National Bureau of Standards

Library, N.W. Bldg

m^vr 1964

^ecUnlcai ^iote 207

QUANTUM FIELD THEORETIC TECHNIQUES

AND THE ELECTROMAGNETIC PROPERTIES

OF A UNIFORMLY MAGNETIZED ELECTRON GAS

LEON A. STEINERT

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

Technical ^otc 207

ISSUED APRIL 6, 1964

QUANTUM FIELD THEORETIC TECHNIQUES

AND THE ELECTROMAGNETIC PROPERTIES OF

A UNIFORMLY MAGNETIZED ELECTRON GAS*

Leon A. Steinert

Radio and Microwave Materials Section

National Bureau of Standards

Boulder, Colorado

* Submitted as a thesis to the University of Colorado in partial

fulfillment of the requirements for the Ph. D. degree.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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TABLE OF CONTENTS

PAGE

INTRODUCTION 2

I. ELECTROMAGNETIC THEORY AND

MACROSCOPIC SYSTEMS 5

II. FUNCTIONAL SERIES EXPANSIONS FOR THE

CHARGE AND CURRENT DENSITIES OF

MACROSCOPIC MEDIA 13

III. NON-RELATIVISTIC QUANTUM ELECTRODYNAMICS AND THE ELECTRON GAS 28

IV. "EXPECTATION VALUES" IN THE

DENSITY MATRIX FORMALISM:

ENSEMBLE AVERAGES 41

V. TEMPERATURE DEPENDENT GREEN'S

FUNCTIONS FOR THE ELECTRON GAS

AND THEIR RELATION TO THE CHARGE

AND CURRENT DENSITIES 46

VI. SOLUTION FOR THE ONE-PARTICLE

GREEN'S FUNCTION 84

VII. THE EQUILIBRIUM ENERGY DENSITY

AND THE CHEMICAL POTENTIAL

OF THE ELECTRON GAS 103

ii
PAGE

VIII. INTERACTION OF TIME-DEPENDENT

ELECTROMAGNETIC FIELDS

WITH THE ELECTRON GAS 116

IX. "DIELECTRIC SCREENING" BY THE

ELECTRON GAS 150

X. SPIN MAGNETIZATION OF THE

ELECTRON GAS 1 6 1

XI. DIAGRAMMATIC TECHNIQUES 168

XII. SUMMARY AND SUGGESTED LINES

OF FURTHER STUDY 17 7

LIST OF REFERENCES 179

MATHEMATICAL APPENDICES . 183

Mathematical Appendix 1 184

Mathematical Appendix II 189

Mathematical Appendix III 193

Mathematical Appendix IV 201

Mathematical Appendix V 202

Mathematical Appendix VI 218

Mathematical Appendix VII 229

iii
FOREWORD

The principal objective of this research was the formulation

of a many body theory of electronic charge transport for uniformly

magnetized systems which could be utilized with some degree of

practicality to predict quantum effects of the electron-electron

interactions.

This objective has been achieved to the point where quantum

exchange contributions may be calculated in addition to the "selfconsistent field" approximation already considered. As a byproduct of this research, expressions were obtained for the spin

magnetization with quantum exchange interaction included. The

results which have been obtained are applicable to the solid state as a model semiconductor and to the gaseous state as a model

fully ionized gas (non-relativistic) with stationary positive charges.

IV
Quantum Field Theoretic Techniques

and the Electromagnetic Properties of a Uniformly Magnetized Electron Gas

L. A. Steinert

Field theoretic operators for the charge,

p, and current, j, densities of the electron gas

were obtained using the Darwin Hamiltonian in

"second quantized" form. Temperature dependent

Green's functions were formed from certain averaged combinations of the field creation and annihilation operators. Equations of motion were obtained

for these Green's functions, and p and j were written in terms of the Green's functions.

In order that p and j might be expressed

explicitly in terms of the perturbing field, formal

functional series expansions were obtained for p and j in terms of the potentials. Connecting relations obtained from the requirements of gauge invariance and charge conservation were used to show the general result that p and j reduce to functional of the electric field alone.

Using the "self -consistent field" approximation, calculations were performed determining

j explicitly to an order linear in the perturbing

electric field. Specific calculations are given for a zero temperature gas and for classical high temperatures. Discussed briefly was the "dielectric

screening" as induced by a magnetized electron gas.

The Green's functions equation of motion for the

"spin magnetization" was obtained. Brief mention

is made of diagrammatic techniques.
INTRODUCTION

Formal functional relations are derived for the charge and

current densities of macroscopic systems acted upon by electromagnetic fields. Quantum field theoretical techniques are combined with

the results of equilibrium statistical mechanics for a "Many-Body"

theory of the electromagnetic properties of a uniformly magnetized

electron gas in the non-relativistic limit. The functional relations for

charge and current density are utilized for the case of an "additional"

electromagnetic disturbance acting upon the electron gas. The results

comprise a theory of charge transport and electromagnetic fluctuations

for the electron gas.

The method of approach used here for the electron gas involves

the Martin- Schwinger techniques concerning temperature dependent

Green's functions (reference (1)). Some discussions and examples of

the concept of temperature dependent Green's functions are given in

the references (1) through (24) with applications to various physical

problems. The references (1) through (8) specifically involve the

^'causal" Green's functions defined by Martin and Schwinger (1), while

the review papers listed as (20), (22), and (23) include discussions of

related Green's functions ("advanced" and "retarded" Green's functions).

A brief review of electromagnetic field theory as related to

macroscopic media is given in Chapter I. It is assumed that Maxwell's

electromagnetic equations are valid for quantum mechanical systems

with electromagnetic field quantities suitably defined as "probabilistic

averages" or "expectation values" of appropriate operators (see reference (25) concerning this point). A formal development of the charge
and current densities of macroscopic systems as functional series

expansions in the disturbing electromagnetic fields is obtained in

Chapter II. This derivation constitutes one of the new results of the

present work. Charge and current densities are defined to include

polarization charges and currents of material media. A "conductance

tensor" is defined which includes both the usual conductivity due to

the motions of charges and the effects of polarization currents. The

quantum field theoretic Hamiltonian for the electron gas interacting

with applied electromagnetic fields is given in Chapter III, along with

a derivation of the charge and current density operators for the gas.

In Chapter IV the definition of macroscopic "expectation values" of

quantum field theoretic operators is discussed in terms of the density

matrix formalism. Temperature dependent Green's functions are

defined in Chapter V; these functions are essentially space-time

"correlation functions" formed from the "expectation values" of certain combinations of the electron field creation and annihilation

operators. The general relationship of the Green's functions to the

charge and current densities of the electron gas is developed.

Equations of motion are derived for the Green's functions, and

the Green's functions are "renormalized" from functionals of the

applied electromagnetic fields to functionals of the total electromagnetic fields. Functional derivatives are obtained in the "selfconsistent field approximation" for the charge and current densities to

the order linear in the disturbing electromagnetic fields in terms of

the "one-particle" Green's functions. In Chapter VI a solution is

developed for the "one-particle" Green's function in the "selfconsistent field approximation. " The chemical potential of the electron

gas appears as a parameter in the Grand Canonical Ensemble and in

Chapter VII the relation of the chemical potential to the equilibrium

electron number density is considered briefly. The equilibrium electron energy density is also discussed in Chapter VII. In Chapter VIII
the current density of the electron gas to the order linear in the

disturbing electromagnetic fields is considered in some detail in

terms of the Fourier transforms (equivalent to plane waves) of the

current density and of the electromagnetic field (this also is given in

the "self-consistent field approximation"). Attention is given to the

Fourier transform, of the "conductance tensor, " the dispersion relations for propagation, plane wave propagation in the direction of the

applied static magnetic field, and complex frequencies. The "dielectric screening" of a static Coulomb charge by a magnetized electron

gas is discussed briefly in Chapter IX. In Chapter X the "spin magnetization" of the electron gas is discussed. The equation of motion

for the "spin magnetization" is formally developed directly from the

"Many- Body" point of view utilized here. A discussion of some

topological methods relating to the solution of integral equations

appearing here is given in the final chapter (XI).

The c. g. s. Gaussian system of units is used throughout this

paper.
CHAPTER I

ELECTROMAGNETIC THEORY AND MACROSCOPIC SYSTEMS

The discussion in this chapter is a brief summary of classical

electromagnetic theory for macroscopic media. The formal development of this theory may be found in a number of excellent standard

textbooks (for example, see reference (26)). It is assumed here that

the Maxwell equations are applicable to quantum mechanical systems,

in which case the electromagnetic field quantities must be defined as

"probabilistic averages" or "expectation values" of the corresponding

operators for the quantized electromagnetic field (see reference (25)

for discussion related to this point).

The fields in a macroscopic medium are given by Maxwell's

electromagnetic equations:

V- E

T

(7,t) = 4ir p T

(r,t)

V • B

T

(r", t) = vxE T

,7.t, . -iiS^a (I " 1)

c at

vxffV,.i ,i^Ip), 4f rV.t). —

T

—

T

In the equations (1-1) E and B are the basic electric and magnetic

T —-T tield vectors, respectively, while the quantities p and j are the

total charge density and the total current density, respectively. These

charge and current densities include the polarization charges and polarization currents of the material media. The non-linear

5
contributions to the electromagnetic field equations (I-l)(see referT —

ence (27)) are incorporated in the quantities p and j —

—T

The field vectors E and B may be obtained from vector and

scalar field potentials as follows: —

T 1 8A T

E = " VU " c IT

d-2) —-T —

B = V XA , —T— T —

where A (r , t) is the vector potential and U (r , t) is the scalar

potential. The introduction of the field potentials, together with the

Lorentz condition,

V.A T

+ I^!=0, (1-3,

C 3t

results in the reduction of the four Maxwell equations (1-1) to a pair of

T —T

wave equations for U and A :

2

-VEU

T

(T,t) + -| -2 U

T

(7,t) = 4Trp

T

(T,t)

c 8t

d-4)

- V A (r , t) + -j —2

A (7, t) = — f (*\ t) c 8t

The field potentials are not unique; the "gauge transformation"

i c at

r-T —T _ T A = A1 - VA ,T

leaves the equations (1-2, 3,4) invariant if A is an arbitrary function

d-5)
satisfying the equation

-V 2 A

T

+ 4 ijJ = Q (I . 6) c at

T t~T The charge density p and the current density j are

connected by the equation of continuity,

^ + v . r

T

= o . a-v

which follows from equations (1-1) through (1-6).

The equations (1-1) through (1-7) apply both to a given phvsical

system and to its environment. If there exist fields not produced by

this given system, then these fields are designated as "externally"

applied fields. Also, if there exist field producing sources (charges

and currents) not considered as properly a part of the given system,

these sources are designated as "external" sources. The sources con- sidered as part of the given system and the fields produced by them are

designated by the term "internal" (or "intrinsic").

