0 − y0 ) = i

[(x, y)](1) = i [(x, y)](1) = [(x, y)](2) =

 −∞ (2π)4 (2E p ) z −E p +iε d4 p e−ip(x−y) i −∞ , (2π)4 (2E p ) p0 −E p +iε θ (y0 − x0 ) < 0| φf (y)φ(x) |0

∞

,z = p ,

> =i

∞

d4 p e−ip(x−y) −∞ (2π)4 (2E p ) p0 +E p +iε ,

(x, y) = [(x, y)](1) + [(x, y)](2)  −ip(x−y)  ∞ d4 p e e−ip(x−y) (x, y) = −∞ + = 4 0 p0 +E p +iε (2π)  (2E p ) p −E p +iε   2 ∞ d4 p e−ip(x−y) ⇒ E p2 = p2 + m2 ⇒ p0 − E p2 + iε = p2 − m2 + iε = −∞ 4 2 2 0 (2π) (2E p ) (p ) −(E p ) +iε  −ip(x−y)  ∞ d4 p e (x, y) = −∞ 4 2 2 (2π) (2E p )

R

(p) −m +iε

The Fourier component of the Feynman propagator applied to a particle with momentum p:

(p) =



d4 e−ip(x−y) (p) (2π)4 i p2 −m2 +iε

(x, y) =

Yukawa’s force -carring particles R

Particles interact by exchanging virtual,force carring particles

R

Hideki (23 January January1907 1907––8 8September September 1981) a Japanese theoretical physicist Hideki Yukawa (23 1981) waswas a Japanese theoretical physicist and the the first first Japanese Nobel laureate forfor hishis prediction ofofthe and Japanese Nobel laureate prediction thepipimeson. meson.He Hewas wasborn born in in a family of geologists, geologists, university university professors. professors.He of He conducted conducted research research and and teaching teaching work work in in Kyoto Kyoto and and Osaka universities, universities, as as well well as as in in US US scientific scientific and and educational educational centers. In 1935, 1935, he he put put forward forward Osaka centers. In hypothesis about about the the existence existence of of aa new new type type of of elementary elementary particles particles with with aa mass mass intermediate intermediate aahypothesis

44

Chapter 3. Binary collisions. Scaterring matrix

Chapter 3. Binary collisions.Scaterring matrix

44

between the masses of an electron and a proton. By the end of the 40s. this hypothesis was confirmed, and in 1949, the Nobel Prize in Physics was awarded for predicting the existence of mesons and theoretical studies of the nature of the nuclear forces of Yukawa. The first Japanese who won the Nobel Prize. Member of the Japanese Academy of Sciences (1946), member of the Pontifical Academy of Sciences (1961), foreign member of the National Academy of Sciences of the USA (1949), the Royal Society of London (1963), and the Academy of Sciences of the USSR (1966).

Definition 3.2.7 Definition 3.2.7— —Yukawa Yukawapotential. potential. ItIt should should pointed pointed out out that that vitual vitual particles particles are are really really exist exist and live a very short time allowing to have aa corresponent corresponent value value of of energy energy due due to to the the Heisenberg Heisenberg uncertainty principle principle and and Yukawa Yukawa guessed guessed how the vetual uncertainty vetual particles particles mediate mediate interactions interactions with with aa of the the following following potential, potential,called help of called Yukawa potential:

|r| e a U(r) ∼ − 4π |r| −

.

R −

|r|





a ⇒ ∇2 − m2 U(r) = δ (3) (r) U(r) ∼ − e4π|r|   (p) = p2 −mi 2 +iε = 0 2 i 2 2 , p0 = ( p ) −(p) −m +iε



p2 + m2 |r|

 − a −ipr 3 A(scatterring amplitude) ∼ U(r)e−ipr d 3 r ∼ e4π|r| e d r∼ 

1 −|p|2 −m2

Definition 3.2.8 — Interaction involve the creation or annihilation of particles.

John Archibald Archibald Wheeler, Wheeler, (born (born July July 9, 9, 1911, 1911, Jacksonville, Jacksonville, Florida, Florida,U.S. U.S.—died — diedApril April 13, 13, 2008, 2008, first American Hightstown, New Jersey), Jersey), physicist, physicist, the the first American involved involvedin inthe thetheoretical theoreticaldevelopment developmentofofthe the field theory atomic bomb. He also originated a novel novel approach approach to to the theunified unified field theoryand andpopularized popularizedthe theterm term field theory, black hole. In later years he turned turned his his attention attention to to the thestudy studyof ofunified unified field theory,the thespace-time space-time continuum, and gravitation. His books include Gravitation Gravitation Theory Theory and and Gravitational GravitationalCollapse Collapse(1965), (1965), Einstein’s Vision (1968), Frontiers of Time Time (1979), (1979), and and Gravitation Gravitation and andInertia Inertia(1995, (1995,with withIgnazio Ignazio

3.2 S-matrix

45

3.2 S-matrix

45

Ciufolini), as well as a major textbook on Einstein’s theory of relativity, Gravitation (1973, with Charles W. Misner and Kip S. Thorne), and an autobiography, Geons, Black Holes, and Quantum Foam: A Life in Physics (1998, with Kenneth Ford). He was awarded the Niels Bohr International Gold Medal in 1982. Definition 3.2.9 — Time evolution operator.

1 (interaction part) =H 0 (Free part) + H H ∂ ψ(x,t)  i ∂t = Hψ(x,t)   2 ,t1 ) ⇒ A time − evolution operator ψ(t2 ) = U(t2 ,t1 )ψ(t1 ), U(t

Time evolution operator properties: • If two slots of time are the same,then nothing will happen  1 ,t2 ) = 1 U(t • If you fixed three slots of time ,you may reach the third slot gradually  3 ,t2 )U(t  2 ,t1 ) = U(t  3 ,t2 ) U(t • Time evolution operator obeys the Schrodinger equation d   i U(t 2 ,t1 ) = HU(t2 ,t1 ) dt2 • The inverse time operator returns you back at initial moment of time  1 ,t2 ) = U  −1 (t2 ,t1 ) U(t • The time evolution operator unitary  f (t2 ,t1 )U(t  2 ,t1 ) = 1 U   f  d f   f dU = − U f HU + U f HU = 0  +U U (t2 ,t1 )U(t2 ,t1 ) = dU U dt2

dt2

dt2

i

i

Definition 3.2.10 — Time evolution operator.  2 −t1 )  2 ,t1 ) = e−H(t U(t

Theorem 3.2.1 Time evolution operator obeys the Schrodinger equation

i Proof.

d  U(t  2 ,t1 ) U(t2 ,t1 ) = H dt2

 (t2 ,t1 ) ψ(t1 ) ψ(t2 ) = U

 2 ,t1 ) dψ(t2 ) dU(t dt2 = dt2 ψ(t1 ) dψ(t2 )  U(t  2 ,t1 ) i dt2 = Hψ(t2 ) = H

ψ(t1 ) 

Another favorite Wheelerism is "one can only learn by teaching." Wheeler has been the supervisor for some 50 PhDs in physics during his career, an "enormous number," according to Jeremy Bernstein, a physicist and science writer. Wheeler’s most famous student was the late Richard P. Feynman, who received a Nobel Prize in 1965 for his work in quantum electrodynamics. Technically, Wheeler can teach no longer. "If you know of a school that lets its professors teach after they reach 70," he says, "let me know."

46

Chapter 3. Binary collisions. Scaterring matrix

Chapter 3. Binary collisions.Scaterring matrix

46

3.3

S-matrix

Definition 3.3.1 — Interaction.

1 (t2 → ∞,t1 → −∞) S = U  ∞ 4   S = T e−i −∞ d xH1 (x)   2   1 (z) + (−1) d 4 yd 4 zH 1 (y)H 1 (z) + ... S = T 1 − i d 4 zH 2!

Definition 3.3.2

1 (interacting part) =H 0 (non − interacting part) + H H    t2   2 ,t1 ) = T e−i t1 H1 dt U(t  S = U(t  2 ,t1 )

A = ψ(a f ter)| S|φ (be f ore)



4. Gravitation fields

4. Gravitation fields

4.1

Gravitational forces result from the properties of spacetime itself The achievements of modern science testify to the preference of the relational approach to the understanding of space and time. In this regard, first of all, it is necessary to single out the achievements of the physics of the 20th century. The creation of the theory of relativity was a significant step in understanding the nature of space and time, which allows us to deepen, clarify, specify philosophical ideas about space and time. What are the main conclusions of the theory of relativity on this issue? The special theory of relativity, the construction of which was completed by A. Einstein in 1905, proved that in the real physical world the space and time intervals change when moving from one frame of reference to another. The frame of reference in physics is the image of a real physical laboratory, equipped with clocks and rulers, that is, the tools with which you can measure the spatial and temporal characteristics of bodies. Old physics believed that if the reference systems move uniformly and rectilinearly relative to each other (this movement is called inertial), then the spatial intervals (the distance between two nearby points) and the time intervals (the duration between two events) do not change. The theory of relativity disproved these ideas, or rather, showed their limited applicability. It turned out that only when the speeds of movement are small relative to the speed of light, we can approximately assume that the sizes of bodies and the course of time remain the same, but when it comes to movements with speeds close to the speed of light, then the change in spatial and time intervals become noticeable. With an increase in the relative speed of the reference system, the spatial intervals are reduced, and the time intervals are stretched. This is a completely unexpected conclusion for common sense. It turns out that a rocket, which had at the start some fixed length, when moving at a speed close to the speed of light, should become shorter. At the same time, in the same rocket, both the clock and the astronaut’s pulse, his brain rhythms, the metabolism in the cells of his body would slow down, that is, the time in such a rocket would run slower than the observer’s time at the launch site. This, of course, contradicts our everyday ideas, which were formed in the experience of relatively low speeds and therefore not sufficient to understand the processes that unfold at near-light speeds. The theory of relativity has discovered another significant aspect of the space-time relationship of the material world. She

47

48

Chapter 4. Gravitation fields

Chapter 4. Gravitation fields

48

revealed a deep connection between space and time, showing that there is a single space-time in nature, and separately space and time act as its original projections into which it is split into different ways depending on the nature of the movement of bodies. The abstracting ability of human thinking divides space and time, setting them apart from each other. But for the description and understanding of the world, their compatibility is necessary, which is easy to establish by analyzing even the situations of everyday life. In fact, to describe an event, it is not enough to determine only the place where it occurred, it is important to also indicate the time when it occurred. Prior to the creation of the theory of relativity, it was believed that the objectivity of the space-time description is guaranteed only when, during the transition from one frame of reference to another, spatial and separate time intervals are maintained separately. The theory of relativity has generalized this position. Depending on the nature of the motion of the reference systems relative to each other, different splits of a single space-time into separate spatial and separate time intervals occur, but occur in such a way that the change of one compensates for the change of the other. If, for example, the spatial interval was reduced, then the time interval increased as much, and vice versa. It turns out that the splitting into space and time, which occurs differently at different speeds of movement, is carried out so that the space-time interval, that is, the joint space-time (the distance between two adjacent points of space and time), is always preserved, or in scientific terms, it remains an invariant. The objectivity of the space-time event does not depend on which of the reference system and at what speed the observer characterizes it moving. The spatial and temporal properties of objects are separately variable when the velocity of the objects changes, but the space-time intervals remain invariant. Thus, the special theory of relativity revealed the internal connection between space and time as forms of being of matter. On the other hand, since the change in spatial and temporal intervals depends on the nature of the movement of the body, it turned out that space and time are determined by the states of the moving matter. 4.1.1

Newton’s Gravity Law

Definition 4.1.1 — Newtonian gravity. Newtonian gravitation potential defined as

 → → − → − − → − F g = −mg ∇φN = −mg ∂∂φx i + ∂∂φx j + ∂∂φx k  r = x2 + y2 + z2 φN = −γ Mr ,

Spacetime tells matter how to move;matter tells spacetime how to curve- Jonh Wheeler For not spinning stars and black holes spacetimes are static and spherically symmetric.We say spacetime is static because the coefficients of the differentials are independent of variable t. In Cartesian coordinates flat spacetime has the following metric: dτ 2 = dt 2 − dx2 − dy2 − dz2

We transform Cartesian coordinates to spherical polar coordinates:

.

x = r sin θ cos φ , y = r sin θ sin φ , z = r cos θ , dx = dr sin θ cos φ + dθ r cos θ cos φ − dφ r sin θ sin φ dy = dr sin θ sin φ + dθ r cos θ sin φ + dφ r sin θ cos φ dz = dr cos θ − dθ r sin θ dτ 2 = dt 2 − dr2 − r2 dθ 2 − r2 sin2 θ dφ 2

4.1 Gravitational forces result from the properties of spacetime itself 4.1 4.1 Gravitational Gravitationalforces forcesresult resultfrom fromthe theproperties propertiesof ofspacetime spacetimeitself itself

49

49 49

Wheelerisisalso alsorenowned renownedfor forhis hiscoinages, coinages,analogies analogiesand andaphorisms, aphorisms,both bothself-made self-madeand andcocoRR Wheeler opted. Among the one-liners he bestows on me are, "If I can’t picture it, I can’t understand opted. Among the one-liners he bestows on me are, "If I can’t picture it, I can’t understandit" it" (Einstein); (Einstein);"Unitarianism "Unitarianism[Wheeler’s [Wheeler’sofficial officialreligion] religion]isisaafeatherbed featherbedtotocatch catchfalling fallingChristians" Christians" (Darwin); (Darwin);"Never "Neverrun runafter afteraabus busororwoman womanororcosmological cosmologicaltheory, theory,because becausethere’ll there’llalways alwaysbebe another anotherone oneininaafew fewminutes" minutes"(a(aprofessor professorofofFrench Frenchhistory historyatatYale); Yale);and and"If "Ifyou youhaven’t haven’tfound found something somethingstrange strangeduring duringthe theday, day,itithasn’t hasn’tbeen beenmuch muchofofaaday" day"(Wheeler). (Wheeler).Lately LatelyWheeler Wheeler has hasbeen beendrawing drawinghis hiscolleagues’ colleagues’attention attentiontotosomething somethingstrange strangeindeed. indeed. ItItisisaaworldview worldview uniting unitinginformation informationtheory, theory,which whichseeks seekstotomaximize maximizethe theefficiency efficiencyofofdata datacommunications communicationsand and processing, processing,with withquantum quantummechanics. mechanics.As Asusual, usual,Wheeler Wheelerhas haspackaged packagedthe theconcept conceptininaacatchy catchy phrase: phrase:"it "itfrom frombit." bit."And Andasasusual, usual,hehedelights delightsininbeing beingahead aheadof—or of—oratatleast leastapart apartfrom—the from—the pack. pack. "I"Ihope hopeyou youdon’t don’tthink thinkI’m I’mtoo toomuch muchlike likeDaniel DanielBoone," Boone,"he hesays saysslyly. slyly. "Anytime "Anytime someone someonemoved movedtotowithin withinaamile mileofofhim, him,hehemoved movedon. on.Accoding AccodingtotoNewtonian Newtoniangravity gravityififyou you are aremoving movingthe thegavitational gavitationalfield fieldaamillion millionlight lightyears yearsaway awaychanges changesinstantaneously. instantaneously.However However according accordingtotospecial specialtheory theoryofofrelativity relativity,signals ,signalscannot cannottravel travelinviniely invinielyfast,so fast,soNewtonian Newtonian gravity gravitycontadicts contadictstho thospecial specialtheory theoryofofrelativity relativity.The .Thestatic staticCoulomb Coulomblaw lawalsoconradicts alsoconradictstoto Lorentz Lorentztransformations transformationsand andneed needtotobe bemodified modifiedwhen whensources sourcesare areininmotion.Spacetime motion.Spacetimeisis no longer the eternal,unchanging stage where al physical phenomena are no longer the eternal,unchanging stage where al physical phenomena aretaking takingplace,they place,they are deeply connected with each other like two sides of coin.moreover,spacetime are deeply connected with each other like two sides of coin.moreover,spacetimeisisaaboss bossand and matter matterisisemployee employeeand andthey theyare arechanges changestheir theirpositions positionscontinuously. continuously.

