Qualitative Methods in Quantum Theory 0738203025, 9780738203027

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Qualitative Methods Quantum Theory

Qualitative Methods in Quantum Theory

A. B. Migdal I. V. Kurchatov Atomic Energy Institute Moscow Russia

,

Translated from the Russian edition by Anthony J. Leggett

University of Sussex

Advanced Book Program

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ISBN 13:978-0-7382-0302-7 (pbk)

A dvanced book C lassics David Pines, Series Editor Anderson, P.W., Basic Notions of Condensed Matter Physics Bethe H. and Jackiw, R., Intermediate Quantum Mechanics, Third Edition Cowan, G. and Pines, D., Complexity: Metaphors, Models, and Reality de Gennes, P.G., Superconductivity of Metals and Alloys d'Espagnat, B., Conceptual Foundations of Quantum Mechanics, Second Edition Feynman, R., Photon-Hadron Interactions Feynman, R., Quantum Electrodynamics Feynman, R., Statistical Mechanics Feynman, R., The Theory of Fundamental Processes Gell-Mann, M. and Ne'eman, Y., The Eightfold Way Khalatnikov, I. M. An Introduction to the Theory of Superfluidity Ma, S-K., Modem Theory of Critical Phenomena Migdal, A. B., Qualitative Methods in Quantum Theory Negele, J. W. and Orland, H., Quantum Many-Particle Systems Nozieres, P., Theory of Interacting Fermi Systems Nozieres, P. and Pines, D., The Theory of Quantum Liquids Parisi, G., Statistical Field Theory Pines, D., Elementary Excitations in Solids Pines, D., The Many-Body Problem Quigg, C., Gauge Theories of the Strong, Weak, and Electromagnetic Interactions Schrieffer, J.R., Theory of Superconductivity, Revised Schwinger, J., Particles, Sources, and Fields, Volume I Schwinger, J., Particles, Sources, and Fields, Volume II Schwinger, J., Particles, Sources, and Fields, Volume III Schwinger, J., Quantum Kinematics and Dynamics Wyld, H.W., Mathematical Methods for Physics

CONTENTS

Page xv

Editor’s Foreword

CHAPTER 1 1

Translator’s Note

xvii

Preface

xix

DIMENSIONAL AND "MODEL" APPROXIMATIONS Order-of-Magnitude Estimation of Mathematical Expressions

6

Estim ation of Integrals

7

The Method of Steepest Descents

13

Properties of Integrals of Oscillating Functions. Estim ates of the Higher Term s in Fourier Series Expansions

17

Methods of Approximate Solution of Differential Equations

25

Atomic Physics

31

Estim ate of the Velocities and Orbit Sizes of the Inner Atomic Electrons

32

Stationary States

33

Distribution of E le ctric Charge in the Atom

39

The Rutherford Formula

41

Inapplicability of C lassical Mechanics for Large Impact P aram eters

43

vii

CONTENTS

viii

Page

3

Estim ate of the Scattering Cross Section for Potentials which Fall Off F aster than the Coulomb Potential

45

Resonance Effects in Scattering

47

Interaction Between Atoms

49

Ionization of Atoms

50

Multiple Scattering

51

Interaction with Radiation

54

Zero-point Vibrations of the E le c tro magnetic Field

54

The Photoeffect

57

Lifetimes of Excited Atomic States

63

Bremsstrahlung

65

P air Creation

69

Creation of Soft Photons in the Scattering of Charged P articles (The "Infrared Catastrophe")

71

The Lamb Shift

80

Asymptotic C haracter of Series in Quantum Electrodynamics

84

CHAPTER 2

VARIOUS TYPES OF PERTURBATION THEORY

86

1

Perturbation Theory in the Continuous Spectrum

89

Scattering of Charged P articles by the Atomic Nucleus

95

Perturbation of the Boundary Conditions

98

The Energy Levels of a Deformed Nucleus

99

2

CONTENTS

ix

Page 3

4

5

6

CHAPTER 3 1

Sudden Perturbations

102

Ionization of Atoms in /3 -decay

105

Ionization of an Atom in Nuclear Reactions

109

Transfer of Energy when a Photon is Emitted by a Nucleus in a Molecule (Mossbauer Effect)