T —T

The total charge density p and the total current density j may each be expressed as a sum of two parts:

T — A — —

p (r,t) = p (r,t) + p(r,t)

>T~ -A (I " 8 >

j (r , t) = j (r , t) + j (r , t) , A —A where p and j are the charge and current densities arising from

the "external" sources, while p and j are the "internal" (or

"intrinsic") charge and current densities of the system. CorrespondT

ingly, the field potentials, the field vectors, and the gauge function A

may all be defined as sums or a linear superposition of terms arising

from the environment of the system (superscript A) and from the system itself (no superscript):
U

T

(r,t) = lAT.t) + U(7, t)

a (T, t) = a (T,t) + a(T, t)

E

T

(7,t) = E A

(7, t) + E(T,t) (1-9)

B

T

(7,t) = B

A

(T, t) + B(7, t)

T —- A —* A (r,t) = A (r,t) + A(r , t)

This splitting (1-8, 9) of quantities into external and intrinsic

parts is by no means unique nor even meaningful in general, but in

many applications the procedure will be clear. All the equations (1-1)

through (1-7) are duplicated for the "external" and "internal" fields and

sources separately. For the "external" fields and sources,

V • E = 4ir p

. V • B

A

= VXEA = -A

c ^at

c at c

J

E A = - VUA - i **t

c at

B

A

= V X A

A

7a i au

A

n ^- 10 ) V '. A + — =

c at 2A i a 2

u

A

A a ~ —r~ =

p c at
2—

A

1 9 2 A A 4tt --A

+ — —r

= — J c at

«* < ± =£

7^A 7~A r,A A

A = A. - VA

2.A i a a n

c at ^

A

+ v-r

A

= c

while for the "intrinsic" fields and sources of the system,

V • E = 4tt p V . B =0

ST 1 B

V X E = —

c at — 1 3E 4tt -t- c at c

J — l a a

E = -VU - -

c —at

B = V X A

c at

d-ii)
2 —

2 t* 1 9 A 4 77 -^ •V A + — —y = T J c at

1 9A U U

l +c -

A = A. - VA

„ 2 . 1 3

2A

V A + T -T= °

c at

If v . r = o

The fact that all the equations (1-1)- through (1-7) have been separated

into the two sets of equations (I- 10) and (I- 11) does not necessarily

imply that they are independent of each other. These two sets of equations, (I- 10) and (I- 11), are generally coupled together through the

A t~A — quantities p , j , p, and j , each of which is in general influenced

by all the electromagnetic fields and all other physical forces present.

The field potentials produced by the sources are given by the

following integrals:

(s) r . r

i2.

u<

s >(T.t,) = Cd3

7. vv =

P ^"^

y 2 r 12

r(s) r

r

i2,

A (r-.t,) = \ d r.

(1-12)

vV - I 2 cr

i2

where the superscript (s) applies generally either to the "total" fields

and sources, to the "applied" fields and sources, or to the "intrinsic"

fields and sources of the given system, and where r = | r - r j. This integral form of the relationship between the potentials

and sources is equivalent to the differential form (1-4). Note, however,

that while in the differential form (1-4) the sources are "local" in the

10
time, in the integral form (1-12) on the other hand the sources contain

a "time retardation" factor f ) which is due to the fact that

electromagnetic signals propagate with finite velocity, i. e. , the electromagnetic interaction occurring between two charges at positions of

finite spatial separation requires a finite temporal interval between

cause and effect experienced by the charges.

For stationary media, or more precisely, for systems

considered to be in a rest frame of reference relative to an observer,

the total charge and current densities intrinsic to the material are

given by

P = Pv + P^ + P,

with

and where

j =

j F

+ j P

+ j M

'

p p

m - V • P

TP

s

1 3P

c at

p m"

TM

S c V X M ,

(1-13)

p.p, = charge density of "unbound" charges

j = current density of "unbound" charges

P = electric polarization density from

"bound" charges and polarization charges

M = magnetic polarization density from

"bound" charges and "intrinsic" magnetic

dipole moments.

1 1
For systems considered to be in a frame of reference moving with constant velocity relative to an observer, p and j are again given by the

equations (1-137; in this case the quantities p , j , P, and M are

related to their "rest system" values by appropriate Lorentz transformations (see reference (26), Chapter 18). It is clear from the equations (1-13) that the total "intrinsic" charge density p and the total

"intrinsic" current density j for macroscopic systems concern all the

electric and magnetic properties, both induced and permanent, of the

system.

12
CHAPTER II

FUNCTIONAL SERIES EXPANSIONS FOR THE CHARGE

AND CURRENT DENSITIES OF MACROSCOPIC MEDIA

In this chapter formal relationships are derived using

electromagnetic field theory for the charge and current densities of

macroscopic systems under the action of an electromagnetic disturbance. While it is generally accepted that there exist in nature other

physical forces between particles besides the electromagnetic interactions, only the electromagnetic forces are considered here.

For dynamical systems involving only electromagnetic

interactions the charge density p and the current density j respond

T -»T directly to the total field potentials U and A . Mathematically — • T —-T speaking, p and j are functionals of U and A :

p = p[r,t;U , A ]

T _T

(II- 1)

j = j [r,t; U , A ].

From the discussion in Chapter I it is apparent that the potentials U , —•A —- T -»T A , U, and A are functionals of U and A (and vice versa); —

A —«A —

therefore, since p and j respond indirectly to U , A , U, and A,

U

A

and A

A

p and j might, in a formal sense only, be regarded as functionals of

. j± —A

p = pfr,t;U ,A ] - -. - A -A (II ~ 2)

j = j [r,t; U , A ],

13
or p and j could be considered as functionals of U and A:

P a p[r7t; U, A]

(H-3)

T = T[r7t; U, A].

The three sets of equations (II- 1, 2, 3) do not imply that a change in the

potentials necessarily produces changes in the charge and current

densities, but they do imply that all changes in the charge and current

densities are certainly produced by the field potentials.

It is desirable to consider the "externally" applied fields E

A

and 3

(or U and A ), or their sources, p and j , as idependently known.

This permits the examination of the behavior of the given system in an

arbitrary physical environment.

The state of the system is presumably known for all times t prior to some given time t , at which time some electromagnetic

perturbation is applied. For the initial state (subscripts (o)) the

following equations apply:

V . e = 4tt p = 4tt (p + pA

) o o o

r o

V o

=

V

—T X E

o

_ 1c

-—

3B

o

8t

V —

X B

o

1c

-*T 8E

o

at + ±? T

T

<H-4)

C O

-H-T

1 o 4-nr „* -^A.

C ot COO

-* T

e

t

= -vuT

- I —

o o c 8t —

—T

B = V X A

o o

14
After the perturbation is applied (t > t ):

T T T

U = U + 6U

o —T —T r -»T A = A + 6A

o _*T —

T

—

T

E = E + 6E

o —T —T —T B = B + 6B

o

(n-5)

T T .A

p = p + Ap + Ap

j = 1 + Aj + Aj ,

where Ap and Aj are the induced charge and current densities of the

A —A.

macr :copic system, while Ap and Aj are the "externally" applied

perturbing sources. From equations (1-1, 2) and (II-4, 5) it is easily

seen that the following equations apply:

• —

T

A

V • 6E = 4tt (Ap + Ap)

V • 6B T

=

(II- 6)

V X 5 E

T

= - - — 6 B

T

c at _ w -*T 13 —T 4 it . A -~A +, VX6B =-—-6E + — (Aj + Aj

c at c 6E T

= - V6UT -i ^- 6A T

c at —

T

—

T

6B = V X 6A

The third and fourth equations in (II- 6) may be combined to result in a —

T

-* wave equation relating 6 E and Aj :

2

V X[V X 6E T

] + -L ±- 6E T

+ i| ±. [aTA

+ AT] = 0. c at c

(II- 7)

15
For future reference the following notations and summation

convention are introduced:

Q(l) = Qtr^tj)

Q*(l) = complex conjugate of Q(l)

d(i) = d

3 r1 dt

1

6

4

(1,2) = 6

3 (r

x - ?2

) 6(t

x - t2

)

(II- 8)

q a im m 'I <sL a im m

m=l

where Q (r, t) represents any function of the space-time co-ordinates

r and t.The quantities p and j are now written in a functional series

expansion (Taylor series; see reference (ZS) and Mathematical

Appendix I) with p and j considered as functionals of the potentials

T -*T U and A (see equations (II- 1)):

p = p [UT

,A

T ]+ Ap[6UT

, 6XT

] o o o r = j^[ U^aJ] + aT[8UT

, 6A T

],

where

Ap = Sp + 5 p + * ' *

A j = 6 j + 6 j +

•••

with

= 1

Spd) = d(2) 5i^_ 6UT

(2)+ ^L_ T

(2) L 6U (2) o

6 V2) o

16
'S S} (1) = \ d(2) ±ni^ 6UT

(E)+ _yjii_ 6AT(2)

L 6U (2) 6 A; (2)

I o

2

6 p(l -M d(2)d(3)

6 2p(l)

6UT

(2)6U

T

(3)

6UT

(2)6U

T

(3)

+ 2 /P(1)

T SU

T (2)6A->

6U (2)6 A^ (3)

Q

(H-9)

+ T

5

P (1)

T 6A^(2)6AT

(3)

6 A, (2)6 A (3)

m

£ mo

6

2

r(D= Yj- C d(2)d{3] — 6UT

(2)6U

T

(3)

- 6U (2)6U

T

(3)

+ 2 ffi

1 * 6UT

(2)6A^3)

6U (2)6A„ (3)

+ /j*' 1 ' 6aJ(2)6A

T

(3)

6A„ (2 6A (3)

I mo

etc. , where the expansion coefficients are "functional derivatives"

(or "variational derivatives"), and where the following defining

relations apply:

6U (2)

6UT

(1)

6AT

(2) m

= (11-10)

17
6aJ(1)

T— = ° 6U (2)

6aJ(D 4

1 = 6

fl 6 (1,2) • 6 A (2) m

The integrations in (II-9) are taken over all space and time. It is quite

possible that there should be some restrictions on the magnitudes of

T -*T 6U and 6 A for the series (II-9) to converge, however,, the mathematical criteria for convergence of such expansions are not discussed

here and in actual calculations in this paper convergence is simplyassumed. — -*T -~T

The dependence of 6p and 5j as functionals of 6 E and 6B

was accomplished by Ashby (reference (3), Chapter III) for the case of

isotropic media with the aid of several connecting relations between the

expansion coefficients (or functional derivatives) in the equations (II-9).

A similar procedure is utilized here in this discussion, resulting in the

development of relations for 6p and 6j of greater generality. These

connecting relations for the functional derivatives of p and j are

obtained from the equation of continuity and from gauge transformations (see Mathematical Appendix II). For the first order "variations,"

these relations are given by the following:

_l md + v . yjj) _

r)t T 1 T

1 5U (2) SU (2) o o

_9_ 5

P

(1) + y

. Sj (1) = Q

9t

l 6A^(2) l 6 aJ(2) S. o i o (H-11)

18
i _a_ 6pd) sp(i) .

C 8t

2 6UT

(2)

U 6A> o I o

1 9 6j (1) _§iiiL =0 . C 8t

2 6UT

(2)

U 6A> o i o

Similar relations may be obtained for the higher order variations. The

T —T —T —1

Fourier transforms for 6U,6A , 6 E ,6B , and the functional

derivatives are now defined in order to make use of equations (II-9).

The required Fourier expansions are given as follows:

rtt

t m , C

d

3 kdo tiV 6U (1) = \

. ^-v-V

j u (k, u )e

J (2ir)

3^ m iflMT-tft ) 6A (1) = \ =| a (k,«)e

J (2ir)

,?",,, - f 6E d kdu -r T /7* \ ! * (1) = \ * e (k,w)e

J (2ir)

.*tm1 r

d

3 kd^T

/ir , ^ k

6B -r-v

(1) = \ rb (k, u)e

J (2ir)

(H-12)

and

6p(d . r '''WX'S .- - . ^i-Wi-VV^

73\7T

= J —r-s fffr.« ;k

2.„2 ).

OU (2) o * (2tt)

19
3— 3— — — — —..

. d k du d k,du i(k . r - w t -k • r +w t 6p(l) f 1 1 2 2 =- ,r* . 11112222 J~^ g (k ,w ;k , «-)e UT„, J „ 8 &m x "l' w

l'"2'~2' 6Am

(2)

Q (2.) (n-l3)

6^(1) dV^d3^ iffri""

r

'iui t ir2' 7

2

+w

2

t 2

> T ~ \

8 P/ k

l ,Wi ;k 2' W2

)e

6U

T

(2) o J (2TT)

8 i

1 1 2 2

3-* 3-* — — — _»

6j„(l) ^ d k.do^d k dco i(k • r

1 -"1 t1 -k2

* r 2

+c°2t 2^ S

I ^ 1

2 2 WkVVr2' W

2

)e

6A

T

(2) (2-rr m o

The use of the equations (11-12) together with equations (H-6)

results in the equivalent equations for the Fourier transforms:

-T-* ico r?-T.-r* . .-* T-* . e (k, w) = — a (k, w) - ik u (k, w)

(n-14)

b (k, w) = i k X a (k, co) .