General Generalstatic staticmetric metricwith withspherical sphericalsymmetry symmetry 2 2 2Φ(r) 2 2Λ(r) 2 2− r 2 e 2B(r) 2 2 dτ dτ 2==ee2Φ(r)dtdt 2−−2A(r)dtdr 2A(r)dtdr−−ee2Λ(r)dr dr − r2 e2B(r)(dθ (dθ 2−−sin sin2θθdφ dφ )2 ) We simplify this equation taking We this equation taking AA=0,B=0. =A=0,B=0. 0, B = 0.Then Then we Wesimplify simplify this equation taking Thenwe wehave: have: 2 2= e2Φ(r) 2 2− e2Λ(r) 2 2− r 2 (dθ 2 2− sin2 2θ dφ 2 ) 2Φ(r) 2Λ(r) 2 dt dr dτ dt − e dr − r (dθ − sin θ dφ 2 ) dτ = e Notice distance and proper time areare as as follows Notice that ,proper distance and proper time Noticethat, thatproper ,proper distance and proper time are asfollows follows 2 22 2 22 ds = rdθ is proper distance and ds − dτ ≡ −dτ ds = rdθ is proper dis tan ce and ds ≡ ds = rdθ is proper dis tan ce and ds ≡ −dτ

Einstein Einsteinfield fieldequations equations

InInNewtonian Newtoniangravity gravitythe thegravitational gravitationalpotential potentialdefined definedby byPoisson Poissonequation: equation: 2Φ ∇∇2 Φ NN==4πGρ 4πGρ → →Newton Newton → −→ − ∇∇EE ==4πρ →Coulumb Coulumb 4πρq q→



 

 

ρ(r ,t)dr ,t)dr → Newton ΦΦNN(r,t)   → Newton (r,t)==−G −G ρ(r r−r r−r ∞ GM(r,x,t) r 2 ∞ GM(r,x,t)d x, ΦΦNN(r,t) = − M(r,t) 22 (r,t) = − r d x, M(r,t)== 0 rρ(r,t)4π ρ(r,t)4π(x) (x)2ddxx r

(x) (x)

0

Theorem Theorem4.1.1 4.1.1— —Density Densityand andPressure Pressureofofthe themedium mediumresponsible responsiblefor forproducing producingthe thegravgravity. ity. 2 2 2Φ(r) 2 2Λ(r) 2 2− r 2 e 2B(r) 2 2 dτ dτ 2==ee2Φ(r)dtdt 2−−2A(r)dtdr 2A(r)dtdr−−ee2Λ(r)dr dr − r2 e2B(r)(dθ (dθ 2−−sin sin2θθdφ dφ )2 )

   −2Λ ==8πGρ(r) rr 11−−ee−2Λ 8πGρ(r)

11 dd r2r2drdr

 22dΦ 1  dΦe−2Λ −2Λ− 1 1 − e−2Λ −2Λ = 8πGp(r) − 1 − e = 8πGp(r) e 2 rr dr rr2 dr

50

Chapter 4. Gravitation fields

Chapter 4. Gravitation fields

50

4.2

Cristoffel symbols and metrics In any inertial reference system the interval equals

ds2 = c2 dt 2 − dx2 − dy2 − dz2

If you want to make transformation in non-inertial system the interval will not be expressed in such a simple form as for inertial systems.Consider the transformation of all coordinates to rotating system of reference.In this case we may write the following set of transformation equations:





x = x cos Ωt − y sin Ωt   y = x sin Ωt + y cos Ωt,  z=z         2   2   2  2  2     2 2 2 dt 2 − dx − dy − dz + 2Ωy dx dt − 2Ωx dy dt ds = c − Ω x + y

There is no way to present this equation as a sum of squars of all coordinates differentials. So in noninertial sistem of reference the transformation of coordinates includinf time might be presented as follows:

ds2 = gik dxi dxk , gik = f (x0 , x1 , x2 , x3 )

Definition 4.2.1 — Non inertial system of reference. So when we are in non inertial system of

reference transformatiom of coordinates are defined by 16 quantities gik = gki .Generally we have four quantitites with equal indices and 4*3/2=6 different indices,totally ten.In inertial systems x0 = ct, x1 = x, x2 = y, x3 = z g00 = 1, g11 = g22 = g33 = −1, gi j = 0, j = i We know this transformation as Galilean.And this transformation is connected only with inertial systems of reference.The geometrical properties are determined by physical phenomena and not existing like independent property of space and time.

4.2 Cristoffel symbols and metrics

51

4.24.2Cristoffel symbols and metrics Cristoffel symbols and metrics

5151

Definition 4.2.2 —— Covarient derivatives of  tensors. us us consider the transformation from Definition 4.2.2 Covarient derivatives of 0 Let 1Let 2consider tensors.    3 the transformation from 

0



1



2



    3 to one coordinate system x0 ,xx01, ,xx12, ,xx23, xto another one coordinate system anotherx x , ,x x , ,x x , ,x x            0  1  2  3 i i i i   0   1   2   3 x = f f x x , ,x x , ,x x , ,x x x = i

k

3

 lm ik =∂ x ∂lxi ∂ x ∂mxkA lm AikA=  mA ) ∂ (x∂ )x∂ (l x∂ ( x )  ( ) ∂ x ∂i x∂ i x ∂mx m l  l AikA= A ik = l l k kmA

∂ (x) ∂ (x) ∂ (x) ∂ (x)



m

Definition 4.2.3 —— Line element in in curvilinear coordinates. Definition 4.2.3 Line element curvilinear coordinates. 2 i dxi k ⇒ k g gkl = kl δ i i (ds) gikgdx (ds)=2 = ik dx dx ⇒ikgik g =l δl k Ai A=i = gklgAklkA , k ,Ai A=i = gikgAikkA  11 0 0 0 0 0 0   0 −1 0 (galilean)→(0) 0  0 −1 0 0  (0)ik gikg(galilean)→(0) == g(0)ik == g  00 0 0 −1−1 0  ik 0 0 0 0 0 0 0 −1−1 eiklm →→ e0123 == 1, 1,e0123 == −1−1 eiklm ei 0123 e0123 ∂ x ∂i xk∂ x∂kxl∂ ∂xlxm∂ xmprstprst iklm prstprst ∂ x E E iklm == e = JeJe     = ∂ x ∂p x∂ px ∂r x∂ rx ∂s x∂ sx ∂t x t e 2 3 ∂ (x∂0 ,xx10,x ∂ xi∂ x∂i xk∂ xk(0)pr ( ,x1,x,x2),x3 ) gik = ik = (0)pr J=   g  0  1  2  3 ⇒⇒ g J=  0 ,x 1 ,x 2 )  3 ∂ x ∂p x∂ px ∂r x r g ∂ (x∂ ,x ,x ,x )  ik   1(x ,x (0)ik √√ 2 , 2 E iklm g  ik= , 1 g  (0)ik   −1, 1 = √1√1eiklm iklm −J−J == , E,iklm = = −ge iklm g = = , E iklm Eiklm −ge g = −1,g 1g = g g, −g−g e iklm √√  0    ∂ ∂ 0 1 1 2 2 3 3 1 1 dΩdΩ= = dxdx dx dx dx = dΩ = −gdΩ , dS = dΩ i i = dΩ dx dx dx = J J dΩ = −gdΩ , dS ∂ xi ∂ xi

52

Chapter 4. Gravitation fields

Chapter 4. Gravitation fields

52

Interpretation: There is a convex quadratic 12 u|| · ||2 that lower bounds f.

Definition 4.2.4 — Covariant differentiation. In Galilean coordinates the defferentials dAi

of a vector Ai form a vector,and derivatives ∂∂ Axki form a tensor.In In curvilinear curvilinear coordinates coordinates the ∂ Ai transformation wouldn’tdo doso. so.dAi is not a vector and ∂ xk is not a tensor.The reason is that dAi transformation wouldn’t being the defference of vectors at different points of space transfom in space differently.Is It comes from the fact that transformation coefficients are also the functions of coordinates. 









2 k   ∂ x ∂x k i ∂x k  ∂x k ∂x k  Ai = A ⇒ dA = dA + A d = dA + A dxl i k k ∂ xi k ∂ xi k ∂ xi ∂ xi k ∂ xi ∂ xl

So the difference of any vector transformation into neighboring point consists on two parts : the fist one due to parallel transformation and the second due the the appearing dependence of transfomation coefficients from curvature of space DAi = dAi − δ Ai ⇒ δ Ai = −Γikl Ak dxl 

Example 4.1

  i k k i i Γk Bl dxm − Bk Γi Al dxm δ Ai Bk = lm lm  Aimδ Bk + Bmkδ Ai =  −A ik l δ A = − A Γml + A Γml dx ⇒⇒ DAik = dAik − δ Aik l ⇔ Aik dxl = ∂ Alk + Γi Amk + Γk Aim dAik − δ Aik ≡ Aik ml ml ;l dx ;l ∂ xl ∂ Ai

i i m l Aik;l dxl = ∂ xlk − Γm kl Am + Γml Ak ⇔ Ai;kl dx = (Ai Bk );l = Ai;l Bk + Ai Bk;l

∂ Aik ∂ xl

m − Γm il Amk + Γkl Aim 

4.2.1

Einstein tensor Definition 4.2.5 infinitesimal closed closedcontour. contour.The find the Definition 4.2.5— —Tensor TensorRicci. Ricci.Let Let us us take take any infinitesimal Thenwe we find A around aroundthis thiscontour: contour: change of vector A 

∂ Ak = Γikl Ai dxl , dxi → d f ki  ∂ xk

   Ai dxi = d f ki ∂∂ Axki = 12 d f ki ∂∂Axik − ∂∂ Axki √   0 1 2 3 dΩ → 1J dΩ  = −gdΩ, dΩ = dx dx dx  dx  i i ∂ (Γikm Ai ) ∂ (Γikl Ai ) lm = 1 ∂ (Γkm ) A − ∂ (Γkl ) A + Γi ∂ (Ai ) − Γi ∂ (Ai )  f Ak = 21 − i i km ∂ xl kl ∂ xm 2 ∂ xm ∂ xm ∂ xl ∂ xl 

Definition 4.2.6 ∂ Ai = Γnil An  ∂ xl i + Γinl Γnkm − Γinm Γnkl

Ak = 12 Riklm Ai  f lm Riklm =

R

∂ Γikm ∂ xl

−

∂ Γikl ∂ xm

Riemann was the eldest son of a poor pastor, the second of six of his children. He was able to start attending school only at the age of 14 (1840). Riemann’s mother, Charlotte Ebelle, died of tuberculosis while he was still at school; his two sisters died from the same disease and, subsequently, he himself would die. Riemann all his life was very attached to his family. The inclinations toward mathematics were manifested in the young Riemann as a child, but, yielding to the wishes of his father, in 1846 he entered the University of Gottingen to study

4.2 Cristoffel symbols and metrics

53

4.2 Cristoffel symbols and metrics

53

4.2 Cristoffel symbols and metrics

53

philology, philosophy and theology. However, fascinated by Gauss lectures, the young man made the final decision to become a mathematician. philology, philosophy and theology. However, fascinated by Gauss lectures, the young man In the 1847, to the University of Berlin, where they taught Dirichlet, Jacobi and made finalRiemann decision moved to become a mathematician. Steiner. In 1849, he returned to Gottingen [8], where he met William Weber, who became his In 1847, Riemann moved to thea University where friend they taught Dirichlet, Jacobi and teacher and close friend; year later, of heBerlin, got another - Richard Dedekind. Steiner. In 1849, he returned to Gottingen [8], where he met William Weber, who became his In 1851, Riemann "Thefriend Foundations the Theory of Functions of a teacher and close friend; defended a year later,his he thesis got another - Richardof Dedekind. Complex Variable"; his supervisor was Gauss, who highly valued the talent of his student. In In 1851, Riemann defendedwas hisintroduced thesis "Thefor Foundations of the Theory of Functions the thesis, the concept the first time, later becoming known of as aa Riemann Complex Variable"; his supervisor was Gauss, who highly valued the talent of his student. In to qualify surface. In 1854–1866, Riemann worked at the University of Göttingen. In order the for thesis, the concept was introduced for the first time, later becoming known as a Riemann the position of extraordinary professor, Riemann, under the statute, was to speak before surface. In 1854–1866, Riemann worked at the University of Göttingen. In order the faculty. In the fall of 1853, in the presence of Gauss, Riemann readtoa qualify historical report for“On the position of extraordinary professor, Riemann, under the statute, was to speak before the hypotheses underlying geometry”, from which Riemannian geometry originates. The the report, faculty.however, In the fall 1853, presence Gauss, Riemann read a historical didofnot helpin- the Riemann wasofnot approved. However, the text ofreport the speech was “Onpublished the hypotheses fromthis which (albeitunderlying very late,geometry”, in 1868), and wasRiemannian a landmarkgeometry event fororiginates. geometry.The Nevertheless, report, however, not help by - Riemann was not approved. However, the of the speech Riemann wasdidaccepted the privat-docent of the University of text Gottingen, wherewas he reads the published very late, in 1868), and this was a landmark event for geometry. Nevertheless, course(albeit of abelian functions. Riemann was accepted by the privat-docent of the University of Gottingen, where he reads the In 1857, Riemann published classical works on the theory of abelian functions and the analytic course of abelian functions. theory of differential equations and was transferred to the post of extraordinary professor at the In 1857, Riemann published classical works on the theory of abelian functions and the analytic University of Gottingen theory of differential equations and was transferred to the post of extraordinary professor at the University of Gottingen

Energy flux:    E 2 +H 2 → Energy flux: − → − ∂ kin dV = − S d f Ei ∂t 8π + ∑  → − → − ∂ → − E 2 +H 2→ −+ ∑ E→ −ikin i dV = − S d f ∂t S =8πc E ×H 4π → − − → − i c → S = 4π E × H

Definition 4.2.7 parts: Definition 4.2.7— —Energy Energymomentum momentumtensor. tensor.The Theaction actionfunction functionSSconsist consistofofthre three parts: Definition S4.2.7 = S—+Energy S + Smomentum tensor. The action function S consist of thre parts: f

m

m f



qk k − Aki dxi , k c ds, Sm f = ∑k c S =SSmf = + Sm∑+k Sm mikf  qkdxdydz k i dΩdt, Sm S=f −=∑k mFkikc F ds, Smk f dΩ = ∑= k c Ai dx ,  ik S f = Fik F dΩdt, dΩ = dxdydz

54

Chapter 4. Gravitation fields

Chapter 4. Gravitation fields

54 R

Lagrange made a significant contribution to many areas of mathematics, including calculus of variations, the theory of differential equations, solving problems on finding maxima and minima, number theory (Lagrange’s theorem), algebra, and probability theory. The formula for finite increments and several other theorems are named after him. In two important works, Theory of Analytic Functions (Théorie des fonctions analytiques, 1797) and On Solving Numerical Equations (De la résolution des équations numériques, 1798), he summarized everything that was known about these issues. in his time, and the new ideas and methods contained in them were developed in the works of mathematics. Langrangian density is equal: L =



LdΩ

Definition 4.2.8 — The Christoffel symbols and metric tensor relation. We remind you,that:

DAi = dAi − δ Ai i k l δ Ai = −Γ  kli A dx  i DAi = ∂∂ Axl + Γkil Ak dxl ⇒ Ai;l = ∂∂ Axl + Γkil Ak   i i DAl = ∂∂ Axl − Γkil Ak dxl ⇒ Ai;l = ∂∂ Axl + Γkil Ak DAi = Ai;l dxl , DAl = Ai;l dxl

In similar way we may express derivative for any tensor: DAik = dAik − δ Aik  l lm k mk i δ Aik = −  A Γml + A Γml dx  DAik = DAik =

∂ Ai + Γiml Amk + Γkml Alm ∂ xl l Aik ;l dx

dxl ⇒ Aik ;l =

∂ Ai ∂ xl

+ Γiml Amk + Γkml Alm

Definition 4.2.9 — Metric tensor property.

DAi = gikDAk , Ai = gik Ak DAi = D gik Ak = gik DAk + Ak Dgik m Dgik = 0 ⇒ gik;l = ∂∂gxikl − gmk Γm il − gim Γkl = ∂ gik = Γk,il + Γi,kl ∂ xl ∂ gii = Γi,kl + Γl,ik ∂ xk ∂ gkl − ∂ xi = −Γl,ki − Γk,li

∂ gik ∂ xl

− Γk,il − Γi,kl = 0

Definition 4.2.10 — Christoffel symbols and metric tensor.

Γikl =





∂ gik + ∂∂ gxilk − ∂∂gxkli ↔ Γi,kl = Γi,lk ∂xl  ∂ gml ∂ gkl 1 im ∂ gmk ⇐ Γikl = gim Γm,kl g + − m l k 2 ∂x ∂x ∂x

Γi,kl =

1 2

4.2 Cristoffel symbols and metrics 4.3 Einstein field equations

55

55

Theorem 4.2.1 — The Christoffel symbols and metric tensor relation.

g = |gik | dg = ggik dgik = −ggik dgik 1 ∂g Γikl = 12 gim ∂∂gximk ⇒ Γiki = 2g ∂ xk √ ∂ ( −ggik ) 1 kl i g Γkl = − √−g ∂ xk

√ ∂ Ai i Al = ∂ Ai + Ai ∂ ln −g + Γ i i li ∂x ∂x ∂ xi √ i 1 ∂ −gA i √ A;i = −g ∂ xi √ ∂ −gAk Aki;k = √1−g ∂ xi i − 12 ∂∂gxkli Akl

Ai;i =

Theorem 4.2.2 — Maxwell’s Equations in Curved Space-Time.

∂µ F µν = jν , 

Fµν = ∂µ Aν − ∂ν Aµ

Example 4.2 — Maxwell’s Equations in Curved Space-Time. The action of electromagnetic

field equals: 



S = 14 F µν Fµν dx + j µ Aµ dx   δ S = 14 F µν ∂µ δ Aν dx + j µ δ Aµ dx ∂µ ∂ µ Aν − ∂ ν ∂µ Aµ = jν , ⇒ ∂µ Aµ = 0 (T he Lorentz gauge) ν − D Dµ A ∂µ ∂ µ Aν = jν ⇒ Dµ Dµ A µ µ  √ µρ gνσ F ν √1 ∂µ −gg = j ρσ −g   √ √ S = 14 Fµν Fρσ gµρ gνσ −gdx + j µ Aµ gdx 

Definition 4.2.11 — The vector change evaluation after parallel displacement around any infenitesimal contour. 