112

Adiabatic Perturbations

115

Ionization of an Atom by the Passage of a Slow Heavy P article

119

Capture of an Atomic Electron by a Proton (Charge Exchange)

122

F ast and Slow Subsystems

127

Vibrational Energy Levels of a Molecule

129

Excitation of Nuclear Dipole Levels by a F ast P article

132

Scattering of a Proton by a Hydrogen Atom (Charge Exchange)

13 7

Perturbation Theory for Adjacent Levels

140

A P article in a Periodic Potential

142

The Stark Effect in the Case of Adjacent Levels

144

The Change of the Lifetime of the 2s 1/2 State of the Hydrogen Atom under an Applied E le ctric Field

145

THE QUASICLASSICAL APPROXIMATION 149 The One-Dimensional Case Asymptotic Series

152 153

CONTENTS

Page Matching of Q uasiclassical Functions

154

The Quantization Condition

160

A ccuracy of the Q uasiclassical Approximation

162

Normalization of Q uasiclassical Functions

163

The Correspondence Principle

165

Mean Kinetic Energy

165

Connection between the Q uasiclassical Matrix Elements and the Fourier Components of C lassical Motion

166

Criterion for the Applicability of P ertu r­ bation Theory to the Calculation of Not Too Small Quantities

168

Calculation of M atrix Elements in the Case of F ast Oscillating Functions

170

B a rrie r Penetration

178

Reflection Above a B a rrie r

186

The Three-Dimensional Case

189

Spherically Symmetric Field

189

Modification of the Centrifugal Potential

191

Energy Levels in the Coulomb Potential

192

Q uasiclassical Representation of Spherical Functions

194

The T hom as-Ferm i Distribution in the Atom

197

Estim ates of Nuclear M atrix Elements

202

Noncentral Potential

206

The Q uasiclassical Scattering Problem

208

CONTENTS

xi

Page

CHAPTER 4

1

2

Cross Section for Scattering of a Proton on a Hydrogen Atom

211

THE ANALYTIC PROPERTIES OF PHYSICAL QUANTITIES

214

Dependence of the Moment of Inertia of a Nucleus on Deformation

218

Dependence of the Frequency of Sound on the Wave Vector

219

Analytic Properties of the D ielectric Constant

221

Analytic Properties of the D ielectric Constant in a Simple Model

226

Analytic Properties of the Scattering Amplitude

229

Unitarity as a Consequence of the Superposition Principle and the Con­ servation of Probability

229

The Dispersion Relation

231

Resonance Scattering at Low Energies

233

Nonresonant Scattering at Low Energies

238

Scattering by a Potential Well

240

Analytic Properties of the Wave Function 242 Single-Particle Wave Functions of the Continuous Spectrum at Low Energy

244

The Use of Analyticity Properties in Physical Problems

247

Theory of Nuclear Reactions with the Formation of Slow P articles

247

Interacting P articles in a Potential Well

251

CONTENTS

xii

Page

CHAPTER 5 1

2

3

Theory of D irect Reactions

255

Threshold Singularities of the Scattering Amplitude

259

METHODS IN THE MANY-BODY PROBLEM

261

The Quasiparticle Method and GreenTs Functions

269

The Transition Amplitude

269

O ne-Particle GreenTs Functions in a System of Noninteracting P articles (Quasiparticle Green’s Functions)

273

The Green’s Function in a System of Interacting P articles

276

Analytic Properties of the O ne-Particle Green’s Function

278

Calculation of Observable Quantities

283

The Ferm ion Momentum Distribution

286

The Graph Method

288

Graphical Representation of P ro cesses

288

The Interaction Between Quasiparticles

302

The Local Quasiparticle Interaction

309

The Solution of Problems by the Green’s Function Method Dyson’s Equation. Model

311

The B asis of the Shell 311

Instability of the Ferm i Distribution in the Case of Attraction. The O ccurrence of a Gap in the Energy Spectrum

316

xiii

CONTENTS

Page

CHAPTER 6 1

The Energy Spectrum of a Bose System. Superfluidity

324

A System in an External Field

333

Change of the P article Distribution in a Field

336

Spin Polarizability and Quasiparticle Magnetic Moments

337

Sound Waves in a Ferm i System ("Zero Sound")