Substitution of the equations (11-13) into the equations (11-11) results in

the'Fourier equivalents of the connecting relations:

- i« f (k

l# « ;k

2

, «

2

) + i k P,Ck »» ;"E^«-«2). =

i.

2 ^

<n- 15 > — f (k^oyk^) - ik2mgm

(k

i

, Wi ;k

2

, W2 )=

10)

2 - - —Pifrl'W^ " ik2m^m(k

l' W

i ;k

2' W

2

>= °'

Substitution of the equations (11-12) and (H-13) into the equations for

6p(l) and 6j~(l) in (II- 9} leads to

20
6p(l) _ ^X^l^Z^Z

(2tt)

8 £(k

1 ,co

1 ;r

2

, W2 )u

T

(k

2

, W2 )

+ gm(k

l' Ui ;k2' U

2

)am

(k2' U

2

)!e

and

«/!>

3— 3—

,d k.doo.d k.du)

(n-i6)

J (2ir)

8 Pi (k

l' W

i ;r2' W

2

)uT(ir

2' C0 2

)

-* — T —

+ q, (k , w ;k ,u>_)a (k , u ) to 1 1 2 2 m 2 2

i(k • r -w t ). V 1 1 11

By combining the last pair of connecting relations (11-15) with the

equations (11-16), one obtains

6p(l) = \ g g_(k i, Wi ;k9

,a, 7

) iu ;

rlW—2 T- a (k , w ) c m 2 2

T — ~ ik o^ u (k

7 »w )

i(k ' r -u> t ) 1 1 1 r

(H-17)

3— 3-*

d kdud k do

1 J (2tt)

8 \

lu2/ trn l l 2 2

rlw—2 T -

a (k , w_)

c m 2 2

ik2m

uT(r

2' W

2

)

i(k

l .ri -co

l ti

).

The substitution of the first equation (11-14) into (11-17) results in the

simplification

21
d k du,d k^dw^ ^_ v m i(k,« r,-»,tj

(2-rr f~ 'T 1 2 2/c \ lTr r* . T^ 1 111

6p(l = \

; b (- )g (k ,o) ]k ,o) )e (k v uJe

J (2*)* \

lw2/ m X

1 2 2 m 2 2- (n-i8)

Now, by substituting the equations for the Fourier coefficients from

(11-12) and (11-13) into the equations (11-18), one obtains

6p(l) = CkT

(1, 2) 6ET

(2)d(2)

J m m

(II- 19)

6V1)= f

SL(1 ' 2)6ETM

(2)d(2)

S. J £m m

where

K

T

(1,2) m

3-* , ,3r* d k^ d k^ i(k - r -« t f

k

2

. r 4^y

njV, s v ^ /-i \ ik* (r«-r^)- i<*i(t -t„)

= rdkd^ d(3) /i£.s _5pm_ e 3 2 3 2

(2ir) ^ w / 6A (3) m o

and

s>2 »

= V _ / -—\q (k , cd. ;k , w_) e J (2u)2ir)

8 WV 112 2

V, •-. 6^(1) i!T-(r*

3

-T

2

)-icl>(t 3-t

2

) e .

22
The equations (11-19) form the results upon which this thesis is based;

they are formal relations for the 6p and 6 j of arbitrary systems in

terms of the basic field vectors, and they give Ap and Aj up to

terms which are linear in the fields. That only the electric field

vector should appear in (11-19) is an interesting result, and is due to —

T

—T

the fact that 6 E and 6 B are not independent, but are linked toT T

gether by Maxwell's equations (or by the field potentials 6U and 6A ).

In principle, the equations (11-19) show that the relations between the

charge density 6 p and the fields, and between the current density 6 j and the fields are"nonlocal" in the space-time co-ordinates.

It is possible to achieve a formal simplification of the equations

T

(11-19) in the case where the time dependence of (6p(l))/(6A (2) ) T mo

and (6j'(l))/(6A (2)) is of the form of the difference ft- t,) of

£ m o 1 Z their time co-ordinates t and t . The time derivatives of 6p and

6T in (H-19) are

at. 6p(i: = j

[stj m

s- 6j *

(1

> fli^ »L/1 - 2)

6E (2) d(2) m

5E (2) d(2) m

(11-20)

where

9t m

3— 3-»

t d k,dco,d k dw .u

- -C ^ fc?

( — ) g (k ,w.;k , w )e \u^ J m 112

i(k

i .r

l-Wi t f

k

2

.T2+^t

2

)

23
^ f?.ir\ V.-2/ (2tt) v -2

The condition that (6p(l))/(6U

T

(2) ), (6r(D)/(6U

T

(2) o

), (6p(l))/(6A

T

(2) ) and (6j"(l))/(6A

T

(2) ) depend upon the time

m o mo

difference (t - t ) requires that f, g , p , and q be of the form: 12 rn £ £m

f(k , w ;k , w ) = (2ir) dCw^ co2

) f (k , k

2

, w )

I (k, w.;k, u ) = (2ir) 6 (u - w_)g (k . k , «_

)

m 1122 1 2 m 1 2 1 (11-21)

Pjg•'(fcj: iu

1 Jk

2

'i w

2

) = (2tt) 6 (co^ <j 2

) £,"0*^1*2' "V

J WV'VV'V = (2ir)5(V 0?2 )q

Jem^l'

ie2' W

l

) -

By substituting the equations (11-21) into (H- 13) and (11-20), one obtains

6p(l) r

d3g

l d3g

2

d" £ = \ — t

7FT f (k ,k

* , W),^1^^ e 6U (2)

q

J (2ir)

X

6AT

(2) J (2k)

7 m * Z m ° (11-22)

= j

7 Pi^i'V^ 6

6U

T

(2) Q " (2ir)

6j (i) d

^

d

3^ dco _ *i vW iw(t r

t2

)

5A

T

(2)

J

(2u)

7 ^m l 2

mo

24
and

3T- k d. 2) - -c

j ——r— «m

(k

i- Vu) e

1 (2 "»

(11-23)

a t r

d k

i d k2dw iP-T-r-7,)-i«(t -t )

From a comparison of the equations (11-22) with the equations (11-23),

it follows that

1 6A (2) m o (H-24)

a t Va)

S, (1,2) = -c

and that

8t i ^ m c A T f>\

1 6A 2 m o

3 6p(1) = -cC 6p

T

(1 L. 6ET

(2)d(2)

8t

l J 6A

T

(2)

m

m o (H- 25)

a r

6j

o (1) t

~-6j,(l) = -c —| 6E (2)d(2).

8t

l i J 6A

T

(2)

m

m o

Since the functional derivatives enter explicitly in the equations (11-25),

these equations (11-25) are of value in calculations involving systems

for which the Hamiltonian is time- independent.

The higher order "variations" of p and j may be developed

by a procedure similar to that which resulted in the equations (11-19)

and (11-25). Thus the series may be considered to any desired order

in the fields. Also, the results obtained here are not strictly dependent upon the use of Fourier expansions. Any complete orthonormal

25
set of basis functions will lead to the same results. Therefore, a representation most convenient to a specific problem may be chosen.

It may be seen without too much difficulty that if one considers —

A —A

p and j as functionals of U and A (H-2), the end result is

6p(l) = f kNi.Z) 6EA(2) d(2)

J m m

(n-26)

6j

i

(1) = I

SL(1 ' 2)5Em(2)d(2)

A A

where K (1, 2) and S (1, 2) have definitions similar to the corresponding quantities in (II- 19). Similarly, if one begins with (E-3) the

final result is

6p(l) = C K (1,2) 6E (2) d(2)

J m m

6i (1) = f S, (1,2)6E (2)d(2) J i J Im m

(11-27)

with K (1, 2) and S (1, 2) again defined similarly to the correspond- m £m^

ing quantities in (11-19). From the requirements of causality, one may

expect the following condition:

.MIL. . .MIL VjVi. . Vl. s (u. 28)

6U (2) 6A (2) 6U (2) 6A (2) o m o m o

2 2 _».-•- 2

for c (t - t 2

) - (r - r

2

) < and (t - t ) < 0. Thus, in

particular, the time integration in equations (11-26) extends only over

the region t > t_.

1 £• Now, from the equations (II- 9) and (11-19) it is clear that 6p(l) — T —T

and 6 j (1) are linear in 6U and 6 A , and the equations (II-9) also

2 2t» show that the second order variations 6 p and 6 j are bilinear in

T —

6U and 6 A , and so forth. Thus, to the first order the equation (II-7)

and the first of equations (II- 6) reduce to

26
, 2

at V X[ViX 6E T

(l)] + -^ A_ 5 e

T

(i) + 1|.^ 6 f(1)+ ll JL ArA

(D= 0, c 1 c 1 (H-29)

and

V • 6E

T

(1) = 4tt 6p(l) + 4tt Ap (1)

or (II- 30)

V^6rT(1

> ^^-6p(l) + 4.A- Ap

A

(l),

with 6p and 6j given either by the equations (11-19) or by equations

(11-25) as appropriate. Since the quantities 6p and 6j are generally —

T

nonlocal in 6 E , the equations (11-29) and (11-30) are inhomogeneous —

T

linear integro-differential equations for the field 6 E . This linear —

T

equation is justified only for small values of 6 E . T

The quantity S (1,2) is called the "conductance tensor. "

In the case that the system is translationally invariant, S„ (1, 2)

T -* = S„ (r - r . t - t' ), and one may define the Fourier transform Im 12 12

s (IT, oj) by the equation im

(ZlT )

The considerations involved in this present chapter apply to the

individual terms p , j , p , y , p and j (1-13) as well as to p and j since the gauge transformations and the equation of continuity

hold true in each individual case

9p F

at j f

9pp + V • i = (H-32)

at J p

9p M -

at j m

27
CHAPTER III

NON-RELATIVISTIC QUANTUM ELECTRODYNAMICS

AND THE ELECTRON GAS

The techniques of quantum field theory are utilized here in the

investigation of the behavior of a uniformly magnetized electron gas

which is unbounded and infinite in extent and is acted upon by electromagnetic disturbances. A background of static positive charge is

superimposed upon the electron gas for overall charge neutrality and

stability. The system is assumed to be initially in a state of

thermodynamic equilibrium at all times t prior to some time t . The references (1), (2), (3), (9), (29) and (30) constitute general

sources on matters' related to quantum field theory and its application

to statistical physics.

This "Many-Body" system of electrons constitutes a particle

field with the creation operator i{j (r, t) and the annihilation operator

ijj (r, t) in the Heisenberg representation. The electrons interact with

each other, with the positive charge background, and with an

"externally" applied electromagnetic field. The interactions between

the particles are characterized by "direct interaction at a distance"

'v\2

(local in time) to the order ( — ] . (See references (31) through (37)

regarding "direct interactions" and "velocity-dependent potentials. ")

This formulation of the quantum electrodynamics of the many-electron

system has its limitations, but it is advantageous in avoiding many of

the complexities and difficulties of a quantized field theory of electromagnetic radiation. The "nonlinearities" of the electromagnetic

28
field (reference (27)) and the effects of creation and annihilation of

particles are not within the scope of the present paper.