Ak =  Γikl Ai dxl , δ Ai = ΓnilAn dxl , ∂∂ Axli = Γnil An ∂ (Γikm Ai ) ∂ (Γi Ai ) Ak = 12 − ∂ xklm  f lm = ∂ xl   i ∂ (Γikl ) ∂ (Ai ) ∂ (Ai ) 1 ∂ (Γkm ) i i Ai − ∂ xm Al + Γkm ∂ xl − Γkl ∂ xm  f lm =2 ∂ xl Ak = 12 Riklm Ai  f lm Riklm =

4.3

∂ Γikm ∂ xl

−

∂ Γikl ∂ xm

+ Γinl Γnkm − Γlnm Γnkl

Einstein field equations



Ak =  Γikl Ai dxl , δ Ai = ΓnilAn dxl , ∂∂ Axli = Γnil An ∂ (Γikm Ai ) ∂ (Γikl Ai ) Ak = 12 −  f lm = l ∂ xm ∂x   i ∂ (Γikl ) ∂ (Ai ) ∂ (Ai ) 1 ∂ (Γkm ) i i Ai − ∂ xm Al + Γkm ∂ xl − Γkl ∂ xm  f lm =2 ∂ xl Ak = 12 Riklm Ai  f lm

56 4.3

∂ Γikm ∂ xl

Riklm =

−

∂ Γikl ∂ xm

+ Γinl Γnkm − Γlnm Γnkl

Einstein field equations

Chapter 4. Gravitation fields

Chapter 4. Gravitation fields

56

Definition 4.3.1 — Properties of the Curvature Tensor .

Riklm = −Rikml Riklm + Rimkl + Rilmk = 0 Riklm = ginRnklm Riklm =

1 2

∂ 2 gim ∂ xk ∂ xl

   2 2 2 p − Γnkm Γilp + ∂∂xl ∂gklxm − ∂∂xk ∂gilxm − ∂∂xig∂kmxl + gnp Γnkl Γim

Riklm = −Rkilm = −Rikml Riklm = Rlmik Riklm + Rimkl + Rilmk = 0

Rnikl;m + Rnimk;l + Rilm;k = 0 ⇒ Rnikl;m = Rik = gim Rlimk = Rlilk ∂ Γl

Rik = ∂ xikl − Rik = Rki

∂ Γlil ∂ xk

∂ Rnikl ∂ xm

=

∂ 2 Γnil ∂ xm ∂ xk

∂ 2 Γn

− ∂ xm ∂ikxl

m l + Γiik Γm lm − Γil Γkm

Definition 4.3.2 — Energy Momentum Tensor.

S=

1 c

δS =

 1 c

√ L −gdΩ   √



1√ 2 −gTik

∂ −gL  ∂ gik



=

√ ∂ −gL ∂ gik ∂ ∂ xl √ ∂ −gL ∂ gik ∂ ∂ xl

δ gik −

√ ∂ −gL ∂ gik

−

   ik

δ ∂∂gxl

 

δ gik dΩ

√ 1  δ S = 2c Tik δ gik −gdΩ 1 1 L = − 16π Fik F ik = − 16π Fik F ikgil gkm  1 1 l Tik = 4π −Fil Fk + 4 Flm F lm gik Tik = (P + E) ui uk − pgik 4.4 Black holes and Schwarzschild metric

57

Definition 4.3.3 — The gravitational field equations.  √ ∂ Γlik ∂ Γlil l m m l Sg = R −gdΩ, ⇒ R = gik Rik , Rik = l − ∂ xk + Γik Γlm − Γil Γkm   √  ik √   ∂ x√ √ √ ik ik δ R −gdΩ = g Rik −gdΩ = Rik −gδ g + Rik g δ −g + gik −gδ Rik dΩ √ √ δ −g = − 2√1−g δ g = − 12 −ggik δ gik    √   √ √ δ R −gdΩ= Rik − 12 gik R δgik −gdΩ + gik δ Rik gdΩ l ∂ δ Γlik − ∂∂xk δ Γlil = gik ∂∂xl δ Γlik − gil ∂∂xl δ Γkik = ∂∂wxl ∂ xl wl = gik δ Γilk − gil δ Γklk  √ gik δ Rik = √1−g ∂∂xl −gwl  ik  ∂ (√−gwl ) √ g δ Rik −gdΩ = dΩ l ∂ x  √ c3  1 c3  √ δ Sg = − 16πk Rik − 2 gik R δ gik −gdΩ ⇔ S = − G −gdΩ 16πk 

gik δ Rik = gik

δ Sg =

c3 − 16πk

   ∂ (G√−g)  

  Rik − 12 gik R = 1



∂ gik

√1 −g

√

  

−

√ ∂ (G −g)  ∂ gik





 √ ∂ ∂ (G −g) δ gik dΩ l ∂x ∂ gik   ∂ ∂ xl  −



 √ ∂ ∂ (G −g) ∂ xl ∂ gik   ∂ ∂ xl

g √δ Rik = g ∂ xl δ Γik − ∂√ δ Γil = g ∂ xl δ Γik − g ∂ xl δ Γik = ∂ xl xk − 12 −ggik δ gik δ l −gik= −i 2√1−gilδ g = k w = √ g δ Γlk − g δΓlk √ − 1l g R δ gik √−gdΩ +  gik δ R √gdΩ δgik δRR −gdΩ ik 2 ik = √1 =∂ Rik−gw  ik

−g ∂ xl

l

∂w ik ∂ l il ∂ k l l g ik δikRik =√gik ∂∂xl δ Γlik ∂−(√∂∂−gw )il = g ∂ xl δ Γik − g ∂ xl δ Γik = ∂ xl k δΓ x g δ Rik −gdΩil = k dΩ l  g δ Γlk1 ∂ x ik √ wl = gik δ Γc3ilk − c3  √   √ δikSg = − 16πk1  − g R δ g G −gdΩ −gdΩ ⇔ S = − 16πk R ik ik l ∂ 2  g δ Rik = √−g ∂ xl −gw    Schwarzschild  √ ∂ (√−gwl ) ∂ (G√−g) √c3 and 4.4 Black metric ik δ Rholes ∂= (G −g) ∂ dΩ −gdΩ g ik − δ gik dΩ δ Sg = − l ik l ∂ x 16πk ∂ x ik √  ∂ g 1  ∂ gik  c3   c3  √ δ Sg = − 16πk Rik − 2 gik R δ g ∂ −gdΩ G −gdΩ ⇔ S = − 16πk  l ∂ x      √   √ ∂ (G  −g) √ ∂ ∂ (G −g) √ ik c3  δ S = − ∂ (G −g) ∂ (G δ −g) − dΩ g 1 16πk 1 ∂ g ik l ∂ x − ik Rik − 2 gik R =√−g∂ g    ∂ gik ∂ ∂∂gxl   ∂ gik   ∂ xl ∂ ∂ xl    √   1  √  δ Sm = 2c Tik δ gik −gdΩ √ ∂ (G −g) ∂ (G −g)   1 1 ∂  √ 3 √ c − gik R =1 8πk ik − R ik ik l − 16πk 2 Rik − 2 gik−g 0 R− δ g ∂ x −gdΩ ∂T g  ∂ gik =  c4 ik  ∂ Rik − 12 gik R = 8πk T ik ∂ xl c4 √ 1  δ Sm = 2c Tik δ gik −gdΩ  ik √  c3  − 16πk T δ g −gdΩ = 0 Rik − 12 gik R − 8πk ik 4 c T Rik − 12 gik R = 8πk c4 ik

4.4

57

Black holes and Schwarzschild metric A black hole is a region of space-time, the gravitational attraction of which is so large that even

4.4 objects Black moving holes and metric at theSchwarzschild speed of light, including the quanta of the light itself, cannot leave it. The

boundary of this area is called the event horizon, and its characteristic size is the gravitational radius. In black the simplest of a spherically symmetric black hole, attraction it is equal to Schwarzschild In A hole iscase a region of space-time, the gravitational of the which is so large radius. that even (t, r, θ , ϕ), which the lastlight 3 areitself, similarcannot to the spherical, the the so-called Schwarzschild objects moving at the speedcoordinates of light, including theofquanta of the leave it. The metric tensor of the most physically important part of the Schwarzschild space-time with topology boundary of this area is called the event horizon, and its characteristic size is the gravitational radius. (the of case a two-dimensional two-dimensional has the form In theproduct simplest of a sphericallyEuclidean symmetricspace blackand hole, it is equal to thesphere) Schwarzschild radius. In (t, r, θ , ϕ), of which the last 3 are similar to the spherical, the theDefinition so-called Schwarzschild coordinates 4.4.1 — Schwarzschild metric. metric tensor ofthe  most  physically important part of theSchwarzschild space-time with topology 1 − rrs 0 Euclidean 0 space 0and two-dimensional  (the product of  a two-dimensional sphere) has the form       rs −1 0 0 0 1 − r Definition 4.4.1 — Schwarzschild metric. gik =   0 0 0 −r2       2  rs 2 0 0 0 −r sin θ 1 − 0   r      2−1    ds = 1 − rr0s c2 dt 2 − 0 2 + dθ 2 , rs = 2GM 1 − rrsdrrs  − r02 sin2 θ dϕ c2 gik = 1− 2   0 0 0 −r   r 58 Chapter 4. Gravitation fields   0 0 0 −r2 sin2 θ   2 2  2  2 2 + dθ 2 , r = 2GM  − r 2 sin c dt −  drrChristoffel 1 − rrs independent θ dϕ Then ds the=non-zero symbols have the form: s c2 s 1−

r

g00 = eν , g11 = −eλ Γ111 = 12 ∂∂λr , Γ010 = 12 ∂∂νr , Γ233 = − sin θ cos θ , Γ011 = 12 ∂∂tλ eλ −ν , Γ122 = −re−λ , Γ100 = 12 ∂∂νr eν−λ , Γ212 = Γ313 = 1r , Γ323 = ctgθ , Γ000 = 12 ∂∂tν , Γ110 = 12 ∂∂tλ , Γ133 = −r2 (sin2 θ )e−λ

Fact 4.4.1 — Einstein equation exact solution. Schwartzschild metric is the only spherically

symmetric exact solution of Einstein’s equations without a cosmological constant in empty space due to the Birkhof theorem. In particular, this metric quite accurately describes the gravitational field of a solitary non-rotating and uncharged black hole and the gravitational field outside the solitary spherically symmetric massive body. Named in honor of Karl Schwarzschild, who first discovered it

Γ110 =

1 ∂λ 2 ∂t

,

Γ133 = −r2 (sin2 θ )e−λ

Fact 4.4.1 — Einstein equation exact solution. Schwartzschild metric is the only spherically

symmetric exact solution of Einstein’s equations without a cosmological constant in empty space due to the Birkhof theorem. In particular, this metric quite accurately describes gravitational field 58 Chapter 4. the Gravitation fields of a solitary non-rotating and uncharged black hole and the gravitational field outside the solitary spherically symmetric massive body. Named in honor of Karl Schwarzschild, who first discovered it

4.5

Standard model and Higgs bosons The Boson Higgs, the Higgs Boson, is an elementary particle, a quantum of the Higgs field, necessarily arising in the Standard Model of particle physics due to the Higgs mechanism of spontaneous breaking of electroweak symmetry. Its opening completes the Standard Model. Within this model, it is responsible for the inert mass of such elementary particles as bosons. With the help of the Higgs field, the presence of an inert mass of carrier particles of a weak interaction (W and Z bosons) and the absence of mass in a carrier particle of a strong (gluon) and electromagnetic interaction (photon) is explained. By construction, the Higgs boson is a scalar particle, that is, it has zero spin. 4.5 Standard model and Higgs bosons

R

Graviton is a hypothetical massless elementary particle - carrier of the gravitational interaction and quantum of the gravitational field without electric and other charges (however, they have energy and therefore participate in the gravitational interaction). It must have spin 2 and two possible directions of polarization. Always moving at the speed of light. The term “graviton” was proposed in the 1930s, often attributed to the work of 1934 by DI Blokhintsev and F. M. Galperin. (especially the Standard Model) in modeling the behavior of other fundamental interactions using similar particles: photons in the electromagnetic interaction, gluons in the strong interaction, W ±, Z-bosons in the weak interaction. Following this analogy - a certain

59

4.5 Standard model and Higgs bosons

R

Graviton is a hypothetical massless elementary particle - carrier of the gravitational interaction and quantum of the gravitational field without electric and other charges (however, they have energy and therefore participate in the gravitational interaction). It must have spin 2 and two possible directions of polarization. Always moving at the speed of light. The term “graviton” was proposed in the 1930s, often attributed to the work of 1934 by DI Blokhintsev and F. M. Galperin. (especially the Standard Model) in modeling the behavior of other fundamental interactions using similar particles: photons in the electromagnetic interaction, gluons in the strong interaction, W ±, Z-bosons in the weak interaction. Following this analogy - a certain elementary particle can also be responsible for the gravitational interaction.

59

5. Electromagnetic fields

5. Electromagnetic fields 5. Electromagnetic fields magnetic fields fields 5. Electromagnetic fields magnetic 5.1

Lagrangian of charged moving particle 5.1.1 Non interacting free particle 5.1 Lagrangian of charged moving particle ingparticle particle5.1 Lagrangian ng of charged moving particle Definition 5.1.1 Lagrangian of free moving non interactingfree particle 5.1.1 Non interacting particle

 Definition 5.1.1 Lagrangian of free moving non interacting particle 5.1.1 Non interacting dt 2 −particle dx2 − dy2 − dz2 ≡ inv ds = c2free emoving movingnon noninteracting interacting particle   bparticle particle 22   Definition 5.1.1 Lagrangian of free moving non ds interacting 2 − dx2 − dy2 − dz2 ≡ inv c2 dt S = ττ1122 Lγdτ, L = − mcγ , S = ττ1122 −mc2 dτ ==−mc a ds   τ2  → −  2 2 2 → − → − → − ∂ L m v mc −dzdz ≡≡inv inv 2 2 ,− 2H 2p≡· inv S = τ1 v22Lγdτ, L = − mcγ , S = ττ12 −mc2 dτ = −mc ab ds → −vc2= p = E = = v − L = , dt − dx dy − dz ds  ∂     v2 −v 2 dτ = −mc b b ds  τ2  b m→ → −p =1−∂vccL22 = −p · → −v − L =  mc ,  −mc2 dτ S== τττ2τ12−mc mc2 2 dτ = −mc a ads S =  τ2 Lγdτ, ,E = H = → → −v a  1−L = − , S = −mc = −mc ds 1  2 ∂ τ τ γ 1 1 2 2  1− v2 → − → − mc c −pp· → − −v c  v ·vv−−LL==mc 2v2, , → −p = ∂ L = m→ → − → − mc ==→  1− ,E = H = p · v −L = , −v  2 1−v 2 ∂→ 1− c2  v2 1− v2 c2c µ −p ), c We may assemble theenergy and momentum in a momentum four-vectorp = ( Ec , → pµ = 1− 2 → − E c ( c ,− p ) We may assemble the energy and momentum in a momentum four-vecto − → V (x) → → −p ), −p ) µ = (EE → E − µ omentum in a momentum four-vectorp , p = − wherepV(x) −→ mentum in a momentum four-vectorp =field (energy p ), and The may electromagnetic be pdescribed a four vector fieldfour-vectorp Aµ (x) = ( µc = , A( E(x)) µ µ= ( c ,by We assemble the momentum in a momentum c c,might µ = c , p ), − scalar A(x) is the magnetic vector potential.In case actionequal: The electromagnetic fieldthis might bethe described by a four vector field Aµ (x) = , −→ p ) potential and (isEcthe − → −→ µ (x) = V → − V (x)magnetic scribedby bya afour fourvector vector field ( V(x)(x) ,AA(x)) (x)) where V(x) potential andAµA(x) vector potential.In this case  field  is the scalar cribed field AAµ (x) where V(x) The electromagnetic might be described vector (x) =is(the  = ( c c ,field → −four t22 −mc22 by a → − c , A (x)) where V(x) µ S = −mcds − qA dx = + q A v − qV dt µ magnetic vector potential.In this case the actionequal: magnetic vector potential.In this case the actionequal:  action2 equal: t11 magnetic is the scalar potential and A(x) is the vectorpotential.In Inthis thiscase casethe the  t2 actionequal: → −→ γ → → − −v − qV dt −mc µ − −  S→ → − −→ − −qAµ dx = −mcds + q A → −p = ∂ L = γm→ −v + q→ ∂ A = t1  µ   2 2 → A , A = V, A , B = ∇ ×  A , E = ∂t − ∇V γ −−→ −v −→ − qV dt −mc  ∂→ → − −mc −→ − − → → − → − − → −v −∂qV µ = t2 −mc2 + q→ ++qqAA→ vv−−qV dt − → −v + q→ ∂A L µ = V, → γ S = −mcds − qA dx A dt γ µ p = = γm A , A A , B = ∇ × A , E = → −  t = ∂ A − ∂ A , F γ µν µ ν ν µ  →  1 → − ∂t − ∂ v − − → −→  − − ∂→ −→ → −→ −→ −→ − ∂AA  1 → 3 − E 2 µ E 3  → − → → − 1→ − 0 −E − −E∂2→ = − ∇V − → − ∂ L A 0 E −E == V,V,AA , , BB==∇∇××AA, ,EE→ = − ∇V AνA−, ∂Eν A=µ , ∂t − ∇V p = B = ∂ ∇µ× ∂t∂t∂ → −v = γm v + q A , A = V, A ,Fµν   1 3 2  1 3 E 2B2 E 3  E 1 0 0 E−B −E 0 −B B 0 −E 1 −E 2 − µν     = F , F = ∂ A − ∂ A ,    µν µ ν ν µ    2 3 1 2 3 1     3 3  2  E−EB1 1 0 0 2 −B −E 33  1 −E B 0 −B3  −B −E1 1 −E −E2 2 −E 0 −B3 00 −E EE3 0 0 3 E 12 E 12 −E 3 B 1  , F µν = E 2 E = F 3 −E 2 −E 1 µν 2 1 3 2 2 3 3    1 3 2 2 −E −B B 0 E −B B 0     −B  B−B3 0B2  −B E B 0 − 1 EE 00 −B BB BB   1 −E  , Fµνµν== −E 0 −B3 B2     , F µν = E −E03 −B 0 µν=12(B 1 , F 2 3 Fµν 1 2 − E 2) 2 1 3 −B2 1  1 2 3 =    F −B E B 0 −B B E B E B 0 −B −B E 2 B3 −E 2 B3 0 −B1  0 −B1  3 −B µν = 2(B2 − E 2 ) 2 2 BB 11 0 3 −B2 B1 F F 3 2 1 EE3 −B 0−E 00 µν 0 E −B B 0 Charge density and current density are also should be included: Fµν F µν = 2(B2 − E 2 ) Charge density and current density are also should be included:  → ∂ρ − → − µ µ ty are also should be included: y are also should be included: = 0, density J = ρ, + ∇ J = ∂µ J current Charge areJalso should be included:  → ∂t density and ∂ρ − → − + ∇ J = ∂µ J µ = 0, J µ = ρ, J   →   → − −→ ∂ρ → − −  ∂t ρ,ρ,JJ + ∇ J = ∂µ J µ = 0, J µ = ρ, J ∂t