338

Plasm a Oscillations. Charge in a Plasm a

Screening of a

340

Conservation Laws and Quasiparticle Charges for Different Fields

343

QUALITATIVE METHODS IN QUANTUM FIELD THEORY

348

Construction of Relativistic Equations

357

Lorentz Invariance

358

MaxwelPs Equations

363

The Klein-Gordon-Fock Equation

365

The Dirac Equation

367

The Green 1s Function of a Spinless Particle

370

The GreenTs Function of a Particle With Spin 1/2

374

The Photon Green's Function

376

Divergences and Renormalizability

379

The Local Interaction Between P articles

380

Feynman Graphs in a Scalar Theory

384

CONTENTS

xiv

Page

3

Estimation of Divergences: The Idea of Renormalization

387

The Condition for Renormalizability

393

The Logarithm ic Approximation and Renormalizability

398

Quantum Electrodynam ics at Small Distances

409

The Local Interaction in Quantum Electrodynamics

409

Vacuum Polarization

414

Radiative Corrections to CoulombTs Law

417

The Electrom agnetic Interaction at Ultra-Sm all Distances

421

Index

433

Editor's Foreword Perseus Publishing's Frontiers in Physics series has, since 1961, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics—without having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty year existence, the series has emphasized informality in both style and content, as well as pedagogical clarity. Over time, it was expected that these informal accounts would be replaced by more formal counterparts—textbooks or monographs— as the cutting-edge topics they treated gradually became integrated into the body of physics knowledge and reader interest dwindled. However, this has not proven to be the case for a number of the volumes in the series: Many works have remained in print on an ondemand basis, while others have such intrinsic value that the physics community has urged us to extend their life span. The Advanced Book Classics series has been designed to meet this demand. It will keep in print those volumes in Frontiers in Physics that continue to provide a unique account of a topic of lasting interest. And through a sizable printing, these classics will be made available at a comparatively modest cost to the reader. The late Arkady B. Migdal was one of the great theoretical physicists of our time, who made significant contributions to nuclear physics, condensed matter physics, astrophysics, and particle physics. He combined keen physical insight with a deep understanding of physics and physical phenomena and believed it truly important that students learn more about quantum theory than about the mathematical manipulation of formulae and equations. It was for that reason that he developed the series of lectures on qualitative methods in quantum theory that are contained in this book. Migdal's originality and careful attention to pedagogy make his book, Qualitative Methods in Quantum Theory, required reading for every scientist interested in learning and applying quantum theory. I am most pleased that its publication in Advanced Book Classics will now make it readily available to the future generation of scientists, who may be expected to profit greatly from reading it.

David Pines Cambridge, England May, 2000

TRANSLATORS NOTE

fi1 the interests of rapid and economical production most of the equations in this book have been photocopied directly from the Russian version.

Generally, notations which may be unfamiliar

to English-speaking read ers have been changed to conform to the English usage, but there are a few cases where this has proved awkward or im possible; the most frequent ones are the following : A simple sequence of two vectors (indicated by boldface) denotes the scalar or dot product. The notation [ a , b ] with a, b boldface denotes the vector product of a and b. 2 2 V , V^

denotes the Laplacian with respect to r and R

respectively. sh indicates hyperbolic sine,

(sinh)

Sp indicates the trace of a m atrix.

xvii

PREFACE

The solution of most problems in theoretical physics begins with the application of the qualitative methods which constitute the m ost attractive and beautiful ch aracteristic of this discipline.

By "qualitative methods" we mean dimensional

estim ates and estim ates made by using simple models, the investigation of limiting cases where one can exploit the sm allness of some param eter, the use of the analytic properties of physical quantities, and finally the derivation of consequences from the symmetry p roperties, that is, the invariance relative to various transform ations (e .g . Lorentz or isotopic invariance).

However,

as experience in the classroom shows, it is just these aspects of theoretical physics which are most difficult for the beginner. Unfortunately, the methods of theoretical physics are usually presented in a form al, mathematical way, rather than in the constructive form in which they are used in scientific work. The object of this book is to make up this deficiency, that is, to teach to beginning students of the subject the co rre ct approach to xix

PREFACE

XX

the solution of scientific problems.