The following definitions and the summation convention for

sums over the spin indices (Greek letter subscripts) are listed for

future reference:

e = )e | (absolute value of the

electronic charge)

e n (r ) = charge density of the positive

charge background

p(Ti'V 5 TVf rA<Vi]

P *<W H "TV f

rA(vv

r i2 "

" r 2l "

r

i

" r 2

12

2

r 2l

3

^1,

Lj a

(m-i)

P

= l

V- = 1 + 2g ,

where 2g is the "radiative correction" for the anomalous magnetic

moment of the electron (g > 0, g « 1, u = 1. 00116; see refer- ence (38)).

29
The Hamiltonian H(t) that we use here is essentially the

second quantized form of the classical Darwin Hamiltonian (reference (31)) generalized to the many* electron system. The Pauli spin

contributions (reference (35)) are included. In a straightforward

manner (see Mathematical Appendix III) one obtains, in the non time

retarded limit (reference (36)), the following expression for H(t):

H(t) = tyt) + H

2

(t) + H

3

(t) , (HI- 2)

where

h (t) = CdV ^(7 t)h A JL,

1 J 1 a 1 | a(3 2m P(r,,t

2 fxfie — s-A -* + ? cr • B (r , t)

1 2mc ap

6

ap

eX,A«7l -«}+? P

i'

t)

1 < •k(r .t)

2 — « n(rj

2 ^}*.*i-*>-

and

H

3

(t) B i(ro) ^1+1^1- ^^^-^-^r^

+ Kffif*!'*)' ^.t) + jxn7 . ap I

# 1 a(3

^XA^.t) -^.t)

30
The operator A (r , t) is given by the definition

A(T.t) s -

1 2mc l r iz

p [r

2

,t) + P*(7

2

, t)-ir 12

+ Hfi

" xlrK<r2< r i2

J

t).

(in-3)

It is also convenient to define another operator U such that

Sffi.t) - JVr2 {- !?,.*)— VV+ -r2

- } • (in - 4)

The operators U and A are so defined for reasons which will become

clear in retrospect. The "radiative correction" 2g is included in

H(t) even though it is small because of some interesting results

arising from its presence. The one-particle operator H is the usual

form of Hamiltonian for electrons acted upon by an "externally"

applied electromagnetic field, the operator H combines the twoparticle electrostatic Coulomb interactions of the electrons and the

Coulomb interactions between the electrons and the positive charges,

while H, is an operator containing the "magnetic interactions"

between the electrons in the lowest order non-time retarded form

(reference (36)). The "magnetic interactions" probably make only a

slight contribution, but they are quite essential to the formulation of

the charge density p and the current density j as functionals of the

T —>T

total field potentials U and A .

31
The particle number operator N for the electrons is given by

the equation:

N = fdV ^(T.t) ij, (T.t). (III-5)

J 1 a 1 a 1

We shall be considering Grand Canonical Ensemble averages

and it is convenient to define a new Hamiltonian ${t) for this purpose.

The operator C,N represents an arbitrary shift in the origin of the

energy for eigenstates of both the operators H(t) and N where t, is a constant taken to be equal to the chemical potential for the electrons

when the system is initially in a state of thermodynamic equilibrium

before the time t . The Hamiltonian for the system may therefore be

redefined as follows:

#(t) = H(t) - &N. (III-6)

An electron number density operator may be defined from

(III-5) by the prescription

N = Jcfe PN

(T.t^) , (III-7)

-so that

VVi> s pn

(1)= +!'^'*i(+. ffi'V:

Similarly, an electron energy density operator may be defined from

(III-2) by the prescription

Hd^) = f^V^y^) (in- 8)

32
so that

v 1 1 = h (1)= ^(r.,t)J6 T

-

f?<'i' t

1 >l

2 v a 1 1 a{3 2m |_ 1 1 J + S|2mcc ap 11

2 n^r" L

-6 eU

ap X'V + I

6aP I^2[4<V^^2^-7^.

+ 6 P(r , t )

• A(r , t ) + 6 A(r .t )• P(T ,t ) ap 4mc * 1 1 11 ap 4mc Ml 1 1

+ l£VLr vI(Vi>j}yVi>-

The field operators vj> and vjj obey the following anticommutation relations (Fermion field):

v|i (r ,t) J(r ,t)+ ^(r,.t) t|j (r , t) "

6 aP 'S - r

2'

+ (*•*) 4>„(r,.t)+ +.(r ,.t) i|> (r ,t) = al P^ P ^ al

4» ff", . t) 4>J (T t) + J(7 ( t) ^t^ > t) = > al p L p <L al

(ni-9)

Time evolution equations for the field operators \\> and \\i are given by

(Heisenberg representation)

ifi

ifi

at.

at.

WV'^V

(HI- 10)

Iall 1

33
where [ F, H] is the commutator:

[F,H] = FH- HF. (HI- 11)

Using the Hamiltonian given by (III-6) in the first of equations (III- 10),

one obtains the equation of motion for \\ii

B *JU r , _ 2 u*e ~ -A A

" ^ eU(1) + V 2^ ?(1) ' r<1) + 6aP I^ 5"'

1

'

' r">

da- u)

The identity — — \7---T--\T-=-:2T- <nl- 13 > 12 r

!2

12 *12

and the anti- commutation relations (III-9) were used in deriving the

above expression. The notation defined in (II- 8 ), Q(T , t ) 3 Q(i ) ,

r is applied her* to operators and functions alike.

TO obtain the time evolution equation for v|i , we assume that

f

ifr and v|i both vanish at the limits of the spatial region for which these

operators are defined:

*<T,t) = *.(£** = o*©* |T| = ». (m-i4)

o (5

In the case of periodic systems periodic boundary conditions are

applicable.

34
The' Hermitian conjugate of equation (III- 12) is the following

t time evolution equation for iji

wla\

71 8t \ (la 2m |_ U,

J + 2mc °>a UJ V

(HI- 15)

-«. eUA

(l)l J(l) + ^(l)eU(l ) - -^ P*(l [5

e 2mc[?* (1)^(1) A(l)11 - -H£- ^(l)T • 2mc v

jB

l

' jkt

VjXAa:

At this point it can be seen that the operator A may be written in the

following form:

(m-H)

Hie charge density operator p for the electrons is given by

p (1) - -evj,'(l)4, (1) • a a (m-i7)

and the total charge daasity operator p for the system is given by

p(l) » Pe (D+ •*^

1

) - (HI-18)

35
A current density operator j may be defined by an equation of

continuity: ^+ V

1

• f(l) = . (111-19)

Since the positive charge background is stationary,

a'p(i) 3p

a

(1)

at at

Also, j will represent the electron current density. By combining

the equations (III- 12) and (III- 15) with (III- 19) one obtains:

»*£--»^VheVK ' a

P*(l)i|i l (l) a

+ 2-'J(l) .+;<i)A(i)+ Ad!

a(i» ,

+ (i) a

(III- 20)

so by comparing equation (111-20) with equation (III- 19), one finds that

j may be identified as

TiD = -~Uu)\P'(i)ty (i)l +|p*(i) ^(d1 + (i)

(111-21)

+ 2-, J(d ^'(1) a(i: Al) q, (1) c a a

As a matter of fact, equation (111-21) is incomplete; the missing term

may be calculated from a variational principle relating a variation in

the field potentials to a variation in the Hamiltonian:

36
6^(t

x

) = ^d3^ 6UA

(1) p e (l)-i 5A A

(l).r(D . (HI- 22)

(See reference (39), Section 128, concerning equation (m-22),V The

first variation of the Hamiltonian (ni-6) for a variation in the

A »A

potentials U and A is given by:

«*<V- jA+IW-t**, S P'1

'

• t

* X^* 6

a„ HT !

5?A

»>' ?(1

2mc a(3 V X 6A A

(1) 6 _ e6UA

(l) (111-23)

+ 6

aB "S 6rA{1)

* A ( 1 )}^3 (1) r mc J r

Equation (III- 2 3) may be rearranged by use of the vector identity

C-(VXD) = V • (D X cf) + D • (VX (f), by several integrations by

parts, and by the use of the boundary conditions (in- 14). The result

is the following equation:

6^(t

1

) = j'd

S7

1 6UA

(l)|-e^(l)4;

a

(l)|

" ild3r

i 6aA(1) - £fe) {^(i)[p(D V1 ^^1 ^^^ *(1)

a

(111-24)

+ HtiV X ^(1)7 . v|i.(l) + 2^ ^(1)^(1)4 (1)} • 1 a ap p c a a

J

37
From a comparison of (111-24) with (111-22) one finds that it is

possible to make the identifications

p (1) x -•11^(1)* (1) , e a a.

and

T (1 > - -2m" {+I(»['a)+.<i)]+[^a>+2ft)]*.c«

+ 2- $ll)fc(l)4 (D + fi-hV.X i^(l)7 a

ib 0)

c a a la ap (r

(ni-25)

J

If we compare (IH-25) with (111-21) we find that in (111-25) there is the

extra term

This term is related' to the 6pin magnetism of the electrons,

since

Kll)s -i£t +I<

i >'.

p y

i > <m- 27 >

is identified as the spin magnetic polarization density operator

(see (1-13)). This term (111-26) apparently failed to appear in (111-21)

because j was identified there from terms of the form V- j (HI- 19),

and because of the vector identity V'(VxS")sJ, Thus, the

operator j (1) defined in (III- 2 6) is the spin magnetic polarization

current density operator for the electron gas. By combining the

38
definitions (III- 2 6) and (HI- 27) one obtains

j (1) = c V X M (1) J M 1 s

x (ni-28)

It is possible to simplify the forms of the equations (III- 17, 25,

26, and 27) by the introduction of the spinor notation for the field

operators (column and row matrices):

ii/te t) = ^l& t)

kji (r, t) =

^(r.t)

+2

(r,t)

and

i|i f (rt t) = *jKt) i|i|(T,t)

(HI- 29)

+ '(r,t) = 4>|(r,t) ^(r, t)

The unit matrix is designated by the term I

'J l)

(I l P

= % (m-30)

The transpose of any 2X2 square matrix y is designated by

Transpose of y = y

where ( Y

^ = (y)^

(ni-3i)

39
Therefore, we may write the .equations (in- 17, 25, 26, and 27) as

follows:

p (1) = -e ^(1) ili(l) .

Td) =-jem {W)[pd)vii(i) P'MD^V) +(i)

and

+ u^ VX?'|"(1) cr i|,(l) +2-|Tp t

(l) A(l)v|i(l)| ,

(III- 32)

& (i) = -r^+^D^qid)

,

s 2mc

r a) = cVjxHd)

40
CHAPTER IV

"EXPECTATION VALUES" IN THE DENSITY MATRIX

FORMALISM: ENSEMBLE AVERAGES

This chapter is devoted to a brief discussion of "macroscopic

expectation values" or "probabilistic averages" of the quantum field

theoretic operators. The density matrix formalism is utilized here

using the results of equilibrium statistical mechanics. Only a brief

sketch is given here of this important and essential aspect of the

overall formalism; more complete discussions may be found in the

references (1), (2), (3), and (9). The following notation is adopted in

general:

"Expectation Value" of F = (F) , (IV- 1

)

where F is an arbitrary field theoretic operator.

The "expectation values" of the field theoretic density operators p and j (III- 18, 32) may be interpreted in a probabilistic sense

similarly to the probability density (the square of the amplitude of the

wave function) in ordinary Quantum Mechanics.