61

Chapter 5. Electromagnetic fields

62

62



Chapter 5. Electromagnetic fields

Example 5.1



Lagrangian and equations of charged particle motion

S = Sp + Sp f  S p = −mc ds,  →  ϕ − α A = c, A , b



b α S p f = −q α dx  a A→  − Aα = ϕc , − A

 b → − → −v − qϕ dt q A · a

α a (−mcds − qAα dx ) , S p f =    −− 2 → v − qϕ dt, γ = S = tt12 − mcγ q A →

S=

2

L=

− mcγ

+

{ f ree particle}

 1   v2  1− c2

→  −− q A→ v −ϕ

Interaction with electromagnetic f ield

Equations of motion





d ∂L ∂L 0 dt ∂ v − ∂ r =   → −→ −v − q∇ϕ ∂L = ∇L = q∇ A −r ∂→    

 − − − → → − → − → − − → − → −v · ∇) → −v ) v + (→ A + −v × ∇ × A + A × (∇ × → ∇ A ·→ v = A·∇ →

=0 − → −p + q→ ∂L = A → − ∂ v  − − − −p + q→ → −v · ∇) → → −v × ∇ × → d → A = q ( A + q A − q∇ϕ dt   → − → − → − −v · ∇) A q dtd A = q ∂∂tA + (→    → − −p − d→ → −v × ∇ × → ∂A + q A = −q − q∇ϕ dt ∂t no dependence on v → − → − → − → − → − − → − −v × → E = − ∂∂tA − ∇ϕ, B = ∇ × A ⇒ ddtp = q E + q→ B

=0

Theorem 5.1.1 — Relativistic mechanics only energy conservation law is valid. General

expression for Lagrangian for free moving object with a mass m equals:   2 2 L = −mc 1 − uc2 → p = ∂∂ Lu 2  mu , E =  mc   2  u  u2   1− 1− 2 Gharged free c2 moving cparticle

p=

5.2

63

mc2 = ∑i mi c2

⇒ m = ∑i mi   T he rest energies o f its constituent particles ∑ mc2 i     T he kinetic energy o f particles E (Full energy) =     T he interaction energy ..............

5.2

Gharged free moving particle Definition 5.2.1

0i = F 0i , i i jk jk E i = −F   B = −ε F ∂L ∂ Aµ

− ∂k

∂L ∂ (∂k Aµ )

= 0,

5.2 Gharged free moving particle

63

mc2 = ∑i mi c2

⇒ m = ∑i mi   T he rest energies o f its constituent particles ∑ mc2 i     T he kinetic energy o f particles E (Full energy) =     T he interaction energy 5.2 Gharged free moving particle ..............

5.2 Gharged free moving particle 0i = F 0i , i i jk jk E i = −F   B = −ε F

Definition 5.2.1

∂L ∂ Aµ

− ∂k

∂L ∂ (∂k Aµ )

= 0,

Definition 5.2.2 — Maxwell equations.

→ − → − L = − 14 Fµν F µν ⇒ ∂k F kµ = 0 → ∇ E = 0, ∇ × B = L = − 14 Fµν F µν − J µ Aµ ⇒ ∂∂ALµ = −J µ , → − → − → − → − ∂k F kµ = J µ → ∇ E = ρ, ∇ × B = J + ∂∂tE 5.2.1

→ − ∂E ∂t

Gauge invariance Definition 5.2.3 — Lorentz gauge.

 → − − → A = A +∇f  ϕ = ϕ − ∂∂tf      S p f = −q ab Aα dxα = −q ab Aα − ∂∂xfα dxα = −q ab Aα dxα → − ∇ · A + c12 ∂∂tϕ = 0 ⇒ T he Lorentz gauge α ∂A ∂ xα = 0

→ − → − −r ) ⇒ A = A + ∇ f (t, →

Definition 5.2.4 — The electromagnetic field tensor.

S=

b a

δS = δ

(−mcds − qAα dxα ) b

α a (−mcds − qAα dx ) = 0, δ ds = dxα d





δ xα ds



δ S = − ab (mduα dδ xα + qdAα dδ xα − qδ Aα dxα ) = 0  δ S = ab (mduα dδ xα + q ∂∂Axβα dxβ dδ xα − q ∂∂Axβα dxβ dxα ) = b

∂ Aβ ∂ Aα α β α α dδ x  + q ∂ xβ dx dδ x − q ∂ xα a (mdu   ∂ Aα ∂ Aα β dτδ xα α = ab m du dτ + q ∂ xβ − ∂ xβ u ∂A Fαβ = ∂ xαβ − ∂∂Axβα   β dτδ xα α δ S = ab m du − qF u αβ dτ

=

Fαβ = −Fβ α

dxα dxβ ) =

= uα d



δ xα c



63

64

Chapter 5. Electromagnetic fields

Chapter 5. Electromagnetic fields

64 R α

= qF αβ uβ m du dτ  → − Aα = ϕc , − A → F10 = ∂∂ Ax10 − ∂∂ Ax01 =   Ey Ez Ex 0 c c c  Ex  0 −Bz By  − Fαβ =  Ecy  − c Bz 0 −Bx  Ez 0 − c −By EBx Ez y Ex 0 −c −c −c  Ex  0 −Bz By   F αβ =  Ecy  c Bz 0 −Bx  Ez −By Bx 0 c

1 ∂ϕ c ∂x

+ 1c ∂∂tAx = − Ecx

Definition 5.2.5 — Lorentz transformation for electromagnetic fields.

Aα = ϕ=

 



ϕ +VAx    V2  1− c2 



− ϕ → c, A

→ −  K ⇒ V = Vex ⇒ K V   Ax + ϕ   2 , Ax =  c V 2 , Ay = Ay , Az = Az ,

Ex = Ex , Ey =

1−





E +V Bz y 2 1− V2

c2

, Ez =

c





E +V By z 2 1− V2 c

V   V  Bz − Ey Ez  2 c c2 Bx = Bx , By =  , Bz =     V2  V2   1− 1− 2 c c2      V Ex , Ey = Ey +V Bz , Ez = Ez −V By c  1  ⇒ Ex =     Bx = Bx , By = By − cV2 Ez , Bz = Bz + cV2 Ey → − → − → − → − → − → − → − → −     E = E + B × V , B = B − c12 E × V − → − → − → − → → − − → − → − → − → − → β = Vc ⇒ E = E + c B × β , B = B − 1c E × β    − K   K −→ → − → → − → − B =0 B = β Ec ⇒ →  − ,E ⊥ B − → − → E =0 E =cB × β  By −

R Fαβ F αβ = inv c2 B2 − E 2 = inv eαβ µν Fαβ Fµν = inv c2 B2 − E 2 = inv → − → − E · B = inv

5.2 Gharged free moving particle

65

5.2 Gharged free moving particle

65

Definition 5.2.6 — Energy density and energy flux of electromagnetic field.

→ → − − − − → − − → − − dW = F d → s = q E +→ u × B ·→ u dt = q E · → u dt   → → − − → − − q = Volume ρ(x, y, z)dxdydz → dW = ρ E ·→ u dxdydz = J · E dV dt   → − → − → − → − → − J · E = µ10 ∇ × B − µ0 ε0 ∂∂tE · E =  2 → − → − → − → − → − → − = µ10 ∇ × B · E − ∇ × E · B + µ10 ∇ × E · B − ε20 ∂∂tE = → 2 2 − → − = − µ10 ∇ E × B − 2µ1 0 ∂∂tB − ε20 ∂∂tE = →   − → − ∂ = − µ10 ∇ E × B − 12 ∂t ε0 E 2 + µ10 B2

Definition Poyting flux flux vector. Definition5.2.7 5.2.7—— Poynting vector.

→ − P = 

− → − 1 → µ0 E × B →  − → − − → − 1 1 2 + 1 B2 = −→ ε E ∇ E × B J · E − 0 2µ0 µ0 2  → − → − → − 1 ε0 E 2 + 2µ1 0 B2 + J · E = −∇ · P 2 

∂ ∂t ∂ ∂t

  1 2 + 1 B2 dV + ε E 2 0 2µ

∂ ∂t



0

f ield energy

ε

particle kinetic energy

− → − =− → P ·d S

Definition 5.2.8 — Momentum density.Maxwell stress tensor. d dt



(pm+ p f ) = Ti j n j dS

   Ti j = ε0 Ei E j δi j + µ10 Bi B j − 12 ε0 E 2 + µ10 B2 U Theorem 5.2.1 — Field of uniformly moving charge. 

ϕ(Laboratory f rame system − K) = γϕ(system o f re f erencemoving with velocity v − K ) ϕ= 

x =



 ϕ   V2  1− c2 x−V t  2

1− V2 c

=

e   V2  R 1− c2 

, y = y, 2



z =z

(x−V t)2 +(1− V2 )(y2 +z2 ) c R = V2 1− c2   → − → − V2 e R E = 1 − c2 R∗3  2 ∗ R = (x −V t)2 + (1 − Vc2 )(y2 + z2 ) → − → − → − → − → − B = 1c V × E = ec V R×3 R 2

66

Chapter 5. Electromagnetic fields

Chapter 5. Electromagnetic fields

66 Definition 5.2.9 — The electric dipole moment.

− qa  → → − ϕ = ∑a − → →  , R  ra − R − r  0 a − → − − → − − → F(R0 − → ra ) = F(R0 ) − → r gradF(R0 )

∑q

a

−r · grad 1 − ∑a qa → ϕ= a R0 → − → − d = ∑a qa r a ⇒ dipole moment o f the system o f qa charges charges i f ∑ qa = 0 → − → − → − ϕ = − d · ∇ R10 = d R· 3R 0 0 → → − → − − → − → −  −  → → − d ·R0 E = −grad R3 = − R13 grad d · R 0 − d · R 0 grad( R13 ) 0 0  → − − − 0 → → − n−d n·d → 3 → −  1 → − → − E = ⇒ E = d · ∇ ∇ R0 R3 a R0

0

5.3

Electromagnetic waves Definition 5.3.1 — The wave equation.

then I f density o f charges and electric currents are zero,then → − → − H≡ B → − → − → − ∇ × E = − 1c ∂∂tB , ∇ B = 0 → − → − → − ∇ × B = 1c ∂∂tE , ∇ E = 0 → − → − → − → − E = − 1c ∂∂tA , H = ∇ × A → − → − → − → − 1 ∂2 A ∇ ×∇× A = − A + graddiv A = − 2 2 c ∂t → − − ∂A ∂ → ∇ ∂t = ∂t ∇ A = 0 − → − 2→ ⇒ wave equation  A − c12 ∂∂tA2 = 0 Definition 5.3.2 — Plane waves. ∂ 2ψ ∂t 2 ∂ ∂t

2

− c2 ∂∂xψ2= 0  ∂ +c ∂ ψ = 0 − c ∂∂x ∂t ∂x ψ = ψ1 (t − xc ) + ψ2 (t + xc ) → − → − → − → − E = − 1c ∂∂tA , H = ∇ × A → − → → − H =− n ×E → − − → − c → E ×H = S (Poyting (Poyntingvector) vector) = 4π → − − c H 2→ c 2→ n = 4π n E − S = 4π

− c → 4π E ×

 → − → − n ×E

5.3 Electromagnetic waves

5.3 Electromagnetic waves

67

67

Definition 5.3.3 — Dipole radiation.

→ − → − A = cR1 0 ∑i qi V i → −˙ → − → − ∑i qi V = dtd ∑i qi r i = d → −˙ → − A = cR1 0 d → −˙ − → − n H = c21R d × → 0 → 2 −˙ → 1 dI = 4πc d ×− n dΩ 3 → − 2 2 − I = 2e a , → a = dv 3c3

dt

In 1864, the great Scottish theoretical physicist James Maxwell created a unified theory of the electromagnetic field. This theory is a generalization of the Gauss theorem, the law of electromagnetic induction, etc. It is presented in the form of 4 equations Maxwell. We formulate the main consequences of them. 1. The electric and magnetic fields are manifestations of a single electromagnetic fields. 2. A changing electric field generates a magnetic field, and a changing the magnetic field generates an electric field, and these phenomena interconnected. 3. Time-varying currents or accelerated charges may be sources of electromagnetic waves. Electromagnetic wave is called electromagnetic field propagating in space. 4. Electromagnetic wave is transverse. When propagating e / m wave oscillations of the electric field strength vector and magnetic induction vector occur. Moreover, these two vectors are mutuallyperpendicular and perpendicular to the velocity of propagation. R

Hertz experiments. Henrich Rudolf Hertz - German physicist. He graduated from Berlin University, where his teachers were Hermann von Helmholtz and Gustav Kirchhoff. From 1885 to 1889 he was a professor of physics at the University of Karlsruhe. Since 1889 Professor of Physics at the University of Bonn. The main achievement is the experimental confirmation of the electromagnetic theory of light by James Maxwell. Hertz proved the existence of electromagnetic waves. He studied in detail the reflection, interference, diffraction and polarization of electromagnetic waves, proved that the speed of their propagation coincides with the speed of propagation of light, and that light is nothing but a kind of electromagnetic waves. He built the electrodynamics of moving bodies, based on the hypothesis that the ether is carried away by moving bodies. However, his theory of electrodynamics was not confirmed by experiments and later gave way to the electronic theory of Hendrik Lorentz. The results obtained by Hertz, formed the basis of the creation of radio. In 1886-1887, Hertz first observed and described the external photoelectric effect. Hertz developed the theory of a resonant circuit, studied the properties of cathode rays, investigated the effect of ultraviolet rays on an electric discharge. In a number of works on mechanics, he gave the theory of impact of elastic balls, calculated the time of impact, etc. In the book Principles of Mechanics (1894), he drew general theorems of mechanics and its mathematical apparatus, based on a single principle (Hertz principle). Since 1933, the Hertz frequency unit is the Hertz frequency unit, which is part of the International System of Units (SI).