This goal largely determines

the character of the presentation; the general results are always obtained first in special cases or with extrem ely simplified models. It seem s to me that a form al exposition, which leaves no tra c e s of a gradual approach to the problem, no tra ce s of the "sw eat" involved, can often leave the beginner in scientific research with a sense of something lacking.

I have therefore endeavoured as far

as possible to indicate the general method of approach to the problem, especially at the first stage of the work.

Of cou rse, this

means I have had to sacrifice rigour in the exposition, and in return disclose some "trade s e c re ts " , that is,the little tricks which shorten the derivation of the resu lts. A common mistake of beginners is the desire to understand everything completely right away.

In real life understanding comes

gradually, as one becomes accustomed to the new ideas.

One of

the difficulties of scientific research is that it is impossible to make progress without clear understanding, yet this understanding can come only from the work itself;

every completed piece of research

represents a victory over this contradiction. will inevitably occur in reading this book;

Similar difficulties

I hope that by the time

it is read to the end they will be overcome. Each of the six chapters of the book begins with a detailed introduction, in which the physical meaning of the results obtained in the chapter is explained in a simple way.

The first three

chapters are devoted to dimensional and model-based estim ates in atomic physics, the applications of various types of perturbation theory and the quasiclassical approximation respectively.

These

chapters are a revised version of the book by A. B. Migdal and

xxi

PREFACE V. P. Krainov "Approximation Methods in Quantum Mechanics"

(Nauka, 1966 : translation published by W. A. Benjamin, Inc., New York, 1969).

The fourth chapter is devoted to various problems

solution of which requires the use of the analytic properties of physical quantities.

The fifth chapter develops the graphical

method and its application to the many-body problem.

Finally, the

sixth chapter is devoted to questions connected with the interaction of elementary particles at short distances;

in this problem of

quantum field theory it is precisely the application of qualitative methods which plays the main role. The author is deeply grateful to A. A. Migdal, A. M. Polyakov and B. A. KhodeP for numerous discussions and sugges­ tions, and to V. P. Krainov for his help in the selection of m aterial for the first three chapters.

He also thanks his friends and

students G. Zasetskii, D. Voskresenskii, N. Kirichenko, O. Markin, L Mishustin, G. Sorokin and A. Chemoutsan for help in the preparation of the manuscript.

A . B . Migdal

CHAPTER 1

DIMENSIONAL AND "MODEL" APPROXIMATIONS

No problem in physics can ever be solved exactly.

We

always have to neglect the effect of various factors which are unimportant for the particular phenomenon we have in mind.

It

then becomes important to be able to estimate the magnitude of the quantities we have neglected.

M oreover, before calculating a

result numerically it is often n ecessary to investigate the pheno­ menon qualitatively, that is, to estimate the order of magnitude of the quantities we are interested in and to find out as much as possible about the general behaviour of the solution. To this end we first consider the problem in the most simplified form possible.

F o r instance, in the case of a particle

moving in a Coulomb field, we replace the problem by that of its motion in a square well potential with an appropriately chosen depth and width depending on the particle energy, and so on. M oreover, we should consider all the limiting cases in which the solution is simplified.

F o r instance, if it is required to solve the

problem of scattering of particles of arbitrary energy, we should 1

QUALITATIVE METHODS IN QUANTUM THEORY

2

first consider the limits of small and large energy and tra ce out how the corresponding expressions match up in the intermediateenergy region.

The aim of this chapter is to instruct the reader

in the art of obtaining approximate solutions from dimensional estim ates with the aid of a simplified model of the phenomenon to be investigated. B

Fig. 1 In some cases dimensional techniques actually enable one to get quantitative rath er than m erely qualitative resu lts.

For

instance, we can prove Pythagoras’ theorem purely from dimen­ sional considerations (see Fig. 1).

It follows by dimensional

reasoning that the area of the triangle ABC can depend only on the square of the hypotenuse, c , multiplied by some function f(o') of the angle a .

The same applies to the areas of the two sim ilar

triangles ABC and BCD, but for these the hypotenuse is resp ec­ tively the sides AB, BC of the large triangle.