The field theoretic "expectation values" are defined in terms

of the density matrix formalism:

<F) = Sp (fF) , (IV-2)

where the "Sp" refers to the spur, or trace of the products of the

operators f and F, and f is the density matrix operator (this operator is usually designated by the symbol p in the literature, however,

41
since we have already introduced the symbol p for the charge density

operator, we thus introduce the notation f for the density matrix to

avoid confusion). The meaning of the trace in equation (IV-2) is

3imply that, given an arbitrary complete set of orthonormal state

vectors | a )

, one may write equation (IV-2) as sums of ordinary

expectation values over the set |a):

(F) = J, < a l fF l a > '

or

a

(IV- 3)"

(F> = ££ (a |f |a«)<o«|F|o>

a a' by the "chain rule" for operator products. The single parameter a actually represents all possible parameters associated with the state

vector | a)

For operators in the Heisenberg representation the density

matrix is time independent, (see reference (9)), hence one may use

the operator f (t < t ) for the system in the initial state of

thermodynamic equilibrium. From equilibrium- statistical mechanics,

we have the result that for systems in a state of definite temperature

T with chemical potential t, the density matrix is given by the Grand

Canonical Ensemble (see: reference (40)):

-0(H -£N ) f (t< t ) = f =Ze °

, (TV-4) o o

where Z is the Grand Partition Function

Z a Sp e

42
and £ = —- where T is the temperature in degrees absolute, and

k = 1. 38 X 10" erg per degree absolute is the Boltzmann constant.

The operators H and N are the time- independent Hatniltonian and

r o o the electron number operator given by (111-2} and (111-5) prior to the

application of an electromagnetic perturbation at the time t . We let

# = H - ;N (IV-5) o o o

after (M- 6), then from equation (IV-4)

-1 - p *o

f = Z • o and - (IV- 6)

Z = Sp e

It has been shown by Martin and Schwinger (reference (l))(see also

reference (3)) that for systems in a state of definite energy and having

a. definite number of particles, the expectation value e of operator*

may be approximated by averaging ©v«r the appropriate Grand

Canonical Ensemble (equivalence <tf a. Ittlcroeanbnical ensemble and a grand canonical ensemble) if the number oi particles is very large.

Thus, in either case, the operator f given by (IV-4, 6) is the density

o matrix we require and

(F) = Sp (f F) (IV-7)

for operators F in the Heisenberg representation. This result (IV-7)

is not applicable to isolated systems originating in nonequilibrium

states.

43
The "expectation values" for p and J are found with the

definition (IV-7) from the equations (III- 18, 32)

P(D = (P(D> = (P e (D> + (en(T)>

= (Pe

(1)

> + en(T

i

)

= -e ($*UH(1)) + en^)

= -e (i|A (1)^(1)) + en(r ) ,

and

where

(IV- 8)

= ~|p(l)<'Jt(2)4>(l))+P >:{ (2)<'$ t

(l)4'(2)>

+ 2 7<1%)A(i)\(;(l)>l +TM(D

c h^\

+ 2|(^(i)r( i)^

Q

(i)>| + rM

(D

(IV-9)

"m(1) = <"m(1) >

= c V, X M (1)

1 s

(IV- 10)

44
with

M (1) = <M (D>

s s -^

(?

! ^

i* a)7w) u,<r vu,> (iv-ii)

T^" ^(1)7 A

v|; fl (l)> 2mc ^ Ya ap T

p '

The "subscript" notation introduced here with the brackets,

{•••} '

*• J 2—1

indicates the designated change of variable after the differential

operation within the brackets is performed.

We introduce here the following notation for a "trace" over

quantities with .spin index:

2 Tr (AB) a > (AB)__ = (AB) aa aa

a=l

2 2

(IV-12!

-II

a=l3=l

A B s A fl- ap pa ap pa

Notice the use of the symbol Tr here as contrasted with the symbol Sp

as used in conjunction with the density matrix (IV-4); this distinction

in the designation of the two traces is observed throughout this paper.

45
CHAPTER V

TEMPERATURE DEPENDENT GREEN'S FUNCTIONS FOR

THE ELECTRON GAS AND THEIR RELATION TO THE CHARGE

AND CURRENT DENSITIES

Green's Functions and Their Time Evolution Equations

In this chapter we consider the Green's functions as defined by

Martin and Schwinger (reference (1)). These functions are temperature dependent space-time correlation functions formed from the

"expectation values" of certain combinations ("Wick products) of the

t field operators- ip and tjj . The one- particle Green's function for the electron field is

given by (according to the definition for a Fermion field):

lap 112 2 lap 1

4» (1)4(2)

a . p

where e (1,2) is the "time- ordered" function

(V-l)

f+ 1 for t > t jU.2) = 4

and r...

- 1 for t < t

1 2

is a time- ordering symbol indicating that the operators

within the square brackets are ordered from right to left in the order

of increasing times. That is

44
4i (1) J (2)] = «< a BH J+

4i (1)^(2) for t > t ^^a(2) 4j (1) for t < t

1 <i

(V-2)

Thus

.i<+

a

(l)+JC2». t

2 >t

2

G (1,2) = .

I +i(i|,J(2)i|ia

(l)>, t

i <t

2

(V-3)

Similarly, the "two-particle" Green's function is defined by

G-> a «(12j34) ^ (-i) e U.2;4, 3)

2a.py6 2

4; (1)^(2)^(3)^(4)

a p v o

J +

where

(V-4)

e 2

(l,2;4.3) =

+ 1 if an even number of permutations is required

to rearrange the ordered set of times

*S (*, > t , t , t ) in the order of increasing times

from right to left.

-1 if an odd number of permutations is required.

In general, the "n-particle" Green's functions are designated by

na a • •

• a , a • • • a 12 n n+1 2n

= (-i) e (1, 2, • • • n;2n, • • • n+ 1)

<

(1, 2, • • • n; n+ 1, • • • 2n)

(V-5)

4j (1)4j (Z)- - - vp (n)^ 1 (n+lJ-.-vJj' (2n)

- 12 n n+1 2n

where e is defined similarly to e and e . Notice that in all cases, n 12

the Green's functions have an equal number of creation and annihilation

operators.

47
Equations of motion can be obtained for the Green's functions.

In the case of G :

l*^<W1 - z>--1 «i

(1 ' 2»

+ i:

Bib (1)

*t»)

8t

l V5

.m \ dt

-o

+-'

: V6

a

at

1

J +

(V-6)

'p (2>

The last term on the right hand side of equation (V-6) arises because of

the discontinuity in G at t = t (see Mathematical Appendix IV).

The integral in (V-6) may be evaluated as follows:

V6

fTMo

+ ^

dtl

~k

(' i£l(1 ' 2)

V6

* (1)^'(2)

a (3

(V-7)

= lim / • — t f _

6 _

+ (-ie

1 (2 +

,2)ip(r

1 ,t

2 +6)^(2)+ ^

(2_, 2) i|,' (2) ^(r^ t £

- 6)

= -i (+.?!. V^(2) + ^(2)+^.^)

=

- i63 (vV

from (III- 9) and since

e (2 +

, 2) = -e (2_,2) = 1 .

(The subscripts (+) and (- ) refer to a time ordering, i. e. , 1 implies

t ± 6 where 6 is an "infinitesimal. ") Therefore, from (V-6) and

(V-7) one obtains

48
l *^ G

la P

(1 ' 2) - ^£

1 {1 ' Z) ({^^ 4

2

\

(V-8)

= 6 6(t - t )6

3

(r - r ) = 6 6

4

(1, 2)

ap 1 Z 1 Z ap

With the aid of equation (III- 12) and the definitions (V- 1

) and (V-4) for

G. and G , equation (V-8) may be written in the form

6 in— - h (1)

aA. dt. aA. G (1,2) = *S 6 (1,2)

lA.p ap

dir

3 7^{ieG2 YaYp

(r 3' t

l +irl'

t

l ;r 3' tl ++; V t2

) + en(r

3,G Up(1 ' 2) -I

2ma c J

d

3^ _ (ie) Jp(r_,t)G (r t +;T , t ;r t ++;r t ) 3 2mcr (^3 1 2vayP 3 1 114 1 2 2

+ p * ( vV%aY p^'

v

; vvvV+;V2

:

+ uhV

3

X v

G

2XQYp

(r*3

, V;r

l ,tl ;r

3 ,t

l++

;r 2,t

2 )^^ (V-9)

4 3

*-fd*rl J

ie) JrgBt tf )-SglG, .Pl.t1 +;rl.t,;'?Jl .t,++-r:.tJ * 3 2mcr

13 | *3"1' 2m ~2-ya-YP*"3*

"1

" '"l' "l'*4' "1' ''"2' "2*

+ P*(r ,t )•

P^G «(r"A.t,+;T.t ;T,t.++;T,t_

]

i 1 Zm Z\a\p 41 1131 22

+ ^(V.XaJ. ^iil C (r , t +;r , t ;r, , t++;r_, t ) 3 y\' 2m Z\ayB K 3' 1 ' 1* 1' 3' 1 ' 2' 2'|-~ — r 4-r

3

ja.n'e —

2mc VMd^. -

(le)

3 2mcr

13

P(r 3' t

l )G

2 Y ,YP

(r 3' t

l + Jr4'V

r

5' t

l++;r2' t 2

)

49
+ p *< v v%wP ( v v;r

4' vvv+; vy

+ n* V

3

x7

K

G (r

3

, t 1+; r 4

, tl ;r3, t1++:.

a

. y

where

ap a (3 2m

VT

3

2 P(l) f

J^T . B A

(1) - 6 £- 6 e UA

(1) 2mc a£ a (3 af3

The + signs following the time co-ordinates t in (V-9) indicate the

proper time ordering arising from the original ordering of the field

t operators ijj and ijj :

t + - t + 6

t ++ = t + 26 ,

(V-10)

where the 6 is an "infinitesimal" quantity. For the Green's function

G (2,1) =

l(3a

•1^(2, 1)

p a

+ (V-Il)

an equation of motion can be obtained which is "conjugate" to equation

(V-9):

\a at \a G18X (2 '" " *

6

B„

6 f 2' 1 '

- e \ d r 13

i ieGo« (r,.t_;r _,t 7 ;r t ++;r ,t + ) + en(r_)G_ (2,1)

3 r,„ f 2£yci.y 2 2 3 111 31 3 lpa

+ ^-!^3^{^V^W<r2

,t

2

;r 3

,t

1 ;r

1 ,t 1++ ;r 4

,t 1+ )

50
+ P *(r*,, t ) G (7 t ;T t ;T , t ++;T t-+)

3 1 2Svav 224111 31 (V-12)

+ |rf V

3

X 7 G

2pXaY (7

2

, t2

;T

3

, t i;7, t 1++S r

3

. t Jrr 4"* .—r_

3

d

3^ „ (l6)

-fp(r ,t)- ^iilG,: , (7 ,t ;T ,t ;r.t ++;"?' t +) tf cr |_*

v "3' 1' 2m 2pyaY

2 2 3 111 41

+ P*(T,,t )•

P_"C(1)

G (7 t :7 t ;7 t ++;r t +)

3 1 2m 2pyaY 224111 31

— P #tll

(r„, t ;r . t :r . t ++;r . t +

3 v\ 2m 2p\av 2 2 3 111 3 1 h* —

fie -* r. 2mc %a-{V

l X

I

,3— (ie; a r.3 2mcr

13

P(r v

t.)G (3VWWt

l++Jr*' t1+) 3 1 2(3yt)y 223141 51

+ P"*(r~ 1

> t )G tt ,t ;r,,t ;r , t ++;r , t +) 3 l' 2P\Tf9 2 2 5 14 1 3 1

'

+ u*iV- X cr G (r t ;r t ;r t ++;r t +) r 3 y^- 2p\r)Y 223141 31

where

75^r

3

iJ (1) = 6 t^—

(3a

V

' pa 2m

2 P*(l) + ^—7 -B A

2mc pa

(l)-6a £-6„ eA'

pa pa

There is an equation of motion for G similar to equation (V-9) for

6 i*T— -h (1)

arj at an G x (12;34) = *6 6 (1,3)G_. (2,4)

2ti(3y^ aY IP*

.*8 .6 d,4)G 1A (2,3) a\ ip y

51
'^s ^7 {' e 'Witts' h^i'V7/ VV V+;~3' WV

(V- 1 3)

+ en(7

5

)G

2a ?VX

<I2:34»}

+ (integral terms involving G which arise from

the "magnetic interactions" between electrons).