R

Radio Hertz (spark) In order to catch the emitted waves, Hertz came up with the simplest resonator — a non-closed wire ring or a rectangular open frame with the same brass balls as the “transmitter” at the ends and an adjustable spark gap. As a result of the experiments, Hertz discovered that if high-frequency oscillations occur in the generator (a spark jumps in its discharge gap), then in the discharge gap of the resonator removed from the generator even by 3 m, small sparks will also slip. Thus, the spark in the second circuit arose without any direct contact with the first circuit. Having conducted numerous experiments at different

68

68

Chapter 5. Electromagnetic fields

Chapter 5. Electromagnetic fields mutual positions of the generator and receiver, Hertz came to the conclusion that there are electromagnetic waves propagating at a finite velocity. Will they behave like light? Hertz conducts a thorough check of this assumption. After studying the laws of reflection and refraction, after establishing polarization and measuring the speed of electromagnetic waves, he proved their complete analogy with light. All this was set forth in the work "On the Rays of Electrical Power," published in December 1888. This year is considered the year of discovery of electromagnetic waves and experimental confirmation of Maxwell’s theory. Thanks to his experience, Hertz came to the following conclusions: Maxwell waves are “synchronous” (the validity of Maxwell’s theory that the speed of propagation of radio waves is equal to the speed of light); It is possible to transmit the energy of an electric and magnetic field without wires. In 1887, at the end of the experiments, the first article by Hertz “On very fast electrical oscillations” was published, and in 1888 - even more fundamental work “On electrodynamic waves in the air and their reflection”. Hertz believed that his discoveries were not more practical than the Maxwell ones: “This is absolutely useless. This is only an experiment that proves that Maestro Maxwell was right. We only have mysterious electromagnetic waves that we cannot see with the eye, but they exist. ” “And then what?” One of the students asked him. Hertz shrugged, he was a modest man, with no complaints and ambitions: "I guess - nothing"

6. Thermodynamics and Statistical Physics

6. Thermodynamics and Statistical Physics

6.1 6.1.1

Internal Energy Avogadro number Ideal Gas State Equation Presure ,Volume,Temperature.Ideal Presure, Volume, Temperature. Ideal Gas GasAvogadro Avogadro number number is is the the number number substance substance species species within within one matter and and equals equals to to6.0210 6.02102323.. One quantity of of matter matter corresponding corresponding its its one mole of matter One mole mole is the quantity position in Mendeleev periodical table taken in grams. For hydrogen, we take one 1 gram, for oxygen 16 grams grams and and for for uranium uranium isotope isotope 235 235 one one mole mole is is 235 235 grams. grams. From From this this we we may may easily easily evaluate evaluate the the 16 hydrogen atom mass: hydrogen atom mass: mH =

1g = 1.6610−24 g 6.021023

Second Newton’s law for an atom striking against the wall of the vessel which contains some amount of gas is equal: −p d→ → − F = dt Then let’s assume that vessel is a cube with leg equal Land the number of atoms in the vessel N. The simple evaluations then give the following: p = mVx − (−mVx ) = 2mVx 1 2 1 mV 2 2 t = 2L Vx ,Vx = 3 V , F = 3 L 2 mV 2 P = NF = 13 mV N = 13 Volume N L2 L3 Introduce temperature T of gas as:: mV 2 2

= 32 kT k = 1.3810−23 KJ

69

70

Chapter 6. Thermodynamics and Statistical Physics

Chapter 6. Thermodynamics and Statistical Physics

70

Finally, we come to ideal gas state equation: PV = NkT N = NA = 6.021023 PV = NA kT = RT PV = RT The internal energy has two major components, kinetic energy and potential energy. The kinetic energy is due to the motion of the system’s particles ( translations, rotations, vibrations), and the potential energy is associated with the static constituents of matter, static electric energy of atoms within molecules or crystals, the static energy of chemical bonds. The internal energy of a system can be changed by heating the system or by doing work on it. If you wll take into account ,that atoms or molecules of the gas within the vessel have some definite sizes and the inteactions due to electromagnetic forces give the presence of potential energy then you get Van-de-Vaalce equation: (P +

a )(V − b) = RT V2

Then the P-V diagramm also looks more realistic:

6.2

First Law of Thermodynamics Definition 6.2.1 — Heat. Heat Heat(joules) is energy that is spent or released in a process of energy

transfer between a system and its surroundings, other than as work or with the transfer of matter. In thermodynamics, finer detail of the process is unknowable. When there is a suitable physical pathway, heat flows from a hotter to a colder body. The pathway can be direct, as in conduction and radiation, or indirect, as in convective circulation.The Fist law of thermodynamics states that the increase in internal energy is equal to the total heat supplied except work done.Though it is a macroscopic quantity, internal energy can be explained in microscopic terms by two theoretical virtual components. One is the microscopic kinetic energy due to the microscopic motion of the system’s particles (translations, rotations, vibrations). The other is the potential energy associated

6.2 First Law of Thermodynamics

71

6.2 First Law of Thermodynamics

71

with the microscopic forces, including the chemical bonds, between the particles; this is for ordinary physics and chemistry. If thermonuclear reactions are specified as a topic of concern, then the static rest mass energy of the constituents of matter is also counted. There is no simple universal relation between these quantities of microscopic energy and the quantities of energy gained or lost by the system in work, heat, or matter transfer. So we may state, that the internal energy has two major components, kinetic energy and potential energy. The kinetic energy is due to the motion of the system’s particles (translations, rotations, vibrations), and the potential energy is associated with the static constituents of matter, static electric energy of atoms within molecules or crystals, the static energy of chemical bonds. The internal energy of a system can be changed by heating the system or by doing work on it dQ = dU + dW dU = cv dT dW = PdV Definition 6.2.2 — Entropy.

dS = dQ T dU = T dS − PdV Entropy is a thermodynamic property that is a measure of the energy not available for work in a thermodynamic process The Entropy is a measure of how much of the energy of a system is potentially available to do work and how much of it is potentially manifest as heat 

Example 6.1

W=

 V2 V1

PdV =

 V2 RT V1

V

dV = RT

 V2 dV V1

V

= RT Ln(

V2 ) V1

This is is how This how to to evaluate evaluate work work done done over over aa system system in in isotermic isotermic trasition trasition causing causing the the volume volume change change whithin which which the the gas gas is is occupied. occupied.Also,remind whithin Also, remind that that the the mole mole isis aa unit unit of of measurement measurement used used in chemistry to to express express amounts amounts of a chemical substance, defined as an amount of a substance chemistry substance, defined substance that that contains as many elementary entities (e.g., atoms, molecules, ions, electrons) as there are atoms in contains 12 grams of pure carbon-12 carbon-12 (12C), (12C), the the isotope isotope of of carbon carbon with withatomic atomicweight weight12 12  Internal Energy and Enthalpy Definition 6.2.3 — Internal Energy-U. N

mVi 1 + ∑ Ui, j 2 i,J i=1 2

U=∑

is Internal Energy. Definition 6.2.4 — Adiabatic process. Adiabatic process is one that slow to allow the system

to always be near equilibrium ,but fast compared with time it takes the system to exchange heat

72

Chapter 6. Thermodynamics and Statistical Physics

Chapter 6. Thermodynamics and Statistical Physics

72

with its surroundings dS = dQ T , T dS = dU(internal enegy) +W (work) dS = 0 S = cons tant, Process is adiabatic or isentropic dU = cv dT,W = pdV, cv dT + PdV = 0, dT = PdV R+vdP cV +R dV cP dP γ R V + P = 0, PV = const, γ = cV Definition 6.2.5 — Enthalpy. Enthalpy is a thermodynamic potential of a thermodynamic sys-

tem.It is equal to the system’s internal energy plus the product of its pressure and volume.In a system enclosed so as to prevent matter transfer, for processes at constant pressure, the heat absorbed or released equals the change in enthalpy.

R H = U + PV dU = δ Q − δW ,dS = δTQ dU = T dS − PdV d(U + PV ) = T dS − PdV + d(PV ) dH = T dS +V dP dH(S,P) = T dS +V dP, ∂∂HS = T , ∂∂H P =V

Definition 6.2.6 — Free Energy. Free energy is that portion of system’s internal energy that is

available to perform thermodynamic work

R •

•

F = U − T S,dF = dU − d(T S) dF = dQ − PdV − T dS − SdT dF = −PdV − SdT ,S = − ∂∂ TF ,P = − ∂∂ VF n

dU = T dS − PdV + ∑ µ i dN i i=1

Definition 6.2.7 — Chemical potential. Chemical potential defines internal energy change due

to particle number alteration n

dU = T dS − PdV + ∑ µ i dN i → µ i = i=1

G = H − T S,H = U + PV dG = dU + PdV +V dP − T dS − SdT n



dG = −SdT +V dP + ∑ µ i dN i → µ i = i=1

∂U ∂ Ni





∂G ∂ Ni

S,V,Nj=i



T,P,Nj=i

6.2 First Law of Thermodynamics 6.2 First Law of Thermodynamics

73

73

Definition 6.2.8 — First Law of Thermodynamics-epigragh. The first law of thermodynamics

is a version of the law of conservation of energy, adapted for thermodynamic systems. The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but cannot be created or destroyed. The first law is often formulated by stating that the change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings. Equivalently, perpetual motion machines of the first kind are impossible. Definition 6.2.9 — Heat. Heat is energy that is in a process of transfer between a system and its

surroundings, other than as work or with the transfer of matter. In thermodynamics, finer details of the process is unknowable. When there is a suitable physical pathway, heat flows from a hotter to a colder body. The pathway can be direct, as in conduction and radiation, or indirect, as in convective circulation.No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work

P

V

Theorem 6.2.1 — Second Law of Thermodynamic. The second law of thermodynamics states

that for a system, the intensive thermodynamic quantities such as temperature, pressure, chemical potential tend to become more uniform as time goes by, unless there is an outside influence which works to maintain the differences. Definition 6.2.10 — Entropy. In the microscopic interpretation of statistical mechanics, entropy

expresses the level of disorder or degree of randomness of the constituents of a thermodynamic system

74

74

Chapter 6. Thermodynamics and Statistical Physics

Chapter 6. Thermodynamics and Statistical Physics

Figure 6.1: Carnot cycle

Definition 6.2.11 — Carnot Cycle. The Carnot cycle when acting as a heat engine consists of

the following steps: Reversible isothermal expansion of the non gas at the "hot" temperature, TH (isothermal heat addition). During this step (A to B on Figure 1, 1 to 2 in Figure 2) the expanding gas makes the piston work on the surroundings. The gas expansion is propelled by absorption of quantity Q1 of heat from the high temperature reservoir. Isentropic (reversible adiabatic)expansion of the gas (isentropic work output). For this step (B to C on Figure 1, 2 to 3 in Figure 2) the piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, working on the surroundings. The gas expansion causes it to cool to the "cold" temperature, TC. Reversible isothermal compression of the gas at the "cold" temperature, TC. (isothermal heat rejection) (C to D on Figure 1, 3 to 4 on Figure 2) Now the surroundings do work on the gas, causing quantity Q2 of heat to flow out of the gas to the low temperature reservoir. Isentropic compression of the gas (isentropic work input). (D to A on Figure 1, 4 to 1 in Figure 2) Once again the piston and cylinder are assumed to be thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to TH. At this point the gas is in the same state as at the start of step 1.

6.2 First Law of Thermodynamics

75

Carnot Cycle Efficiency

6.2 First Law of Thermodynamics Efficiency of Carnot engine is

6.2 First Law TC W of Thermodynamics

First Law ηCycle = QH of = 1Thermodynamics − 6.2 Carnot Efficiency 6.2 First Law TH Efficiency of Carnot engine is

Carnot Cycle Efficiency Carnot TC = 1 −engine η =ofQWHCarnot TH Efficiency is

η=

W QH

75

of Thermodynamics

75

75

Cycle Efficiency

Efficiency of Carnot engine is

= 1 − TTHC

W

TC

η = QofH this = 1fact: − No TH engine operating between two heat reservoirs Carnot’s theorem is a formal statement can be more efficient than a Carnot engine operating between the same reservoirs. Definition 6.2.12 — Heat Engine. Heat engine is a system that performs the conversion of heat Carnot’s theorem is a formal statement of this fact: No engine operating between two heat reservoirs or thermal energy to mechanical work can be more efficient than a Carnot engine operating between the same reservoirs. Carnot’s theorem is a—formal of this fact: is Noa engine between two heat reservoirs Definition 6.2.12 Heat statement Engine. Heat engine systemoperating that performs the conversion of heat can be more efficient than a Carnot engine operating between the same reservoirs. or thermal energy to mechanical work Definition 6.2.12 — Heat Engine. Heat engine is a system that performs the conversion of heat

Carnot’s theorem is a formal statement of this fact: No engine operating betwe can be more efficient than a Carnot engine operating between the same reser

or thermal energy to mechanical work

Definition 6.2.12 — Heat Engine. Heat engine is a system that performs t

or thermal energy to mechanical work

Theorem 6.2.2 — Second Law of Therdynamic. The second law of thermodynamics is an

6.3

expression of the fact, that over time, differences in temperature, pressure, chemical potential tend to equilibrate in an isolated physical system. It explains the phenomenon of irreversibility Theorem — Second Law whose of Therdynamic. of thermodynamics is an in nature. 6.2.2 No process is possible sole result isThe the second transferlaw of heat from a body of lower 76 Chapter 6. Thermodynamics and Statistical Physics expression of the fact, that over time, differences in temperature, pressure, chemical potential temperature to a body of higher temperature.If heat flows by conduction from a body A to another Theorem 6.2.2 — in Second Law physical of Therdynamic. The second law of thermodynamics is an tend to equilibrate an isolated system. It explains the phenomenon of irreversibility body B , then a transformation whose only final result is to transfer heat from B to A is impossible expression of process the fact,isthat over time, differences pressure, chemical in nature. No possible whose sole resultinistemperature, the transfer of heat from a body potential of lower tend to equilibrate in an isolated physical system. It explains the phenomenon of irreversibility temperature to a body of higher temperature.If heat flows by conduction from a body A to another in nature. No process is possible whose sole result is the transfer of heat from a body of lower Statistical temperature Physics to a body of higher temperature.If heat flows by conduction from a body A to another Definition 6.3.1 — The probability P(s) that the system occupies microstate s is.

Ps = Z1 e−β ES ,β = N

1 kT

Z = ∑ e−β Es s=1

Definition 6.3.2 — First quantization. Particles might be treated like waves

Theorem 6.2.2 — Second Law of Therdynamic. The second law of th

expression of theWaves fact, might that over time, Definition 6.3.3 — Second quantization. be treated likedifferences particles

in temperature, pressure

Chapter 6. Thermodynamics and Statistical Physics

76

76body B , then a transformation whoseChapter Thermodynamics and B Statistical Physics only final6. result is to transfer heat from to A is impossible 6.3

Statistical Physics

Definition 6.3.1 — The probability P(s) that the system occupies microstate s is.,

Ps = Z1 e−β ES ,β = N

1 kT

Z = ∑ e−β Es s=1

Definition 6.3.2 — First quantization. Particles might be treated like waves Definition 6.3.3 — Second quantization. Waves might be treated like particles

Interpretation: λ (wave length) =

h momentum)

p(particle s

h = 6.62610−34 j · s 

Example 6.2

h = 6.62610−34 j · s me = 9.110−31 kg, c = 3 ∗ 108 ms me V 2 E0 = me c2 , β = Vc 2 = K, 2 K = √me c 2 − me c2 , K = √ E0 2 − E0 1−β  1−β E02 E02 2 1 − β = (K+E )2 , β = 1 − 2 (E0 +K)

0

E0 = 9.1 ∗ 10−31 ∗ 9 ∗ 1016 = 8.1 ∗ 10−16 J K = 1 Mev = 1. ∗ 106 1.610−19 = 1.6 ∗ 10−13 Ve = 2.3 ∗ 106 ms , p = meVe = 9.110−31 2.3106 = 2.0910−24 kgm/s −34 λ = 6.62610 = 3 ∗ 10−10 m = 0.3nm 2.0910−24 

Definition 6.3.4 — Free energy and statistical sum.

  En En F Q = ∑n e− kT = e− kT , F = −kT LnQ = −kT Ln ∑n e− kT F = U − T S, dF = dU − SdT  −  T dS = T dS − PdV − SdT − T dS dF = −PdV − SdT, S = −

∂F dT

∂ 

F 

S = −k ∑n Pn LnPn En U = Q1 ∑n En e− kT =

∂

1 T

T

En

V

, Pn = Q1 e− kT

6.3 Statistical Physics

77

6.3 Statistical Physics 

77

Example 6.3 — System of Harmonic oscillators. M

Eni

F = ∑ Fi i=1



Qi = ∑n e−

 hωi n+ kT



1  2

Fi = −kT LnQi =

Ui =

1 Qi

Eni = hω(n + 12 )

Qi = ∑n e− kT

Ei − kTn

=

hωi 2

∑n Eni e

e

−

hωi 2kT → hωi

1−e− 2kT

hωi

1

hωi 1−e− 2kT hωi − kT

+ kT Ln(1 − e  Fi  ∂

= 1 + e− kT + e−

2hωi kT

+ ...