Hence

c2/ (a) = a2/ (a) + b*f (a), which proves the th eo rem . As a second example we consider the problem of finding the resistive force when a body moves in a viscous medium with arb itrary velocity.

We sta rt with the limiting case of small

DIMENSIONAL AND "MODEL" APPROXIMATIONS velocities;

3

then the resistive force will be determined by the

viscosity of the medium.

The param eter which defines the notion

of "sm all" as opposed to "larg e" velocities may be found by forming a quantity with the dimensions of velocity from the viscosity, the density of the medium and the dimensions of the body.

We assume that the body has all its dimensions approxi­

mately equal; then for the purposes of dimensional estim ates it can be characterized, just like a sphere, by a single length R. From the viscosity 77 , density p, length R and velocity v we can form only one dimensionless combination, the so-called Reynolds number

where v - 77/p -

Since the momentum current is given by 77 £ v ,

the order of magnitude of the force acting on unit surface area is P ** 77 v/R .

(The estim ate of the velocity gradient as v/R

is

made as follows: at the surface of the body the velocity of the liquid is v , while far from the body (at distances of order R) the liquid is at re st.

Hence vv ~ A v/R ~ v / R .)

surface area of the body as

If we estimate the

2

4 7rR , then the total resistive

force is given by F ~ knv\vR.

We may note that an exact solution of the problem for a spherical body gives in the case of small velocity F = frrcr)vR,

hi the case of arbitrary velocity this expression must be multiplied by some function of the dimensionless param eter Re : F =

•

4

QUALITATIVE METHODS IN QUANTUM THEORY Now let us consider the limit of very large velocities.

In

this case the resistive force is independent of the viscosity and is determined by the momentum which is transferred per unit time to the column of liquid lying in front of the body; the base area of this column is just the cro ss-sectio n al area of the body, and so we find F ~ ftR 2pv2.

Thus, for large velocities the function $ (x ) satisfies the estim ate $ (x )~ ~ x . The rough character of the solution for all velocities is determined by the interpolation formula F ~ 6itr\vR (l + 4 “ ^ " ) *

According to this estim ate, the transition from one regim e to the other should take place for R e ^ 6 .

In reality, the transition to

the turbulent regim e, where the resistiv e force is independent of viscosity, takes place for R e ~ 1 0 0 .

Here we have run up against

a rather unusual case - usually, the transition from one limiting case to the other is characterized by a value of the relevant dimensionless param eter of order unity. Another example we shall give relates to the possibility of constructing a theory which will connect gravitation and electro ­ dynamics.

Such a theory, if it existed, would have to fix the value

of a dimensionless param eter relating the gravitational constant g to the quantities which characterize electrom agnetic p ro cesses, viz. the electron charge e and m ass m , the speed of light c and PlanckTs constant ft.

From these quantities it is possible to

construct two dimensionless ratio s:

DIMENSIONAL AND ’’MODEL" APPROXIMATIONS

5

p = i £ 2- = 2 •10-46. As already mentioned, the dimensionless param eters which occur as a result of solution of the equations usually turn out to be of order unity.

Thus the quantity £ must enter in such a way that

one obtains a number of order unity, e. g. a In (1/E) -

1.

It is indeed in just such a form that the param eters a

and £ come

into those estim ates which give some hope of establishing a con­ nection between gravitation and electrodynamics. We shall give one more example of the way in which o rd erof-magnitude estim ates help one to orient oneself in complicated problems.

L et us answer the question: beyond what strengths of

the electric field £ and magnetic field K do Maxwell1s equations in free space become nonlinear ?

The reason for the nonlinearity

is the perturbation of the vacuum by the external field.

Let us

then construct a quantity with the dimensions of field from the quantities which ch a racterize the vacuum fluctuations of the electron-positron field.

Since e £ has the dimensions of energy

divided by length, we find

It is clear from this expression that the critical field 8

is c determined by the particles with the sm allest m ass, that is, by the electron-positron field.

We see that the quantity

is the field

strength at which the potential difference acro ss a Compton wave­ length is of the order of the energy n ecessary for pair creation. Substitution of the numerical values of e , m , fi and c gives £ ~ 10 16 V /cm . c

6

1.