The equation (V-13) can be converted from the differential form in G.

to a direct equation for G if we first pre-multiply both sides of the

equation by G (5, 1), then sum over a, and finally integrate over

the co-ordinates (r , t ):

fd(l)G. (5, 1)[~6 i*-^--h (1)1 G, (12;34)

J lva an at an ZrjpvA.

= *6 Cd(l)64

(l-.3)G. (5,1)G (2,4)

ayJ 1 va 1 (3\

- *i

aXy*h)»

4

(i.4iialMat.i)a

1 ^(2.

, 's)

(V- 14)

VWY^VWV

+ en(r ) G (12;34)L + (integral term* containing G and G

5 2a(3vA. j 1 3

which arise directly from the "magnetic interactions" between electrons).

After performing some integrations by parts on fV-14), and making

use of the boundary conditions (III- 14) (we assume that G (1,2)

vanishes for infinite time separations and infinite spatial separations,

|t -t_ | —• oo and | r ,- r ,|-*oo). one obtains

52
I

d(1)CW12;34) -6 i-R-^-h1' (1)

1 G. (5,1) lva

(V-15)

iG

lvv

(5 ' 3)G

l?X (2 ' 4 >-* G

lvX

(5 ' 4>G

lpY

(2 ' 3)

+ (integral terms involving G , G , and G which

arise directly from the interactions between electrons).

By comparison of (V-12) with (V-15) we see that equation (V-15)

further reduces to

fd(l)G, a , (12;34)-n6 6

4

(5, 1) = tfG, _ , (52;34)

.) 2npv\ vr) 2vp\\

(V-16)

= ^G, (5,3)0, (2,4)-hG i (5,4)G (2,3)

Ivy IP* lv ^ 1 ?Y

+ (integral interaction terms which contain G , G , and G ).

The equation (V-l6) expresses G as a Hartree-Fock type of combinations of G as shown explicitly, plus some terms arising directly

from the interactions between particles which are designated as "correlation" terms. The first term on the right hand side of

equation (V-l6) is the "direct term" of the Hartree-Fock expansion,

while the second term is the "exchange" contribution.

If we substitute for G from (V-l6) into equation (V-9) and

show explicitly the results of the "Hartree-Fock" contributions from

G , we obtain the following equation:

6aX

i*8^- h

aX (1) G iV' 2> = *% 64(1 ' 2)

••<6(i)>a {i.a|+^.f(Sli)>a o.2j

53
P(l \±fte —

+ -/A(1)\ • V^G (l,2)+f cr • VX/A(1)\

c

K '' 2m lap 2mc a\ ! G

ixp a; «

r'.'T^G, (r,t:r t+)G (r ,t ;r ,t ) 3r lay 1 1 3 1 lyp 3 1 22

(V-17)

+ ("exchange" contributions of order f— ] and all

other "correlation" interaction contributions) ,

where (U(l)) and (3t(l)) are obtained from the definitions (III-4, 16)

and (V-l) and are given by

(Ua^-Jd^^l-a^^.tj)^.^)) + en^j

tot'K (T.t :T t +) + en(F)l

Y7 3 1 3 1 3

J

<r(1) ) . - jWj -ga— {P(7

3

. v<^(T4

. ti ) +v(?

3

, v> (V-18)

4 3

= i"

dV

3 ll% {

P K' V

G- -.«'.• '- *" f - +» l^Y 3' 1' 4' 1

+ „< V

3

XT^G^ti ;T

3

, t i+)^

r4^r

3

54
In a similar manner, equation (V-12) reduces to

6

x ^^-- hI

&>'

\a at Aa ^BX 12 ' 1

' = *

5

Ba

5 »»- •<«(»> G (2.1)

+ ^-^f1 ') G

l P a

(2 ' J

» + f <A(1)

>

' T^^pa'2 ' 1 '

ujie -* 2mc \a v

x x <£•(!)) <V(2,l) (V-19)

I

+ \ d r 3— (ie ) G. . (r , t :r, t, ) G. . (r* , t •?' t +)

3 r 10V 1 2' 2' 3' 1' lyd v 3' 1* l' 1

2 + ("exchange" terms of order ( — 1 and all

other "correlation" interaction contributions).

Charge and Current Densities

The charge and current densities p and j are given from

equations (IV- 8) through (IV- 11) with the aid of the definitions of G

and G (V-l) and (V-4) as follows:

l

p(l) = ie G. (1,1 ) + en(T)

lyy + 1

= ieTrG^l, 1 +

) + en^) ,

(V-20)

and

r(l)-^{?(l)G

lYY

(l,2) + r*(2)G

lvv

(l (2)i + J M

(1)

m c J

3 2mcr |_

(V-21]

55
+ P *{r ., t ) G (r t +;r , t jr . t +++jr . t ++)

4 1 2\aav 311111 41

+ ul^iV X "a , G„ x (r , t +;r , t :r , t +++;r .t ++)

3

with.

y\ 2\aay x 3 11111 3 1 'J— _j~ r 4

r

3

j (1) = cV X M (1) M 1 s M (1) = 3^7 G. (1,1 ) (V-22)

s Zmc y^- 1^-V +

. Ma T.7G.K1.1J . 2mc 1 +

The symbol (n ) represents the set of co-ordinates (r ,t +). If we

+ n n

substitute for G from (V-16) into (V-21), and show explicitly only

the terms derived from the "Hartree-Fock" combinations of G , we

obtain

r(1) = -s{p(1 >°iw

(I - 2> + r*'2 » G

i YY

f1 ' E >

+ ("exchange" contributions of order — ) and all other "interaction" terms).

From a comparison of (V-18) and (V-23) one finds a similarity

between the form of (V-23) and the integrand in (V-18). That is,

except for the term 2 — (A (1)\ G, (1, 1 ), the integrand in (V-18)

c

x ' lyy + appears to be basically of the form

ffflitj. cr

i3

-3-1

56
Specifically, if the interaction term

(

2/7<t(1 »%v(1 'V)-

order f — i is neglected, then to the lowest order in ( — ) equa- of oraer is negietiea, uien 10 tut; iuwcsi uiuci j.h * — i \

c / V

c /

tion (V-23) is given by

f(l) S^-(p(l).G, (1,2) + P*(2)G. (1.2)1 + J[\) 2m 1 lyy *YY Jo , M

(V-24)

^irl Tr

{

?(1)G

l (1 ' 2) + ^

(2)G

l (1 ' 2)

}2 ^1

+ rM

(1) •

This "basic" form of J" (V-24) recurs because of the electron inter'

action terms in (A^ . From (V-18) and (V-24) one has the result that

3^ j <*3'V

<A(1)> £ \d r 3-^_± (V-25)

13

Moreover, from (V-18) and (V-20) one also finds that (U) takes the

form

<U(1)> = \ d r

*

. (V-26)

J 13

Now, in the equations (1-12) U and A are given in the "time-retarded"

form. If one makes infinite series expansions for U and A with the

"retardation" term

r 12

in the integrands as the "expansion parameter, " the results take the

form

57
3_ P< r 2 'V 1 8 r.3- U(r 11 J 2 r c3t J 2 2 1

(V-27)

rc^i.tj

*<vV =I

dV2-7^-;^dV(T2'V + --

Comparing the equations (V-27) with (V-25) and (V-26) we find that

U^,^) = (U(l))

and (V-28)

Aflyt^ = (X(l))

to the lowest order of [ —•] . Thus, from (V-28) one may state

(V-2J) in the form

3"u) S —-/tFdG. (1,2)+TT

*(2)G (1,2)1 + 37,(1) 2m ^

1 YY l Yy J2—1. M

= ~~ TrJT

T

(l) 0^1,2} +TT

*(2)G

1 (1.2)| +1M

+

(V-29)

(1) .

where

T

T

(1) = P(U+£ A(l) = | V

L +| A

T

(l)

(V-30)

7

T

*(i) E p*(i)+|X(i) = -|v +£a T

(i).

The "exchange" terms of order ( — j as well as all "correlation"

terms from G are omitted from this final equation for the current

density j , since we are interested here in only the lowest order

58
contributions. In any case, the terms in are incomplete, as we have seen from the discussion of the Hamiltonian in Chapter III (time-retardation terms and other relativistic contributions, both of

order (W

— i , were omitted).

We return now to a consideration of equation (V-17). From a comparison of (V-17) with (V-28) we see that in the lowest order

equation (V-17) may be written

6

v i£~-- h

T

(1)] G (1,2) = #6 _ 6

4

(1,2)

aX. 3t a\ l\p a|3

(V-31)

C 3— (i e ) — — — —

+ \ d r„ i *-G (r ,t ;r .t+)G (r,,t ;r ,t ) J 3 r lay 1 1 3* 1

' 1 Y P 3 I 2 2

where

h

ap

(1) = 6 ap.IrrT

i r—t

TT (1) + £~ o- . B

T

(1) - 6 Q £ - 6 o eUT

(l) 2mc aj3 ' a.0 ap*

The relations

and

B

A

(1) = V

x X A

A

(1)

B(l) = V

2 X A(l)

B

T

(1) = V

L X A

T

(1)

(V-32)

were used in (V-31). Similarly, the "conjugate" equation (V-19)

reduces to the form

Xa at -- h^«1 >]

G

1 ^

2- 1 ' -*«p.«4

B.l)

(V-33)

I

dV

3

^v^'VVi'Vvvvv'

59
where

h

pl

(1)s «p.S

"-*T

TT *(1) t^T, B

T

(1)-6 A

C - 6. e U

T

(1) . 2mc pa £a (3a

It should be pointed out that in equation (V-31) a term —- X(l) . - aT(1)G, Jl.2) 2m c

v

' c

x

' la (3

T

was inserted into the operator h Q (l); while in equation (V-33) the

a P

quantity —-- A(l) • - A(l)G lfl (2,1) 2m c c lpa

tT

was similarly incorporated with h ' G (2, 1). This insertion is

Xa 1 pK

justified partly by the contention that such a term would have appeared

if the Hamiltonian had included an equivalent operator of order

( — ) , and by the fact that the inclusion of the terms "round off"

V c / , > • f v

equations (V-31, 33) up to terms of order i —

In retrospect, we see now that while the equations (V-9),

(V-12), (V-20), and (V-21) represent G , p, and j* correctly up to

order ( — ) as fvinctionals of U and A , the equations (V-20),

(V-29), (V-31), and (V-33) represent G , p, and j correctly up to

/v \ „ T —

order f — \ as functionals of U and A . Thus a "renormaliV

C / *

zation" of G , p, and j from functionals of the "externally" applied

field potentials to functionals of the total electromagnetic field

potentials has been accomplished. The interactions between particles

are accounted for up to the lowest order of the exchange contribution;

the most significant contribution from the interactions appears now in

the formulation in terms of the "self-consistent fields" (total field

potentials) of the system. One may consider the "renormalized"

60
values of G , p, and j represented by equations (V-20, 29, 31, 33) to

»T T

constitute a new theory in which A and U satisfy Maxwell's equations (Chapter I). The "renormalization" of G , p, and j* from functionals of

the applied potentials to functionals of the total potentials was established in the lowest order for U and A" (V-28). In actuality the

renormalization is valid at least to the second order for U, as can

easily be seen since

I

(V-34)

3— —

d r p(r ,t ) = total charge of the system

= constant

= (by definition the system

is electrically neutral).

Functional Derivatives of the Charge and Current Densities — —T T Now that we have p and j as functionals of A and U given.

by equations (V-20) and (V-29), we can in principle solve explicitly for — —

T

T —T T

p and j if A and U are independently known. If A and U are time dependent, this problem may be quite difficult to solve directly.