)

= ∂ (1/T T hωi − hωi e kT hωi i i Ui = hω + = hω 2 2 + hωi hωi − − kT 1−e e kT −1   Ui = ni + 12 hωi 1 ni = hωi − e kT −1 

6.3.1

Black body radiation

Definition 6.3.5 — Black body radiation. ak

akz y x nx = ak 2π , ny = 2π , nz = 2π 3 3 dk dk dk dnx dny dnz = abc x y 3 z , dabcn = d k3  (2π)  (2π)  hω(k) 2d 3 k F = kT Ln 1 − exp( 3 V kT F V

U V

U V

(2π)



= 2kT Ln(1 − exp( hω(k) kT ) =2



hω(k)exp(− hω(k) kT )  

= σ T 4,

Uν dν =

hω(k) ) kT π 2 k4 σ = 15h 3 c3

1−exp(− 8πν 2 c3

4πk2 dk (2π)3

Ln

exp

Ln

exp

8π kT 4  ∞ e−x 2 4πk2 dk 3 = 3 h3 c3 0 1−e−x x d x (2π)

(2π)

hνdν

hν e kT −1

This relation between internal energy radiation and system temperature is called Stefan Boltzmann law. R

Albert Einstein (March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics). His work is also known for its influence on the philosophy of science.He is best known to the general public for his mass–energy equivalence formula , which has been dubbed "the world’s most famous equation".He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect", a pivotal step in the development of quantum theory.Near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led him to develop his special theory of relativity during his time at the Swiss Patent Office in Bern (1902–1909),

78

Chapter 6. Thermodynamics and Statistical Physics

Chapter 6. Thermodynamics and Statistical Physics

78

Switzerland. However, he realized that the principle of relativity could also be extended to gravitational fields, and he published a paper on general relativity in 1916 with his theory of gravitation. He continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the thermal properties of light which laid the foundation of the photon theory of light. In 1917, he applied the general theory of relativity to model the structure of the universe. Except for one year in Prague, Einstein lived in Switzerland between 1895 and 1914, during which time he renounced his German citizenship in 1896, then received his academic diploma from the Swiss federal polytechnic school (later the Eidgenössische Technische Hochschule, ETH) in Zürich in 1900. After being stateless for more than five years, he acquired Swiss citizenship in 1901, which he kept for the rest of his life. In 1905, he was awarded a PhD by the University of Zurich. In 1933, while Einstein was visiting the United States, Adolf Hitler came to power. Because of his Jewish background, Einstein did not return to Germany. He settled in the United States and became an American citizen in 1940.On the eve of World War II, he endorsed a letter to President Franklin D. Roosevelt alerting him to the potential development of "extremely powerful bombs of a new type" and recommending that the US begin similar research. This eventually led to the Manhattan Project. Einstein supported the Allies, but he generally denounced the idea of using nuclear fission as a weapon. He signed the Russell–Einstein Manifesto with British philosopher Bertrand Russell, which highlighted the danger of nuclear weapons. He was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 300 scientific papers and more than 150 non-scientific works. His intellectual achievements and originality have made the word "Einstein" synonymous with "genius".[16] Eugene Wigner wrote of Einstein in comparison to his contemporaries that "Einstein’s understanding was deeper even than Jancsi von Neumann’s. His mind was both more penetrating and more original than von Neumann’s. And that is a very remarkable statement." Definition 6.3.6 — Bose-Einstein statistics. The chemical potential of a component µ is defined

as the energy that comes or goes out when a single particle is added or left the system. Then the expressions for the differentials of thermodynamic potentials described in previous sections. Now we take system of N non interacting identical particles which are distributed with nα particles on the εα energy level. To define the statistical sum of this system we confine system by addition condition: ∑α nα = const, nα = 0, 1, 2, 3, 4, .. Bose Einstein statistics Fermi Dirac statistics nα = 0, 1 Generally statistical sum might be presented as follows:   nα εα ∑   α Q = ∑ exp −  kT n1 ,n2 ,....

6.3 Statistical Physics 79 To fulfill the condition of number conservation of all particles the statistical sum should include the chemical potential:   Qµ

 = ∑n1 ,n2 ,... exp −

∑n α

α (εα −µ

kT

 



g  1 LnQµ , e− kT = Qµ = ∑n1 ,n2 ,... exp − g = − kT

∂ Qµ ∂µ



 1 = ∑n1 ,n2 ,... kT ∑α nα exp −

N =

1

µ

∑



 Nexp −

kT



kT



∑ nα (εα − µ  α

α

∑ nα (εα − µ  α



∑ nα (εα − µ  



 1 Nexp −  = ∑n1 ,n2 ,... kT

 = kT

∂ LnQµ

= − ∂g



∑ nα (εα − µ  α

kT



the chemical potential: 

 Qµ = ∑n1 ,n2 ,... exp −

∑n α

α (εα −µ

kT

6.3 Statistical Physics 1 −g µ g = − kT LnQ , e ∂ Qµ ∂µ

 1 = ∑n1 ,n2 ,... kT ∑α nα exp − 1 Qµ

 



 = Qµ = ∑n1 ,n2 ,... exp − 

N =

R

kT





 ∑n1 ,n2 ,... Nexp −



∑ nα (εα − µ  α



kT



∑ nα (εα − µ  α

α

kT



 1 Nexp −  = ∑n1 ,n2 ,... kT

kT

∑ nα (εα − µ

79





∑ nα (εα − µ  α

kT

 ∂g ∂ LnQµ  = kT ∂ µ = − ∂ µ



Satyendra Nath Bose (Satyendra Nath Bose) (January 1, 1894, Calcutta - February 4, 1974, Calcutta, India) is an Indian physicist who specialized in mathematical physics. One of the founders of quantum statistics (Bose-Einstein statistics), the Bose-Einstein condensate theory. The boson was named after him. He graduated from the University of Calcutta (1915). In 1924-1925, he worked in Paris with M. Sklodowska-Curie. In 1926-1945, a professor at the University of Dhaka, in 1945-1956 - in Calcutta. One of the founding members (1935) of the Indian National Academy of Sciences (until 1970 - National Institute of Sciences of India). Member of the Royal Society of London since 1958. National Professor of India (1958-1974). He derived Planck’s formula for the distribution of energy emitted by an absolutely black body, based on the assumption that two states of the system, differing in the rearrangement of identical quanta in the phase space, are considered identical.

Definition 6.3.7 — Fermi Dirac statistics. In this case we take system of N non interacting

identical particles which are distributed with nα particles on the εα energy level. To define the statistical sum of this system we confine system by addition condition: ∑α nα = const, nα = 0, 1, 2, 3, 4, .. Bose Einstein statistics Fermi Dirac statistics nα = 0, 1 Generally statistical sum might be presented as follows:   nα εα ∑  α  Q = ∑ exp −  Chapter 6. Thermodynamics and Statistical Physics 80 kT n1 ,n2 ,....



 Qµ = ∑n1 ,n2 ,... exp −

∑n α

α (εα −µ

kT



  , but at this case n1 = 0, 1; n2 = 0, 1; ...

    εi −µ εi −µ g 1 g = − kT ∑i Ln 1 + e− kT , e− kT = ∏i 1 + e− kT   2 1 g = − kT

nα  =



∂g ∂ εα

Ln 1 + e−

=



exp − 

1+exp

p 2m −µ kT

(εα − µ kT

2d 3 p V (2πh)3 

(ε − µ − α kT



=

exp



1

(εα − µ kT



+1

. R

Enriko Fermi (September 29, 1901, Rome, Italy - November 28, 1954, Chicago, USA) Italian physicist, best known for creating the world’s first nuclear reactor, who made a great contribution to the development of nuclear physics, elementary particle physics, quantum and statistical mechanics. It is considered one of the "fathers of the atomic bomb." During his life he received several patents related to the use of atomic energy. Winner of the Nobel Prize in Physics in 1938 "for proving the existence of new radioactive elements obtained by irradiation with neutrons, and the associated discovery of nuclear reactions caused by slow neutrons." Fermi was one of the few physicists who succeeded both in theoretical physics and in experimental physics. Member of the National Academy dei Lincei (1935), foreign

1  g = − kT Ln

nα  =

∂g ∂ εα

80 .

R

=



1 + e− 

exp − 

1+exp

p2 2m −µ kT



(εα − µ kT

2d 3 p V (2πh)3 

(ε − µ − α kT



=

exp



1

(εα − µ kT



+1

Chapter 6. Thermodynamics and Statistical Physics

Enriko Fermi (September 29, 1901, Rome, Italy - November 28, 1954, Chicago, USA) Italian physicist, best known for creating the world’s first nuclear reactor, who made a great contribution to the development of nuclear physics, elementary particle physics, quantum and statistical mechanics. It is considered one of the "fathers of the atomic bomb." During his life he received several patents related to the use of atomic energy. Winner of the Nobel Prize in Physics in 1938 "for proving the existence of new radioactive elements obtained by irradiation with neutrons, and the associated discovery of nuclear reactions caused by slow neutrons." Fermi was one of the few physicists who succeeded both in theoretical physics and in experimental physics. Member of the National Academy dei Lincei (1935), foreign corresponding member of the USSR Academy of Sciences (1929). He created the theory of beta decay, neutron moderation. In 1939, he introduced the concept of a chain reaction and later took part in an atomic project. The Fermi-Dirac distribution, the Thomas-Fermi model, the chemical element Fermi, etc. are named in his honor.

Definition 6.3.8 Quantum Mechanics. Mechanics.Consider Definition 6.3.8 Symmetry Symmetry Requirements Requirements in Quantum Consider aa gas gas in in ampule ampule containing number ofofidentical,non-interacting,structureless identical, non-interacting, structureless particles within the volume V. containing N number particles within the volume V.Qi are Q all coordinates of ith particle. The probability of an oservation of thefinding system finding the it alli are coordinates of ith particle.The probability of an oservation of the system the ith particle particle dQi is equal: in rangeinQrange dQQi i is+ equal: i to i to QQ i+

|ψ(Q1, Q2, ...QN )|2 dQdQ2 ...dQN of the the fundamental fundamentalpostulates postulatesofofquantum quantummechanics mechanics that paticles of the same species One of is is that paticles of the same species are are indistinguisable.Note such constraint doesnot notarises arisesininclassical classicalmechanics. mechanics.Suppose indistinguisable. Note thatthat such constraint does Suppose we swap ith particle by jth: jth: Qi ↔ Q j If the particles are really indistinguishable then nothing has been happened we don’t notice any change.The change. The probability probability of of observing observing the system also can not be changed:     ψ(...Qi ...Q j ...)2 = ψ(...Q j ...Qi ...)2 6.3 Statistical Physics 81 internal coordinates coordinates like like spin, spin,then If we want to consider internal then relationbetween relation between wave wave functions functions will will look look like: like:

ψ(...Qi ...Q j ...) = Aψ(...Q j ...Qi ...) A2 = 1,then A = ±1

brings totwo twotwo typetype of wave functions:symmetric and antiWe also also may mayconclude concludethat thatinterchange interchangebrings of wave functions:symmetric and symmetric: anti-symmetric:

ψ(...Qi ...Q j ...) = +ψ(...Q j ...Qi ...) ψ(...Qi ...Q j ...) = −ψ(...Q j ...Qi ...) To first equation particles with with integer integer spin spin(0,1,2) To first equation particles (0,1,2) satisfy satisfy without without no no restriction restriction on on how how many many particles can occupy a given single particle quntum state.These particles are called bosons.On occupy a given single particle quantum state. These particles are called bosons. the On contrary particles with halfhalf integer spins areare not able the contrary particles with integer spins not abletotooccupy occupyone oneand andthe thesame sameenergy energy state state property of of their their wave wave functions functions and and called calledfermions. fermions. due two antisymmetric property There are are three three sets of rules that can be used to enumerate the states of a gas presented by identical There particles: Bose-Einstein Bose-Einstein statistics-particles statistics -particlesare areindistinguishable indistinguishable and and no no limits limits to to occupy occupy one energy particles: state,spin-integer state, spin-integer number number Fermi Fermi Dirac statistics-particles are indistinguishable and no more the one particle energy state,spin-half-integer state,spin-half-integer number Maxwell-Boltzmann Maxwell-Boltzmann statisticsparticle can can to occupy one energy statisticsparticles are distinguishable and no limits to how occupy a given quantum state particles are distinguishable and no limits to how occupy a given quantum state 

Example 6.4 — Illustrative example.

1 2 3 AB ... ... ... AB ...

→ Maxwell Boltzmann statistics

particles can occupy a given single particle quntum state.These particles are called bosons.On the contrary particles with half integer spins are not able to occupy one and the same energy state due two antisymmetric property of their wave functions and called fermions. There are three sets of rules that can be used to enumerate the states of a gas presented by identical particles: Bose-Einstein statistics -particles are indistinguishable and no limits to occupy one energy state,spin-integer number Fermi Dirac statistics-particles are indistinguishable and no more the one 6.4 Ideal Gas one energy state,spin-half-integer number Maxwell-Boltzmann statistics81 particle can Bose to occupy particles are distinguishable and no limits to how occupy a given quantum state 

Example 6.4 — Illustrative example.

1 2 3 AB ... ... ... AB ... → Maxwell Boltzmann statistics A B ... B A ... ... A B 1 2 3 AA ... ... ... AAAAA ... → Bose Einstein statistics A A ... AAA A ... ... A AA 1 2 3 A ... ... ... A ... → Fermi − Dirac statistics A A ... A A ... ... A A 

Chapter 6. Thermodynamics and Statistical Physics Definition 6.3.9 — Formulation of Statistics. The particles are assumed to be non-interactive

82

and at state r we have nr particles ER (total energy) = ∑ nr εr r

N(total number o f particles) = ∑ nr r

ER

Z(statistical sum) = ∑ e− kT = ∑R e− ns =

6.4

∑R ns e ∑R e

n ε +n ε +... − 1 1 kT2 2

n ε +n ε +... − 1 1 kT2 2

n1 ε1 +n2 ε2 +... kT

R

1 = − kT

∂ LnZ ∂ εs

Ideal Bose Gas opened and and the the number number of of particles particles in in the the system system changes. changes.The Consider the system hich is opened The chemical potential µ is the thermodynamic potential used to describe the state of systems with a variable variable potential µ is thermodynamic potential used to describe the number thermodynamic potentials (Gibbs energy, energy, internal internal number of particles. Determines Determines the change in thermodynamic energy, the system system changes. changes. It is the the energy energy of of adding adding energy, enthalpy, enthalpy, etc.) etc.) as as the the number number of of particles particles in in the It is one particle to the system without performing work. Used to describe the material interaction. The one particle to the system without performing work. Used to describe the material interaction. The definition of chemical potential can be written in the form: definition of chemical potential can be written in the form: dE = T dS − PdV + µdN

number of particles. Determines the change in thermodynamic potentials (Gibbs energy, internal energy, enthalpy, etc.) as the number of particles in the system changes. It is the energy of adding one particle to the system without performing work. Used to describe the material interaction. The definition of chemical potential can be written in the form:

82

dE = T dS − PdV + µdN

Chapter 6. Thermodynamics and Statistical Physics

If the the particles particles in in aa given given system system are are indistinguishable indistinguishable and and the the number number of of them them equal equal nnααthen the the If statistical sum sum might might be bepresented: presented : statistical

Qα =

∑

e

∑ nα (Eα − µ) α

kT

n1 ,n2 ,...

6.4 Ideal Bose Gas

83

It is convenient to introduce another type of statistical sum in the following way: e

g − kT

∑ nα (Eα − µ) α

kT = Qα = ∑n1 ,n2 ,... e ∑ nα (Eα − µ)

α

∂ Q(µ) ∂µ

kT = ∑n1 ,n2 ,... β (∑α nα ) e− ∑ nα (Eα − µ)   α kT N = Q1(µ) ∑n1 ,n2 ,... nα e−

N=

1 ∂ Q(µ) ∂µ (µ) Q

=

1 ∂ µ kT ∂ µ Ln(Q )

, N = ∑α nα ,

= ∑n1 ,n2 ,... β Ne−

∑ nα (Eα − µ) α

kT

= − ∂∂ µg

kT Fact 6.4.1 The support integrals which are to be known:

2        ∞ −x2 ∞ −x2 ∞ −y2 ∞ ∞ −(x2 +y2 ) e d x = e d x e d y = −∞ dxdy −∞ −∞ −∞ −∞ e    2π  ∞ −r2   √ 2 2 = 0 0 e rdrdθ = − 12 02π 0∞ e−r (−2rdr)dθ =  π 



∞

√

−x2 d x = −∞ e  ∞ 2 −αx2 dx −∞ x e

π

=−

∞

d −αx2 −∞ dα e

d = − dα

  π α

1√ = 12 α − 2 π

For ideal Bose gas we have:   n (ε1 −µ) n2 (ε2 −µ) − 1 kT + kT +... e = ∑n1 ,n2, ... e =   n (ε1 −µ) n (ε2 −µ) − 1 kT − 2 kT = ∑n1 e e ..., nα =  ∑n1 α −µ) − nα (εkT 1 = ∑nα e εα − µ g − kT

−

0, 1, 2, ...