QUALITATIVE METHODS IN QUANTUM THEORY

ORDER-OF-MAGNITUDE ESTIMATION OF MATHEMATICAL EXPRESSIONS Before explaining techniques for estimating physical

quantities, we first review a rath er simpler sort of estimation problem, namely the estimation of mathematical expressions. The basic idea involved consists in determining the region of the variables which gives the principal contribution to the resu lt, separating the part of the mathematical expression which is fast varying in this region from the slowly varying p art, and also using the asymptotic form s of the expression.

Estimation of a derivative.

In the simplest ca s e , when the

important region of variation of the function F (x) is characterized by a single length ft, then the order of magnitude of the derivative F f(x) is simply F (i)/L

2

2

F o r instance, if F (x) = exp (-x /ft ), then

the derivative F ?(x) = - (2 x/j02 ) exp (-x 2 / f 2 ),

so that Tp' (ft)** F (ft)/L

However, for x much larg er than ft this estim ate is clearly invalid.

F o r a power function F(x) - x 11 the M region of appreciable

changeM is defined by the variable x itself.

In fact we have

F ' (x) = nxn~x ~ n F (x)lx .

In some cases the relevant length ft is different for different regions of variation of the variable x.

Then in each

region of x the derivative F*(x) is of order F (x )/f (x ), where f (x) is the length over which F(x) changes appreciably in this region.

F o r instance, suppose F(x) has the form shown in fig. 2.

Then we have F *(x)~ F (x 1 ) / f 1 for x ^ x - ^ b u t F f(x )^ F (x 2 ) / f 2

DIMENSIONAL AND "MODEL” APPROXIMATIONS

for x ~ Xg*

7

In more complicated c a s e s , if F(x) can be sketched,

even roughly, the best way of estimating its derivative is from the graph.

Estimation of integrals.

We will demonstrate some methods of

estimating integrals by various examples: 1.

Often one can obtain approximate values of integrals by

expanding the integrand in a power series. X

F o r instance, we write

x

^exp (— t2) dt = ^{ 1 — t2 + ^4/2

o

••»)dt =

a8 £5 - x - T + TO

0

This integral converges for all x.

To estim ate the integral we

may r e s tric t ourselves to the first few term s of the se rie s; the resulting estimate

will of course be appropriate only for x oo the factorial (2 n -l )!I

increases faster

than the power term This series is an example of a so-called asymptotic series (for details see p.153).

Since it diverges, it does not pay

to take a very large number of term s when one is estimating the integral; this actually makes it less accurate. the optimum number of term s to keep ?

How do we find

We notice that for large

x the term s of the series first d ecrease in absolute magnitude and then subsequently begin to increase.

The optimum number of

term s is evidently defined by the requirement that the rem ainder of the series should be a minimum.

It is easy to see that the

rem ainder is of the order of the (n + l)-th term of the series. Hence the co rre ct prescription is to sum as far as the sm allest term of the series.

The condition for the minimum to be reached

at the n-th term can be approximated by setting the n-th and (n+l)-th term s equal: (2/i — l)!!

(2/i + l)H

2n+l/p2n+l

2n+2£2n*:l

Hence we find n ^ x 2 .

Problem.

Show that, in estimating the integral

exp(-t)dt x in the case x » 1 , the optimum number of term s to keep in the asymptotic se ries is equal to x. 2.

Many integrals can be estimated by separating out the most

important part of the integrand. X

Case 1)

= 0

Consider the following examples:

DIMENSIONAL AND "MODEL" APPROXIMATIONS If x «

9

2

1 , then the exponential exp (t ) in the integrand is approx­

imately equal to 1.

Consequently

5 V **-** = 5 y r = & ' 0 r 0 Since this integral contains no p aram eters, we have I(x)~ 1. i (An exact calculation of the above integral gives [ __ j V i — z2

0

z

If x » 1 , then, in view of the exponential increase of the 2

factor exp (t ), the principal contribution to the integral comes from the region near the point t = x.

If we write £ = x - t, then

we have I(x) = \ exp 0.

0 oo

(2 )

(3)

S” p( - « W

, 4

r

-

sin (a;/a)

J *(*»+&«)

+ t) ■ «.»>. a’ & > 0 -

Solutions. (1)

a ^ > b : y^Vt/Sa; a

The exact value of the integral is

b: aj^jt/1663.