In this case the utility of the perturbation theory developed in

Chapter II becomes quite apparent. One may first consider p and j for arbitrary perturbing fields, and finally fix the values of the fields

through Maxwell's equations.

The initial equilibrium state of the system is assumed to be

known in terms of the Green's functions and their functional derivatives.

61
From (V-20) and (V-29), one obtains the functional

derivatives

x m 6G (1,1) ^-p^- = ieTr ^ -^- 6UT

(2) 6UT

(2) o o

r m SG (1,1)

6AT

(2) 6AT

no no(2)

631D ie, r-T„.

6G

1

(1 ' 3

> -T.._, »°1»- S

>1 <V" 35)

TrJTT (1) = + t, *(3)

6UT

(2)

2m I ° 6UT

(2) ° 6UT

o

(2) h-V 'o o +

. 6rM

(1>

6UT

(2) o

«33a- - ^Tr Jr^64

(i, 2)G,(i.i) + -T

u>

^£i.'

6AT

(2)

2m \n c ' 1 +o ° 5AT

no no(2)

_T 60(1.3). srM(D

+ it *(3) ° 6AT

(2) J 6AT(2)

n o 3-.-1 n o with

- 6 rM

(i) sff (i) = c V X

6UT

(2)

IT6U (2) o o

6j M

(1 » „ v 6Ms (1) = c y x

T

'! T 6Ano(2) 6Ano(2)

6M (1) . 6G (1,1 ) s lufie m — 1

(V-36)

Tr cr 6UT

(2)

2mC 6UT

(2) o o

62
6M (1)

s 6AT

(2) n o

- * 6 0(1,1)

2mc 6AT

(2) n o

It should be understood that all expressions displayed in equations

(V-35, 36) represent the first variations of the expressions (V-20, 29)

given in the limit of no perturbations (e. g. , the initial state of the

system at times t < t ): o

T 6U — —

T

8 A —

(V-37)

The applied magnetic field is taken to be aligned along the

z-axis making the z-axis a preferred axis of symmetry. Accordingly, one may define several useful combinations of operators and combinations of the current density components:

—

T

TT (1) —

T

IT *(1)

Ml)

7*(i)

with "equilibrium" —

T

—

T

value A for A

o

TT (1) = TT (D + iTT (1) + x y

TT (1) = TT (l)-iTT (1) x y

71 *(1) = TT *(l)-itr *(1)

TT *(1) = TT ^(1) + i TT *(1) x y

= x + i y

x = x - ly

63

(V-38)
ax 2 I 9x 8y

8x 2 [ 3x " * dy

cr = cr + i cr + x y

a = cr + i cr x y

T T T A+

(l). h A

x

(l) + iA (1)

T T T A

X (1) = A (1) - iA (1)

x y

5 . 6

- l 6 A" (2) LsaT

(2) 6AT

(2). + x y

6AT

(2)

5 + i, 6

6AT

(2) 6AT

(2). x ' y '

J + (D s JX

(D + ij (1)

j (1) = j (1) - ij (1)

x y

M (1) = M (1) + iM (1)

s + s x s y

M (1) = M (1) - iM (1)

s s x s y

64
From (V-30) and (V-38), the following combinations are possible

h a l —

j.1 — zz — TT - TT '' 1 ax

+

4 +

J

h a 1

-| — z: — TT - TT '• 1 dx 4 +

-

h a 1

-I — » — — TT - TT *

1 az 2 Z Z

- A

T

(l) = -i [tr + tt *

c +

v o 2 +

(V-39)

^AT

(1) c - o

C z o

12

TT + IT *

.1

1 1 2

TT + TT *

Z Z

With the above definitions (V-38), one may write equations (V-35, 36) as follows:

* m 6G (1,1 ) -^- = ieTr L

+ 6U (2) 6U (2) o o

6pd)

5A^(2)

+ o

12

r 6p(l) 6p(l) I -5AT

(2) 6AT

(2) . x o y o

= i e Tr

6A^(2)

+ o

6pd)

6AT

(2) o

12

r 6 P (i)

|

. 5 P (i) - -6AT

(2) 6AT

(2) - x o y o

= i e Tr

6AT

(2) o

65
SPOU = ieTr 1 1-

6AT

(2) z o 6AJ< 2 z Lo

j (1) - 6J (1) 6J (1)

6UT

(2) L5UT

(2) 6U T

(2) o o

i e 2m Tr ir+

(l)+ w*(3)

6 ^M {1) +

- 60^1,3)

- 6UT

(2) -»3-~l. 6UT(2) o + . o

6j, (1)

T

6aJ(2) + o

_____ -t- i

- T

- i ————— -f — T T T -6A (2) 6A (2) 6 A (2) 6A (2) J x o x o y o y o

6j(l) 6j (1) 6j (1) + i i i £ + l

i fi f 2 & 4

= 5_ T*tT

8

' (1,z,G

» a,1^H ir+

(l) + ir*(3)

- SG^l.3)

8A (2) ->3—

+ o +

6j + —

6A T

+ < 2

>o

6j+ d) 12

r syi)

6AT

(2) -6AT

(2)

5j (I) 6j (1) 6i (1) + i = +i

6AT

(2) X " 'O X o 6AT(2) 6AT

(2) - y o y o

(V-40)

ie it i

5Gi^« 3h

5AT

(Z)

66
6i (1) r 6j (1)

z o

5j (1) n y

L6AT

(2) 6AT

(2) - z o z o

+ i

l e 2m Tr tt+

(1)+tt_*(3)

. 50^1,3) 6 V1} + 6AT

(2) h—l, 6 A (2) z o + z o

6j (1) r

6j (1) 6j (1) _,

• y - i - 6UT

(2) L6UT

(2) 6UT

(2) o o

i e 2m Tr TT_(l) + TT+

*(3)

-, SG^l, 3)

- 6UT

U (2) -»3— 1. 6U

6 JM (1)

(2).

i r

6 J X

(1) 6j (1)

y

6jx

(l) 6jx

(l) n

- 1 6AT

(2) 6AT

(2) 6AT

(2) 6AT

(2) _ xo xo yo y°

6j_(l)

6A^(2)

+ o

i e 2m Tr TT_(l) + TT+

*(3)

6 6.(1,3). 6 J>^ 1 ) 1 1 + M - 6aJ(2) ^3-1 5aJ(2)

+ o + + o

6j (1) . r6j (1)

2 6AT

(2)

- l 6j (1) Si (1) 6j (1) —* + i 2 + 2

L6AT

(2) 6AT

(2) 6AT

(2) 6AT

xo xo y° (2) y°

= f^- Tr{- 6

4

(1, 2)G.(1,1 ) + [* (l) + u*(3) 2m c 1 + o - +

1 6G

1 {1,3)

6A (2) J 3^1

o +

6 JM (1)

- 6AT

(2)

67
6j (1) r 6j (1) x 6A T

(2) z o

6J (l)

L6AT

(2) 6AT

(2) J z o z o

i e 2m Tr TT_(l) + TT+

*(3)

-, SG^l.3)

+

6 V1J . 5AT

(2) J 3—1 6AT(2) z o + z o

6 J z ^ i e 6UT

(2)

2m

o

Tr 7T (1) + TT*(3)

z z SG

l (1 ' 3)

j

+

5 ^M (1) z

J 6UT

(2) ^3—1 6UT(2) o + o

6j

2

(1) = i r 6j

z

(1)

.. 6j

z

(1 »

6A^(2)

+ o

- 1 L6AT

(2) 6AT

(2) x o y o

i^.Tr||.

z

(l) + ^(M

-, 5G (1,3) 6 ^M (1) z 6aJ(2) J3-*l, 6aJ(2)

+ o + + o

6j (1) z 6AT

(2)

6j (1) 6j (1) _, - + I. 2

L6AT

(2) 6AT

(2) J x o y o

i e 2m Tr TT (1)+TT *(3)

z z

60,(1, 3)

6A (2)_ J 3-l.

M z 6AT

(2) T O

6j

z d)

v =s; rMT 64|u' G

6A i<

uA ,

< 2 z Uo

ir (l) + w *(3)

z z

-. SG^l.3) 5 ^M(1)z 6AT

(2) -b—1 6AT

(2) z o + z o

+

68
•with

M+—. _ 9 szd s- = - ic Z — + ic — —

.T.„. 9x, _„T,_ 9z, _„T 6U (2) 'I- 6U (2) 1 6U (2)

and

6 'm (1 > + "£«„

«M(I

>- 6UT

(2) o

6 J M"»-

6AT

i (2)

"M

P>. 6UT

(2) . o

6 VX) Z 6AT

(2)

6Ms (l) + 6UT

(2) o

6M (1)

6AT

+

(2)

5M (1)

s 6UT

(2)

= - ic2

a 6M (1) 5M(1)

d s z . d s +

"alT t .„. + 1C

"aTT . . t

i- -v} -

1 "»«•

„ 6M (1) SM (1)

. 9 s z . 9 s ic 2 — - ic

8X

1+ 6UT

(2)

9Z

1 5UT

(2)

= ic2

6M (1) 6M (1)

s z . 9 s — - ic

1 6A^ (2)

= ic

= ic

8

6M (1)

s - 8x

i. 6UT

(2) o

3

6M (1)

s 8X

1- 6A^ (2)

IC

a 5M (1)

9 s +

ic

3X

1 T 6UT

(2)

8

5Ms

(1 » +

fi}

9x. A T ._. 1+ 6A (2) ® ° (V-41)

-j- 6G,(1, 1 ) 2mC + 6U T

(2)

u* 6G.(1,1 ) — I r a — 2mc + . T

6 A.

±

J2)

{^} • z °

• • 6G, 1,1

Tr a 2m c 6UT

(2)

69
6M (1) ., 5G,(1, 1 ) s - iM>1ie _ 1 + 6A 7(2)

it}

° 2mc i r tr 6A J (2)

{1} ° 6M (1)

s z ijxne

2mc Tr 0" z

SGjd, 1 +

) T

6U (2) o T

6U (2) o 6M (1)

s z ijjcfie 2mc Tr 0" z

SG^l, 1 +

) 6A J (2) 6A 7(2)

{z} °

The functional derivatives

SGjd, 3) SG^l, 3) ^ and —

6U (2) 6A (2) o q o

are found in Cartesian co-ordinate form from the equations (V-31) and

(V-33). The first vari

results in the equation

T

(V-33). The first variation of equation (V-31) with respect to U (2)

a T ^ 6G l xs' 1, 3

^ 4

6 . rh^- -h 1

x (l) [ —^ + e 6*(1, 2)G. (1. 3) a\ Bt

x a\ J 6UT

(2)

laB

14 L 6U T

(2)

V^' 3

'^

o F..^{-

+ G. r., t ,;r.;t.+) W

lQ ^ ^ * J 4 J ° 6UT

(2) o

70
To obtain

SG^I, 3)

T

' 5U (2) o

both sides of equation (V-42) are pre-multiplied by G (5, 1) , then

summed over the spin index a, and integrated over the set of

co-ordinates (r , t ), i.e.,

fd(l)G. (5,1) lb i*(-£-- h

T

(I) I

J

It] a o [ a\ 3t a\ oj 6UT

(2) o

+ efd(l)C [5,1 [1.3

•qa o lap o

rrd(1)d - <i£!i{Gi (5ll)

6Gi^rVVV'G jt, v

t

3 JJ 4 r

i 4 I

l ^ a ° KTTT m 1 YP 4 1 3

. v3 O

(V-43)

?

4

(t

l

+) - -

14 * ' 6U (2) o

+ G (5,1) G (?