=

Fact 6.4.1 The support integrals which are to be known:

2        ∞ −x2 ∞ −x2 ∞ −y2 ∞ ∞ −(x2 +y2 ) e d x = e d x e d y = −∞ dxdy −∞ −∞ −∞ −∞ e    2π  ∞ −r2   √ 2 2 = 0 0 e rdrdθ = − 12 02π 0∞ e−r (−2rdr)dθ =  π 



6.4

√

∞

−x2 d x = π −∞ e   ∞ d −αx2 Ideal 2 ∞ Bose 2 −αxGas d x = − −∞ −∞ x e dα e

d = − dα

  π α

=

1√ = 12 α − 2 π

83

For ideal Bose gas we have:   n (ε1 −µ) n2 (ε2 −µ) − 1 kT + kT +... e = ∑n1 ,n2, ... e =   n (ε1 −µ) n (ε2 −µ) − 1 kT − 2 kT = ∑n1 e e ..., nα =  ∑n1 α −µ) − nα (εkT 1 = ∑nα e εα − µ g − kT

g − kT

e

R

84

= ∏α

1

1−e

εα −µ 1−e− kT

,

−

kT g=

1 kT

0, 1, 2, ...

∑α Ln(1 − e−

εα −µ kT

)

Already at a relatively early stage in the study of superconductivity, at least after the creation of the Ginzburg – Landau theory, it became obvious that superconductivity is a consequence of combining the macroscopic number of conduction electrons into a single quantum mechanical state. The peculiarity of electrons connected to such an ensemble is that they cannot exchange energy with the grid in small portions smaller than their binding energy in the ensemble. This means that when electrons move in a crystal lattice, the energy of electrons does not change, and matter behaves as a superconductor with zero resistance. Quantum-mechanical examination shows that in this case there is no scattering of electron waves by thermal vibrations of the lattice or impurities. And this means the absence of electrical resistance. Such a union of particles is impossible in an ensemble of fermions. It is characteristic of the ensemble of identical bosons. The fact that electrons in superconductors are combined into boson pairs follows from experiments to measure the magnitude of a quantum of magnetic flux, which is “frozen” in hollow superconducting cylinders. Therefore, already in the middle of the 20th century, the main task of creating a theory of superconductivity was the development of a mechanism for the pairing of electrons. The first theory claiming to be a microscopic Chapter 6. Thermodynamics and Statistical Physics explanation of the causes of the appearance of superconductivity was the theory of Bardeen – Cooper – Schrieffer, which they created in the 50s of the 20th century. This theory received under the name of the BCS universal recognition and was awarded the Nobel Prize in 1972. When creating their theory, the authors relied on the isotopic effect, that is, the effect of the isotope mass on the critical temperature of the superconductor. It was believed that its existence directly indicates the formation of a superconducting state due to the operation of the phonon mechanism. The BCS theory left some questions unanswered. On its basis, it turned out to be impossible to solve the main task - to explain why particular superconductors have one or another critical temperature. Moreover, further experiments with isotopic substitutions showed that, due to the anharmonicity of zero-point oscillations of ions in metals, there is a direct effect of the ion mass on the interionic distances in the lattice, and therefore directly on the Fermi energy of the metal. Therefore, it became clear that the existence of the isotopic effect is not evidence of a phonon mechanism, as the only possible one responsible for the pairing of electrons and the occurrence of superconductivity. Dissatisfaction with the BCS theory in later years led to attempts to create other models, for example, a model of spin fluctuations and a bipolaron model. However, although they considered various mechanisms of combining electrons into pairs, these developments also did not lead to progress in understanding the phenomenon of superconductivity.

• Positively homogeneous σC (αx) = ασC (x)∀α0 σC (αx) = supu∈C αx, u = α supu∈C x, u = ασC (x) • Sub-linear( a special case of convex, linear combination holds ∀α. σC (αx + (1 − α)y) = supu∈C αx + (1 − α)y, u ≤ α supu∈C x, u + (1 − α) supu∈C y, u



Example 6.5 — Superfluidity. Within the two-fluid model (also known as the “two-component

model”), helium II is a mixture of two interpenetrating fluids: the superfluid and normal components. The superfluid component is actually liquid helium, which is in a quantum-correlated state, somewhat similar to the Bose condensate state (however, unlike the condensate of rarefied gas atoms, the interaction between helium atoms in a liquid is quite strong; therefore, the Bose condensate theory is not directly applicable to liquid helium). This component moves without friction, has zero temperature and does not participate in energy transfer in the form of heat. The normal component is a gas of two types of quasi-particles: phonons and rotons, that is, elementary excitations of a

and a bipolaron model. However, although they considered various mechanisms of combining electrons into pairs, these developments also did not lead to progress in understanding the phenomenon of superconductivity.

• Positively homogeneous σC (αx) = ασC (x)∀α0 C (x) u∈C x, u = 84 σC (αx) = supu∈C αx, u = α supChapter 6. ασ Thermodynamics and Statistical Physics • Sub-linear( a special case of convex, linear combination holds ∀α. σC (αx + (1 − α)y) = supu∈C αx + (1 − α)y, u ≤ α supu∈C x, u + (1 − α) supu∈C y, u

 Example 6.5 — Superfluidity. Within the two-fluid model (also known as the “two-component model”), helium II is a mixture of two interpenetrating fluids: the superfluid and normal components. The superfluid component is actually liquid helium, which is in a quantum-correlated state, somewhat similar to the Bose condensate state (however, unlike the condensate of rarefied gas atoms, the interaction between helium atoms in a liquid is quite strong; therefore, the Bose condensate theory is not directly applicable to liquid helium). This component moves without friction, has zero temperature and does not participate in energy transfer in the form of heat. The normal component is a gas of two types of quasi-particles: phonons and rotons, that is, elementary excitations of a quantum-correlated fluid; it moves with friction and is involved in energy transfer. • The superfluid model of the atomic nucleus was constructed, which describes the experimental data quite well. In 2000, Ian Peter Tohenes demonstrates the superfluidity of hydrogen at 0.15 K Since 2004, based on the results of a number of theoretical works, it is assumed that at pressures of about 4 million atmospheres and higher hydrogen becomes unable to pass into the solid phase at any cooling (like helium at normal pressure) thereby forming a superfluid liquid. Direct experimental confirmation or refutation is not yet available. There are also works that predict superfluidity in cold neutron or quark aggregate state. This may be important for understanding the physics of neutron and quark stars. • In 1995, in experiments with rarefied gases of alkali metals, sufficiently low temperatures were achieved in order for the gas to become Bose-Einstein condensate. As expected, based on theoretical calculations, the resulting condensate behaved like a superfluid liquid. In subsequent experiments, it was found that when bodies move through this condensate with speeds less than critical, no energy transfer from the body to the condensate occurs. • In 2004, it was announced the discovery of superfluidity in solid helium. Subsequent studies, however, 6.4 Ideal Boseshowed Gas that the situation is far from being so simple, and therefore it is still premature 85 to talk about experimental detection of this phenomenon. • In 2005, superfluidity was discovered in cold rarefied fermion gas. • In 2009, superfluidity of the supersolid type in cold rarefied rubidium gas was demonstrated. 6.4 Ideal Bose Gas

• In 2005, superfluidity was discovered in cold rarefied fermion gas.

85



Fact 6.4.2 Superfluidity Nobel prize winners superfluidity liquid helium II below lambda . Thesuperfluidity • In 2009, of theofsupersolid type in cold rarefiedthe rubidium gas was demonstrated.  point (T = 2.172 K) was experimentally discovered in 1938 by P. L. Kapitsa (Nobel Prize in Physics Fact 6.4.2 Superfluidity Nobel prize winners The superfluidity of liquidhelium helium II below the lambda for 1978) and by John Allen. It was already known that when this point passed, liquid pointfrom (T = 2.172 K) was experimentally discovered in 1938 byhelium-I) P. L. Kapitsato(Nobel undergoes a phase transition, moving a completely “normal” state (called a Prize in Physics for 1978) and by John Allen. It was already known that when this point passed, liquid helium new state of so-called helium-II, butundergoes only Kapitsa showed that helium-II flows in general (within a phase transition, moving from a completely “normal” state (called helium-I) to a experimental errors) without friction.new state of so-called helium-II, but only Kapitsa showed that helium-II flows in general (within experimental errors) withoutIIfriction. The theory of the phenomenon of superfluid helium was developed by L. D. Landau (the The theory of the phenomenon of superfluid helium II was developed by L. D. Landau (the Nobel Prize in Physics for 1962).

Nobel Prize in Physics for 1962).

Suppose in a scorching summer day, you are drinking some cold drinks. Suddenly, your cold drink starts leaking through the glass or starts climbing climbing up through the glass walls.

Suppose in a scorching summer day, you are drinking some cold drinks. Suddenly, your cold drink

6.5 Ideal Fermi Gas

85

Suppose in a scorching summer day, you are drinking some cold drinks. Suddenly, your cold drink starts leaking through the glass or starts climbing climbing up through the glass walls.

Superfluids do exactly the same thing. Superfluidity is a state of matter in which the matter behaves like a fluid with zero viscosity; where it appears to exhibit the ability to self-propel and travel in a way that defies the forces of gravity and surface tension. While characteristic was 86 Chapter 6. Thermodynamics andthis Statistical Physics originally discovered in liquid helium, it is also found in astrophysics, high-energy physics, and theories of quantum gravity. The phenomenon is related to the Bose–Einstein condensation, but it is not identical: not all Bose-Einstein condensates can be regarded as superfluids, and not all superfluids are Bose–Einstein condensates.

6.5 Ideal Fermi Gas For Fermi gas the number of particle occupation on any energy level is limited due to Pauli forbidden law: g

−

= ∑n1 e

∑nα e



−

e− kT = ∑n1 ,n2, ... e −

n1 (ε1 −µ) kT

nα (εα −µ) kT



g

e− kT = ∏α

1+e

n1 (ε1 −µ) n2 (ε2 −µ) + kT +... kT

−

∑n1 e

=



1

n2 (ε2 −µ) kT





=

..., nα = 0, 1, 2, ...

εα − µ − kT 1+e 1 1 − εαkT−µ , g = − Ln(1 + e ) ∑ εα −µ α kT − kT

That is, at zero temperature, all particles fall to the lowest state and lose all their kinetic energy. However, for Fermi gas this is impossible. The Pauli exclusion principle allows only one Fermi particle with a half-integer spin to be in one state. R

The Pauli principle helps to explain various physical phenomena. A consequence of the principle is the presence of electronic shells in the structure of the atom, from which, in turn, follows the diversity of chemical elements and their compounds. The number of electrons in a single atom is equal to the number of protons. Since electrons are fermions, the Pauli principle forbids them to accept the same quantum states. As a result, all electrons cannot be in one quantum state with the lowest energy (for an unexcited atom), but successively fill quantum states with the lowest total energy (one should not forget that electrons are indistinguishable from each other, and therefore it cannot be said what kind of quantum state is a particular electron). An example is the unexcited lithium atom (Li), in which two electrons are on the 1s orbitals (the lowest in energy), they have their own angular momentum, and the third electron cannot occupy the 1s orbital, since the prohibition will be violated Paulie. Therefore, the third electron occupies a 2s orbital (the next lowest orbital after 1s).

R

The Pauli principle helps to explain various physical phenomena. A consequence of the principle is the presence of electronic shells in the structure of the atom, from which, in turn, follows the diversity of chemical elements and their compounds. The number of electrons in a single atom is equal to the number of protons. Since electrons are fermions, the Pauli principle forbids them to accept the same quantum states. As a result, all electrons cannot be in one quantum state with the lowest energy (for an unexcited atom), but successively fill quantum 6. not Thermodynamics Statistical Physics states with the lowest total energyChapter (one should forget that electrons and are indistinguishable from each other, and therefore it cannot be said what kind of quantum state is a particular electron). An example is the unexcited lithium atom (Li), in which two electrons are on the 1s orbitals (the lowest in energy), they have their own angular momentum, and the third electron cannot occupy the 1s orbital, since the prohibition will be violated Paulie. Therefore, the third electron occupies a 2s orbital (the next lowest orbital after 1s).

R

Pauli’s finest hour came in 1925 when he discovered a new quantum number (later called spin) and formulated Pauli’s fundamental prohibition principle, which explained the structure of the electron shells of atoms. In the late 1920s, a heavy crisis began in Pauli’s personal life. In 1927, his mother committed suicide. The father remarried, and his relationship with his son deteriorated markedly. In 1929, Pauli married the ballet dancer Kete Deppner (Käthe Margarethe Deppner), his wife soon went to her old friend, and in 1930, the couple separated. Pauli became depressed, it was then that he began to communicate with the psychoanalyst Carl Gustav Jung, sharply broke with the Catholic religion and began to abuse alcohol. In 1928, Pauli went to Switzerland, where he was appointed professor at the Zurich High Technical School. In 1930, Pauli suggested the existence of an elementary neutrino particle, which became its second significant contribution to atomic physics. This all-penetrating particle was experimentally discovered only 26 years later, even during the lifetime of Pauli. In the summer of 1931, Paulie first visited the United States, then went to the International Congress on Nuclear Physics in Rome.

86

7. Hydrodynamics

7. Hydrodynamics

7.1 7.1.1

Boltzmann equation Function of energy distribution Definition 7.1.1 — Function of energy distribution. Function of energy distribition equal to

the number atoms in volume dxdydz and within the given range of velocities Definition 7.1.2

   − − → − → − − −r + → −r , → −r d → F f t + dt, → ξ dt, ξ + m dt = f t, → ξ + (+ − − ) d → ξ dt     → − → −   df ∂f ∂f F ∂f f f1 − f f1 gdbdεdξ1 − = −r + m → dt = ∂t + ξ ∂ → ∂ξ

− − → − → C = ξ −→ u − → − ∞ −r , → → − ξ )d ξ n(t, r ) = 0 f (t, → − − → − → − −r ) = 1  ∞ → −r , → u (t, → ξ f (t, → ξ )d ξ 0 n → − − → − → − → 3 1  ∞ mC 2 2 kT = n 0 2 f (t, r , ξ )d ξ − → − −r ) = m  ∞ C C f (t, → −r , → ξ )d ξ Pi j (t, → 0 i j − → − − → − −r ) = m  ∞ C2 → −r , → q (t, → C f (t, → ξ )d ξ 2 0

Navier Stokes equation.The conservation laws in liquids

7.1.2

Main assumptions The theory for ideal gases makes the following assumptions: The gas consists of very small particles, all with non-zero mass mass. The The number number of of molecules molecules isis large such that statistical treatment can be applied. These molecules are in constant, random motion. The collisions collisions of of gas gas particles particles with with the the walls walls of of the the container container holding holding them them are are perfectly perfectly elastic. elastic. The Except during collisions the interactions among molecules are negligible. The total volume of the the Except during collisions the interactions among molecules are negligible . The total volume of individual gas gas molecules molecules added added up up is is negligible negligible compared compared to to the the volume volume of of the the container. container. This This isis individual

87

88

Chapter 7. Hydrodynamics

Chapter 7. Hydrodynamics

88

equivalent to stating that the average distance separating the gas particles is large compared to their size. The molecules are perfectly spherical in shape, and elastic in nature. The average kinetic energy of the gas particles depends only on the temperature of the system. Relativistic effects are negligible. Quantum –mechanical effects are negligible. This means the molecules are treated as classical objects. Definition 7.1.3 ,momentum Definition 7.1.3— —General Generalequations equationsfor formass mass, momentumand andenergy energytransfer. transfer.   → − → − → − ∂f  ∂f F ∂f d ξ = Icollisions ϕ(ξ ) ∂t + ξ ∂ → − −r + m → ∂ξ    → − → − ∂f → − → −  ∂f F ∂f ∂  ϕ(ξ ) ∂t + ξ ∂ → + ϕ(ξ ) f d ξ d ξ = − −r m ∂→ ∂t ξ    → − f → − → − → −   ∂ ϕ(ξ ) ξ ∂∂→ ϕ(ξ ) ξ f d ξ − −r d ξ = → → − ∂ r   → −  ∂ϕ → − → −   ξ =∞ ∂f ϕ(ξ ) → dξ d ξ = ϕ(ξ ) f |ξii =−∞ d ξ − f dξ − i ∂ξ  → −  ∂  → −  Fi  ∂ ϕ → − ∂  ϕ(ξ ) f d ξ + ϕ(ξ )ξ f d ξ − f d ξ i m ∂t ∂ ri ∂ ξi

Definition 7.1.4 — Anri Navier. After the death of his father in 1793, Henri’s mother gave

further training to his son in the hands of his uncle, Emiland Gothe, engineer of the French Corps des Ponts et Chaussées. In 1802, Navier entered the famous polytechnique École polytechnique, and in 1804 he continued his studies at the National School of Bridges and Roads, which he successfully completed in 1806. As a result, Navier changed his uncle as chief inspector in the corps of bridges and roads. He oversaw the construction of bridges in Choisy, Anyer and Argenteuil in the Seine department, and also built a pedestrian bridge on Cite Island in Paris. He also owns the first project of the Bridge for the Disabled. In 1824, Navier was accepted into the French Academy of Sciences. In 1830 he accepted the post of professor at the National School of Bridges and Roads, and the following year replaced the post of Professor of Mathematics and Mechanics of the Exiled Augustin Louis Cauchy at the Polytechnic School. Navier formulated the theory of elasticity in a mathematical form (1821), making it suitable for use in construction with sufficient accuracy for the first time. In 1819, he determined the zero level of mechanical stress, thereby correcting the results of Galileo, and in 1826 he introduced elastic moduli as a characteristic of materials, independent of the second area moment. Navier is considered one of the founders of the modern theory of elasticity. His most famous contribution to science is the derivation in 1822 of the Navier – Stokes equations, which play a key role in hydrodynamics. Mass and heat transfer in gases and liquids

ϕ(ξ ) = m, ρ = mn ∂ρ → − ∂t + div (ρ u ) = 0 → − ϕ(ξ ) = m ξ − − − ∂→ u + (→ u · ∇→ u ) = − 1 ∇p ⇒ ∂t

→ − → − 2 ϕ(ξ )= m( ξ 2− u )  → − ∂ T ∂ 3 −r 2 Rρ ∂t + u ∂ →



Example 7.1

ρ

∂ ∂s



u2 2

 + ρp + gz = 0

− = −∇→ q − Pi j ∂∂ ruji 

7.1 Boltzmann equation 7.1 Boltzmann equation

R

For the warm-up, let us specify who Osborne Reynolds was at one time in his field. This scientist was a representative of the English (in some sources - the Irish) scientific community in the field of physics, engineering and hydraulics. Born in 1842, after 28 years he became a professor at the Department of Structural Mechanics in Cambridge. His research was conducted in relation to the behavior of fluid in the laminar (layered) and turbulent (vortex) flow. It is to him that the selection of the numerical criterion for the transition of one type of flow into another belongs. This quantity is known to us as the Reynolds number. What is the meaning of the number? This parameter is not a constant and represents a certain ratio of inertia forces and internal friction in the fluid flow. Re is used to designate this criterion. As for the units of measurement, there are no such units, since the Reynolds number for water, gases and any other medium is a dimensionless quantity (the number itself)

89

89

90

Chapter 7. Hydrodynamics

Chapter 7. Hydrodynamics

90

7.2

Sound Definition 7.2.1 — Sound. Sound propagation requires an elastic medium. In a vacuum, sound waves cannot propagate, since there is nothing to oscillate there. This can be seen from simple experience. If an electric bell is placed under the glass bell, then as the air is pumped out from under the bell, the sound of the bell will become weaker and weaker until it stops altogether. It is known that during a thunderstorm we see a flash of lightning, and only after a while we hear thunder. This delay is due to the fact that the speed of sound in air is much less than the speed of light coming from lightning. The speed of sound in gases depends on the temperature of the medium: it increases with increasing air temperature, and decreases with decreasing. In different gases, the sound propagates at different speeds. The greater the mass of gas molecules, the lower the speed of sound in it. Definition 7.2.2 — Speed of sound. 