DIMENSIONAL AND "MODEL" APPROXIMATIONS (2 ) a >> b: In (a/b); a

13

b: Y nalb.

The exact value is exp (b/2a)'K0 (b/2a) where K

is the Macdonald function,

o (3 )

o

b: n!2ab; a < ^ b: n/262.

The exact value is « [ l _ e x p ( - b/a)].

j|{

The method of steepest descents

Consider the integral

r» 00

I =

g(t) exp f ( t ) d t , where f(t) is a function which has a sharp

o maximum for some value t > 0 of t. Suppose that near t the o o function g(t) is slowly varying. Then we can replace the function f ge near the maximum by a simpler function; to do this we expand f in a Taylor series around its maximum tQ: f ( t ) = f(to) + - r ( t - t o ) Y ( t o ) + •••

i i "2 Assume that |f" (tQ) ( » t Q ; this is just the mathematical expres­ sion of the assumption that f has a sharp maximum.

2

values of (t-t ) l / f " (tQ),

®

In fact, the

which are important in the integral I are of order

as we shall see below (eqn. 1 .3 ) ;

2

thus, (t-tQ) / t

2

* This method is discussed in m ore detail in Chapter 3 (p. 170).

1.

QUALITATIVE METHODS IN QUANTUM THEORY

14

This condition makes it legitimate to omit the higher term s in the Taylor series written above for f(t) (cf. below). We have

(1.3) - OO

Here we replaced the limits of integration by [-«> ,«> ], since the integrand d ecreases exponentially in the region

L et us now estimate the correction given by the subsequent term s in the Taylor series.

If we keep only the cubic term and

expand the exponential of it in a s e rie s, then the first term of the expansion gives no contribution, since the integrand is odd.

There­

fore we consider the fourth-order term in the Taylor expansion of f , namely f ^ V 0 )(t-t ) ^ /4!

If we now further expand the exponen­

tial of this quantity in a s e rie s , we find that the correction is of order f ^ V ( f " ) 2

relative to the expression (1.3).

If the function

f(t) is characterised by a single param eter, then estimating the order of magnitude of the derivatives of f , we find

Thus, the condition for applicability of the method of steepest descents is f(t ) » 1 ; this is equivalent to the assumption ft —2 |f"(t )| » t made above. o' 1 o If in (1. 3) we were to make the substitution t “tQ= i £ ,

the

DIMENSIONAL AND "MODEL" APPROXIMATIONS integrand would become an increasing exponential. the point t

15 In other words

is a saddle point in the complex t plane and the

direction of integration is the direction of steepest descent from the saddle point (see Fig. 3).

Hence the name, saddle-point

method or method of steepest descents.

We have actually consi­

dered the special case in which the direction of steepest descent coincides with the real axis;

one can also consider the general

case in which the direction of steepest descent makes an arbitrary angle with the re a l axis.

Fig. 3 L et us use the method of steepest descents to obtain an asymptotic expression, for large x , for the gamma function T (x + l) = ^ exp (— t + x In t) dt.

o We write -t + x Hn t = f(t).

Then the condition that f! (t) be zero

gives us the saddle-point tQ: f (to) = — 1 + 7 7 =

16

QUALITATIVE METHODS IN QUANTUM THEORY

whence we have t = x . Since f(t ) = x Hn x - x , the condition of o 'o ’ applicability of the method of steepest descents, viz. f ( x ) » l , 2

means that x » l .

We further have f"(t ) = - x / t = - 1 /x . v o7 o Using (1 .3 ), we therefore obtain T (x + 1) »

a . 4)

\f2nx (x/e)x.

This asymptotic formula is called the Stirling formula.

To

estim ate its accuracy we use the relation T(x + 1 ) = x T (x ), and write the (unknown) exact expression for T(x + 1 ) in the form

r (x + 1) = Y^u (-f )'r li + , where v is the velocity of the incident 2

electron.

Since (e / # v ) « 1, the minimum angle 0 . is defined v ’ mm not by the classical condition but by the diffraction condition (cf. p.43 ), i* e 0 0 . ~ 1/pp or q . * co /v . mm mm o HE dx

Z n , pv r ln — p* coo

Thus we have

.

As we have proved above (p.40) the bulk of the electrons 1 /3 found in a region of radius a /Z , and the energy per 4 /3 ° is of order Z .The quantity co is of order v / a ~ Z

are to be electron 2 /3 -1 /3 /Z =Z.

Re-introducing dimensional quantities and introducing the ionization potential of hydrogen, d E _ p Zne2 , ~dJ~~ ~tnv*

Iq, we obtain E hZ ’

An exact calculation gives C =* 47T.

Multiple Scattering A narrow beam of electrons passing through a medium will gradually be smeared out as a result of multiple scattering; although we may assume the probability of an electron being

for,

52

QUALITATIVE METHODS IN QUANTUM THEORY

deflected to left or to right is equal, the m ean-square deflection angle is not zero.

We encounter a sim ilar phenomenon in everyday

life if we observe the progressive twisting of a telephone cord with time.

After each telephone conversation the cord is twisted in one

sense or the other with equal probability, but after a large

number

of conversations it ends up twisted by an angle proportional to the square root of the number of conversations. Since the angular spreading of the electron beam is a sum of a large number of independent random p ro cesses, the angular distribution has a Gaussian form :

2

2

5>(0) = A exp - 0 / 0 .

The

2

m ean-square deflection angle 0

is proportional to the number of

collisions N, which is equal to the sample thickness L divided

2

by the mean free path L

If 0^

is the m ean-square deviation

angle in a single collision, then 02 = N0l2 = e 2 L/4 - no-L^2 = I2 - ^ r - 6 (8o +

- &p ) d a =

The cro ss section for the photoeffect is equal to the number of transitions per unit time divided by the photon beam intensity, which is just the velocity of light c , since we have normalized the problem to one photon per unit volume.

Consequently, we get the

order-of-magnitude result liV l'2(2jc)2c• ik r We shall now show that the exponent e ~ occurring in (1.23) may be replaced by unity.

The wave number k = co/c is of

order I /c , where I is the ionization potential of the atom; for the inner electron shells.

2

1^ Z

The radius of these inner shells is

of order 1 /Z , and so kr ^ (Z2/ e ) ( l / Z ) ~ Z /c

and e ^ ^ « 1.

DIMENSIONAL AND "MODEL" APPROXIMATIONS

59

The replacement is even better justified for the outer shells , where we have k r^ 1 /c & 1 /1 3 7 . Since (

.A the interaction of the particle with the photon field, and H^ the Hamiltonian of the photon field (cf. above)

Let us assume that the energy of the photons under consid­ eration is much less than that of the particle.

Then the motion of

the particle can be taken as given and we need consider only that part of the Hamiltonian which contains operators which act on the photon wave functions.

The photon Hamiltonian changes over the

time of the collision (which will be assumed small) from H = H - (l/c)p .A before the collision to H = H - (l /c )p .. A o y ~o ~ 1 y ~ afterwards. L et us go over to a system of coordinates in which the particle was at re s t before the scattering. H 0 = jy Y,

where c^=

! =

Then we have

Hy — - i - q A ,

1

( . 38)

is the change in the momentum of the particle

due to scattering. The perturbation -(l/c) is practically unchanged, so that the transition y

amplitude into the state X^ is given by (X^| XQ) (c^* P*102)*

DIMENSIONAL AND "MODEL" APPROXIMATIONS

73

We represent the Hamiltonian of the electrom agnetic field, which is given by H-

=H^l2+^(rot4)2)dr’

(1-39)

in the form of a sum of Hamiltonians for the various field oscilla­ to rs.

To do this we write the vector potential A in the form A = 2 V 2n°2

exP (i k r ~

+

+ qlx exp ( — i k r +

where k, X, and

i(Ok\t)).

are respectively the wave vecto r, polariza-

tion, and energy of the photons, and vector.

(1. 40)

,.

77

is the unit polarization

Omitting the indices k , X in what follows and writing q = Q + -^ P ,

q * = Q - ^ p’

and substituting in (1 .3 9 ), we get tfY= 2 4 - ( .P 2 + °J2