P = P0 + P, ρ = ρ0 + ρ ⇒ P ≡ P  P0 , ρ ≡ ρ  ρ0  ∂ρ ∂ρ → − → − → − + div (ρ u ) = 0, u  1 ⇒ ∂t− ∂t + ρ0 div ( u ) = 0 −  → − → − ∂→ u 1 ∂→ u 1 = − ρ0 ∇P ∂t +( u ·∇) u = − ρ ∇P ⇒  ∂t     − ∂P ∂P ∂P P = ∂ ρ0 ρ ⇒ ∂t + ρ0 ∂ ρ0 div→ u =0 S S 2  → − u =∇ψ, ⇒ P = −ρ ∂∂tψ ⇒ ∂∂tψ2 − c2 ∇2 ψ = 0   ∂P c= ⇔ speed o f sound wave propagation ∂ρ S

R

The analysis of the solutions of the equations is the essence of one of the seven “millennium problems”, for the solution of which the Clay Mathematical Institute appointed a prize of1 million dollars. It is necessary to prove or disprove the existence of a global smooth solution of the Cauchy problem for the three-dimensional Navier – Stokes equations. Finding a general analytical solution of the Navier – Stokes system for a spatial or flat flow is complicated by the fact that it is nonlinear and depends strongly on the initial and boundary conditions. As of 2014, confirmed solutions of these equations have been found only in some particular cases: there are several situations (caused by simple geometry), which are solved in an analytical form. In other cases, numerical modeling is used. On January 10, 2014, the Kazakhstan mathematician Mukhtarbai Otelbayev published an article in which he claims that he gave a complete solution to the problem, checking the result by the international community is complicated by the fact that the work was written in Russian. In the communities of mathematicians, counterexamples to the main statements are discussed. Against the background of the discussion of the work of Otelbayev, on February 6, 2014, the winner of the Fields Award, Terence Tao, published a preprint stating that it is impossible to solve the “millennium problem” with the means currently available.

Definition 7.2.3 — Mach number. The Mach number (M) - in continuum mechanics is one of

the similarity criteria in fluid and gas mechanics. It is the ratio of the flow velocity at a given point of the gas flow to the local sound propagation velocity in a moving medium - named for the German scientist Ernst Mach Mach has a number of important physical discoveries. His first scientific works relate to optics and acoustics and are devoted to the study of the processes of hearing and vision (an explanation of the mechanism of action of the vestibular apparatus, the discovery of an optical phenomenon — the

7.3 Shock waves

91

7.3 Shock waves 91 7.3 Shock waves 91 so-called rings, or bands, Mach). Among the works of this period are “On the color of double stars so-called rings, or bands, Mach).(1861), Among the works ofofthis are “On the color of double based on the Doppler principle” “Explanation theperiod Helmholtz musical theory” (1866), stars “On based on the Doppler principle” (1861), “Explanation of the Helmholtz musical theory” (1866), “On the stroboscopic determination of pitch” (1873), “On the reflection and refraction of sound” , 1873), the stroboscopic determination of pitch” (1873), “Onofthe refraction(1875). of sound” , 1873), "Optoacoustic Experiments" (1873), "Fundamentals thereflection theory of and kinesthesia" Since 1881, "Optoacoustic Experiments" (1873), "Fundamentals of the theory of kinesthesia" (1875). Since 1881, Mach worked on the issues of gas dynamics (one of the founders of which he is considered). He Mach on the issues of gas accompanying dynamics (one the of the foundersmotion of which he is considered). He studiedworked the aerodynamic processes supersonic of bodies; discovered and studied the aerodynamic processes accompanying the supersonic motion of bodies; discovered and investigated the process of the appearance of a shock wave. In this area, the name Mach is called a investigated the process of the appearance a shockthe wave. In cone, this area, the name series of quantities and concepts: the Machofnumber, Mach the Mach ring.Mach is called a series of quantities and concepts: the Mach number, the Mach cone, the Mach ring.

7.3 Shock waves 7.3 Shock waves Definition 7.3.1 When passing through the front of the shock wave, the pressure, temperature, Definition 7.3.1 When passing through theasfront of the shock wave, thefront pressure, density of the medium’s substance, as well its velocity relative to the of thetemperature, shock wave,

density medium’s substance, as wellindependently, as its velocity but relative to the to front of thecharacteristic shock wave, change. of Allthe these quantities do not change are related a single change. All these quantities do not change independently, but are related to a single characteristic of the shock wave, the Mach number. The mathematical equation relating thermodynamic of the shock wave, number. quantities before and the afterMach the passage of a The shockmathematical wave is calledequation the shockrelating adiabat,thermodynamic or the Hugoniot quantities before and after the passage of a shock wave is called the shock adiabat, or the Hugoniot adiabat. Shock waves do not have the additivity property in the sense that the thermodynamic adiabat. Shock waves do not have the additivity property in the sense that the thermodynamic state of the medium that occurs after the passage of one shock wave cannot be obtained by state of the medium thatshock occurs afterofthe passage of one shock wave cannot be obtained by consistently passing two waves lesser intensity. consistently passing two shock waves of lesser intensity.

Definition 7.3.2 — Shock waves origination. Sound is a fluctuation of density, velocity and Definition — Shock waves origination. is aoffluctuation of density, and pressure of 7.3.2 the medium, propagating in space. TheSound equation state of ordinary mediavelocity is such that,

pressure of the medium, propagating in space.velocity The equation of state of ordinary media is increases. such that, in the high-pressure region, the propagation of small-amplitude disturbances in the high-pressure region, the propagation velocity of small-amplitude disturbances increases. This inevitably leads to the phenomenon of "overturning" of disturbances of finite amplitude, This inevitably leads waves. to the phenomenon of mechanism, "overturning" of disturbances of finite amplitude, which generate shock By virtue of this a shock wave in a conventional medium which generate shock waves. By virtue of this mechanism, a shock wave in a conventional medium is always a compression wave. The described mechanism predicts the inevitable transformation is always a compression wave. The described mechanism predicts the inevitable transformation

92

Chapter 7. Hydrodynamics

Chapter 7. Hydrodynamics

92

of any sound wave into a weak shock wave. However, in everyday conditions this requires too much time, so that the sound wave has time to damp out before the nonlinearities become noticeable. For the rapid transformation of the density oscillations into a shock wave, strong initial deviations from equilibrium are required. This can be achieved either by creating a very high volume sound wave, or mechanically, by transonic movement of objects in the environment. That is why shock waves easily arise in explosions, in near-and supersonic body motions, in powerful electric discharges, etc.

Shock waves at Saturn could reveal secrets of exploding stars Definition 7.3.3 — Shock waves hydrodynamical equations.

ρ1 u1 (be f ore shock wave f ront) = ρ2 u2 (a f ter shoch wave f ront) = j P1 + ρ1 u21 = P2 + ρ2 u22 h1 + 12 u21 = h2 + 12 u22  h2 − h1 +

j2 2

1 ρ22

2 −P1 ) − ρ12 = 0 , j2 = − (P V2 −V1 , 1

1) 1) (V2 +V1 ) = 0 , ε2 − ε1 − (P2 +P (V1 −V2 ) = 0 h2 − h1 − (P2 −P 2 2

Definition 7.3.4 — Nuclear explosion. The shock wave in most cases is a striking factor for

a nuclear explosion. By its nature, it is similar to the shock wave of a blast, it acts for a longer

7.4 Combustion 7.4 Combustion

93

93

time and has a much greater destructive force. A shockwave of a nuclear explosion can, at a considerable distance from the center of the explosion, inflict damage on people, destroy structures and damage military equipment.

7.4

Combustion

Definition 7.4.1 Combustion is a complex physico-chemical process of transformation of the

initial substances into products of combustion in the course of exothermic reactions, accompanied by intense heat generation. The chemical energy stored in the components of the initial mixture can also be released in the form of thermal radiation and electromagnetic radiation. Gore is a complex physico-chemical process of the transformation of precursors into combustion products during exothermic reactions, accompanied by intense heat generation. The chemical energy stored in the components of the initial mixture can also be released in the form of thermal radiation and light. The luminous zone is called the flame front, or simply the flame. Mastering the fire has played a key role in the development of human civilization. The fire opened the way for people to heat treat food and heat homes, and later - to develop metallurgy, energy and create new, more advanced tools and technologies. The control of combustion processes underlies the creation of engines for automobiles, ships and rockets. Combustion is still the main source of energy in the world and will remain so in the near future. Chemical reactions of combustion, as a rule, follow a branched chain mechanism with progressive self-acceleration due to the heat generated in the reaction. The features of combustion that distinguish it from other physicochemical processes involving redox reactions are a large thermal effect of the reaction and a large activation energy, leading to a strong dependence of the reaction rate on temperature. As a result, a combustible mixture that can be stored at room temperature indefinitely may ignite or explode when the critical ignition temperature is reached (self-ignition) or when initiated by an external energy source (forced ignition or ignition). If the products formed during the combustion of the initial mixture in a small volume in a short period of time, perform significant mechanical work and lead to shock and thermal effects on the surrounding objects, then this phenomenon is called an explosion. A special kind of burning is detonation. Definition 7.4.2 Reaction rate, the speed at which a chemical reaction proceeds. It is often expressed in terms of either the concentration (amount per unit volume) of a product that is formed in a unit of time or the concentration of a reactant that is consumed in a unit of time. Definition 7.4.3 — Arrehenius law. Arrhenius law gives the dependence of the rate constant of

a chemical reaction on the absolute temperature, a pre-exponential factor and other constants of the reaction. Ea

k = Ae− RT • Hydrodynamical equations in the presence of chemical reaction ∂ ci ∂t

− u ) = Si+ − Li− + div (ci →

∂ T = κT +W (S+ , L− ) i i ∂t • Heat explosion

8. Complementary materials

8. Complementary materials

8.1

Appendix 1.Vector and tensor calculas Definition 8.1.1 — Levi-Civita tensor. A tensor Levi-Civita looks like this



 0, i f any two labels are the same εi jk =  1, i f i, j, k is an even permutation o f 1, 2, 3 −1, i f i, j, k is an odd permutation o f 1, 2, 3

R

• Repeated indices are imlicitly summed over. • Each index can appear at most twice in any term • Each term must contain identical non-repeated indices

Definition 8.1.2 — Cross poduct of two vecrors presentation with Levi-Civita symbol.

   e 1 e2 e3    → − → − A × B = A1 A2 A3  = εi jk A j Bk ei B1 B2 B3 

Definition 8.1.3

εi jk εimn= δ jm δkn −δ jn δkm 0, j = m δ jm = 1 j=m Some useful examples presented below 

Example 8.1

→ − → − ∇T = ei ∂∂ xTi , ∇ · A = ∂∂Axii , ∇ × A = εi jk ei ∂∂Ax kj −r = 3, ∇ × → −r = 0 ∇→

95

96

Chapter 8. Complementary materials

Chapter 8. Complementary materials

96

 

Example 8.2

→ − → − ∇ · ( A × B ) = ei ∂∂xi (ε jkl Ak Bl )e j = ∂∂xi (ε jkl Ak Bl ) = = εikl ( ∂∂Axik Bl + Ak ∂∂Bxil ) = εikl ∂∂Axik Bl + εikl Ak ∂∂Bxil = = εlik ∂∂Axik Bl − Ak εkil ∂∂Bxil = → − → − → − → − − → → − → − → − = (∇ × A ) · B − A · (∇ × B ) = B·(∇ × A ) − A · (∇ × B )



8.2

Appendix 2 Residue Theorem 8.2.1 — Cauchy’s theorem. For any closed path C the integral of continous complex

function is equal zero 

f (z)dz = 0

f (z) =

1 2πi

f (n) (z) = 8.2.1

 f (ζ ) dζ ζ −z  f (ζ ) n!

2πi

(ζ −z)n+1

dζ

Residue Definition 8.2.1 — Residue. n f (z) = ∑∞ n=−∞ an (z − z0 )  f (ζ ) 1 an = 2πi dζ (ζ −z0 )n+1  1 a−1 = 2πi C f (ζ )dζ



Example 8.3

n f (z) = ∑∞ n=−∞ an (z − z0 )  f (ζ ) 1 an = 2πi dζ (ζ −z0 )n+1  1 a−1 = 2πi C f (ζ )dζ 

8.3 Useful tables

97

8.3 Useful tables

8.3 8.3.1

Useful tables Some fundamental constants Quantity Avogadro number Speed of Light Planck’s constant Boltzmann constant electron-volt Gravitational constant Permeability of free space Permitivity of free space Electron’s charge Atomic mass unit Electron mass

me

Neutron mass

mn

Proton mass

mp

Borh magnetic moment Proton magnetic moment 8.3.2

Symbol NA c h k eV G µ0 ε0 e u

eh µe = 2m e h h = 2π eh µ p = 2m p

Physical data often used Average earth-moon distance =3.84 ∗ 108 m Average earth-sun distance= 1.49 ∗ 1011 m Average radius of the Earth=6.37 ∗ 106 m Density of air =1.29 mkg3 Density of water =(1.00 ∗ 103 mkg3 )1.00 ∗ 103 mkg3 Standard atmospheric pressure= 1 atm = 1.013 ∗ 105 mN2 Mass of the Earth=5.98 ∗ 1024 kg Mass of the Moon=7.36 ∗ 1022 kg Mass of the Sun=1.9981030 kg

Value 6.0221023 2.99 ∗ 108 ms 6.626 ∗ 10−34 J · s 1.38 ∗ 10−23 KJ 1.602 ∗ 10−19 J 2 6.67210−11 Nm kg2 4π ∗ 10−7 AN2 C2 8.85 ∗ 10−12 Nm 2 1.6 ∗ 10−19C 1.66 ∗ 10−27 kg 9.1 ∗ 10−31 kg 5.49 ∗ 10−4 u 0.511 Mev c2 1.675 ∗ 10−27 kg 1.008u 939.6 Mev c2 1.673 ∗ 10−27 kg 1.007u 938.28 Mev c2 9.27 ∗ 10−24 Am2 5.05 ∗ 10−27 Am2

97

Еducational issue

Kunakov Sandybek Kadyrovich PHYSICS FOR UNIVERSITY STUDENTS COURSE OF MODERN PHYSICS

Textbook First edition

Cover design G. Kaliyeva Cover design photos were used from sites www.GHubble-image-2-square-1-1030x1030.com

IB No.13299

Signed for publishing 11.01.2020. Format 70x100 1/8. Offset paper. Digital printing. Volume 16 printer’s sheet. 100 copies. Order No.270. Publishing house Qazaq University Al-Farabi Kazakh National University KazNU, 71 Al-Farabi, 050040, Almaty Printed in the printing office of the Qazaq University Publishing House.