1 .t1 ;T,t+) ilP_±_i- 3

'

lt,a o lav 1 1 4 1 o

g yT

o

After several integrations by parts in the first term on the left hand

side of equation (V-43) and with the use of equation (V-33) one obtains

the following result:

6G (5,3)

6UT

(2) o * lT)X ° 1XP

(V-44)

2 p

„ 6G (r ,t;r t+)

+ if-ffd(i)d 3r -L Q. (5,D

lav

T

l 4 l o. Bg.,v?v

tj . * JJ 4 r 14

1t1 q ° 6U (2)

ly(3 4 1 3 3o

o

71
If the exchange contribution to (V-44) is considered negligible, then

6G (1.3) lap

6UT

(2) — G. (1,2) G (2,3)

-ft laX ol\p o

or, in matrix form, (V-45)

6G (1,3)

i £ _| G (1.2) G (2,3)

6UT

(2) * 1 °

1

The first variation of equation (V-31) with respect to A (2) is

q

aX ot

: aX of 6AT

(2)

q o

i , T^64

(1,2)tt (1) aX 2mc q

+ 6 .^ ( 1)6

4

(1 ,2) + i^L

aX 2mc q 2mc VX V (1 ' 2 » Wh3)

o

(V-46) F^{ 6AT

(2)

q o

G

lv P

(r 4* t

i ;r 3' t 3

) o

6G, „(r„,t ,;r ,t ). lay 1 1 4. 1 o T 6A J

„<2 >« q o

By the same procedure that led to equations (V-44, 45) we find from

equation (V-46) that

SG (5,3)

In (3 e_ 6AT 2mc*

q o

TT (2)+TT *(6)

q q

G. . (5, 6) G (2,3) lnX o lXp oL

72
2mc

_ (V-47)

V XG. (5,2) cr G^ (2, 3)

2 lr|a o a\ iA-p o

+ i£_Jjd(l )d V-L_G 1 (5,1)

laX

i

l 4 * G, % jET,.t,;g,t, 4r n/l In a o 14

T 6A (2)

q o

l\j3

V 4' V 3'Vo

If the exchange contribution in (V-47) is considered negligible, then

5G

ias

(1 ' 3) e rr 1 ££ S ——; J tt (2)+ tt *(4) G. (1,4) G. (2,3)

6 A (2)

2mch^L q q lay o 1-yp o q o

+ ^[V

2

XG

1 aY

(1 ' 2) oV G lVE ' 3!

olJ4

^

or, in matrix form, (V-48)

6G(1,3)

1 e 6AT " 2mch

q o

tt (2) + tt *(4)

q q

0,(1,4) G(2,3)

1 o 1 o

+ u^ V XG.d.2) cr G(2, 3) 2 1 o 1 o

1 9^4—2

With the aid of the definitions (V-38), one obtains from (V-48)

the equations

6G(1,3)

1 e

c »T 4mch

6 A (2)

T O

JTtt (2)+ir*(4)] G (1,,, 0,(2.3)

+ 1 o 1 (

+ u.<fi i2^-G.(l,2) cr G (2,3) -i® (3,(1,2) cr G, (2, 3) ox 1 o z 1 o 3z 1 o - 1 o 4^2

73
5G

l(

l,3)

T, , 4mcfi 6A (2) o

TT+

(2) + 1T_*(4) G.(l,4) G (2,3)

1 o 1 o

+ uh -i2~— G.(l f 2) <r G (2,3) +i-L G (1,2) or G (2, 3) 8x 1 o z 1 o 3z 1 o + 1 o ,

(V-49)

SGfl, 3)

1 e

_ T 2md

6 A (2) z o

TT (2)+ TT *(4)

z z G(l,4) G(2, 3)

1 o 1 o

+ i^f—— G1 (l,2) <r G(2,3) - ~- G. (1,2) or G (2, 3)

1 8x 1 o - 1 o 3x 1 o + 1 o 4—2

With the help of the equations (V-45) and (V-49), equations (V-40, 41)

may be written-in the form

6p(l) ie

6UT

(2)

TrG (1,2) G (2,1) n * 1 o 1 o

Spd)

saJu) + o

i e 4mch

Tr tt_(2)+ it *(4) G_(l,4) G.(2, 1)

1 o 1 o

+ iu-h" 2 -s G(l,2) or G(2,l) -—-G.(l,2) cr G(2,l)

ox 1 o z 1 oz 1 o - 1 o

.J 4—2

Sp(D i e

_ . T 4mch

6 A (2) o

Tr ir+

(2) + w*(4) G(l,4) G(2, 1)

1 o 1 o

iirn 2-^— G (1,2) o- G (2, 1) - -JL G (1,2) cr G (2, 1) ox. 1 o z 1 o 8z 1 o + 1 o | 2- 2 -< J 4—2

74
Spd i e 6AT

(2) z o 2mcft

Tr TT (2) + TT *(4) z z G (1. 4) a p. i) o

+ ifjL-fi -1—0.(1,2) cr G (2,1) --JL-G.U.2) «r,G(2, 1) 3x 1 o - 1 o ox 1 o + o 4—2

6j- (1)

6UT

(2)

|~ Tr /[»(!) + it *(3)]g (1,2) G (2, 3) 1 + -i5j M

(1) +

(2)

6j + (D

6 A (2) 8m eft ^ + o

-^4— Tr /8iiin6

4

(l,2)G 1 (l, 1 ) £ ^ 1 + o

(V-50)

7r+

(l) + TT_*(3) TT (2)+TT *(4) G.(l,4) G i (2,3)

1 o 1 o

i^["ir .(l) + ir niilh^—G (1,2) cr G_(2,3) --L-G.(1.2) cr G (2,3)11 + - ! [_ o x->, 1 o z 1 o dz 1 o - 1 oif + iirft 4—2

5 W

6A

+

(2 »o

6 A (2) 8m eft o

tt,(2) + tt *(4)1 G.(l,4) 0,(2,3)

+ - 1 o 1 o

Hi* ir (l) + w*(31nr 9 ~ „ „ _ ^ /o „ a 12^^— G.(l,2) a G(2,3)-/-G l (l,2)crG(2,:

ox, 1 o z 1 o cz 1 o + 1

6j M

(1 ' +

5 A

1

(2)

75
6j* (1) + i e T 2 "" 1 6 A (2) 4m d LL

z o

Tr JlTT, (l)-tTT *(3) TT (2) + TT *(4) z z G (1,4) G (2. 3)

1 o 1 o

+ i fJ.£[~TT+

(l) + TT_*(3) -i- G (1,2) or G (2,3) --J_ G.(i,2) o- G(2,3)

8x 1 o - 1 o 9x 1 o + 1 ojj.

4—2

6AT

(2) z o

6j (1) . 2

i e 6UT

(2)o 2m^

Tr tt_(1) + tt+

*(3)

6j (1)

G (1,2) G (2.3) J. + "

°f3-l 6UT

(2)

8j (l) ,2i e Tr I

6 A (2) 8m cK L + o

ir_(l) + TT+

*(3) tt (2).+ tt*(4) G t (l,4) G(2, 3)

1 o 1 o

+ i|x"5i TT_(l: ) + TT+

*(3) 2-j4-G.(l,2) o- G (2,3) --2-0(1,2) «r G_(2f 3)"Jl ox 1 o z 1 o 3z 1 o -• 1 oIL

£ T Z -^ J —» i 4—2

6 V1}

- 5A^(2)

o

6j (1)

6 A (2) 8m2cK '- o

Tr J8tfm6

4

(l,2) G_(l, 1 ) V +'o

TT_(l) + ir+

*(3) TT (2)+TT *(4) G.(l,4) G (2,3)

1 o 1 o

i\xft w(l) + TT+

*(3) 2-5^-0(1,2) o-G (2,3) -/- ox 1 o z 1 o oz

G.(l,2) o-G (2,3) 1

1 o+ 1 oJJ^

4—2

76
6AT

(2)

6j (1)

1 e 6AT

(2) 4m2

cii z o

Tr TT_(l) + TT+

*(3) r (2)+- *(4)]G.(1,4) G (2, 3) z z 1 o 1 o

+ i^*["ir (l) + Tr*(3)T~— G(l,2) cr G (2,3)~G(1,2) c- G (2,3)]L

L

" +

JL8x

2- * ° "

l ° 3x

2+ ~

° + l h-~l

4—2

M

T 6A 2) z o

5UT

(2)

"

2mfi

o

Tr TT (1) + TT*(3) z z

1 6j v

(1)

, G.(1.2)

1 G (2,3)1 + ^ °

l °h~l 6UT

(2)

6j (1) z

. 2

i e Tr

6 A?" (2) 8m2

c£

+ o

7T (1) + TT *(3)

Z Z

TT_(2) + TT+

*(4) G (1,4) G (2,3)

1 o 1 o

+ i|x-ft rr U) + ir*(3)

z z 2^—0,(1.2) rG(2,3)~G(l,2) a G (2,3)11 ox 1 ozl o oz 1 o-l °!J3_4,

1 4—2

M z 6aJ(2) + o

6j (1) . 2

z i e

6 A (2) 8m2

cK

o

Tr TT (1) + TT *(3) z z

tt (2) + tt *(4)1g,(1,4) G(2,3)

I 1 o 1 o

ill* TT (1) + TT*(3)

z z 2^—0,(1,2) o- G/2,3) -0(1,2) <r G (2,3)

c)x 1 ozl ol o+l o 3—1

4—2

77
+ M z 6AT

(2) o

6j (1) z

= —-^|— Tr-J4hrn 6

4m rfi ^ T 6A (2) z o

+ pz

(l) + IT Z*(3) TT (2) + TT *(4) z z

r ' +'o

G.(l,4) G(Z,3)

1 o 1 o

+ iu^ TT (1) + TT*(3) z z C(l,2) <r 6(2,3)-^— Gfl.2) cr G (2,3) ax r 'o - r b ax r

' 'o + r 'o 3—1

4—2

6AT

(2) z o

The variational derivatives of j are again given by equations (V-41)

and the variational derivatives of M are given by

s 6M (1) . 2

5 =-^£- Tr. Gfl.2) CA2 > l ) n 6UT

(2)

2mC + .1 0. 1. •

6M(1)

s +

6 A, (2) 8m c + o

.2 rr

= -^Yl Tr °+l

wJ

2

> + Tr+

*{4) GU.4) G_(2, 1)

1 o 1 o

+ iu^ 2—2— G (1.2) o-G(2,l) -—- G.(l,2) cr G (2,1) U 3x2+ l

v 'oz l

x o 3z

2

1 o- 1 °jj4—2_^

78
6M (1) . 2

s + ljx e T

~ 2 2

6 A (2) 8m c o

Tr cr jr-r (2) + tt *(4)~JG (1,4) G (2, 1) + (I + 1 1 ol o

^£ 2^—CAl.Z) cr GJ2.1) -r^-G.(l,2) <r G (2,1) "] ox 1 o z 1 o 9z 1 o + 1 o i

6M (1) . 2

s + i^e

T 2 2

6 A (2) 4m c z o

Tr •.{[• (2) + TT *(4)

z z Gfl.4) G(2,l)1 o 1 o

+ 1^^—0.(1,2) o- G (2,1) --i—G.(l,2) <r G (2,1) n OX 1 O - 1 O OX 1 o + 1 L

6M (1)

s 6UT

(2)

Tr cr G (1,2) G 1 (2,1) 2mc 1 o 1 o (V-51)

5M (1) . 2

s i[j.e

, Tr cr -Mtt (2) + tt *(4)

6 A (2) 8m c

' LL

" + o

G.(l,4) G,(2,l)

1 o 1 o

+ i(j.^i 2——G,(l,2) cr G ,(2,T 8x 1 o z 1 o 9z

G (1,2) o- G (2,1) U 1 ° "

l °JJ 4 ^2

6M (1) 2

s i[j.e T 2 2

6 A (2) 8m c o

Tr o- tt (2) + tt *(4) G(l,4) 0(2,1)

1 o 1 o

ijj.'fi 2-^—0,(1,2) cr 0/2,1) --^-GQ.2) ^.(2,1) 3x 1 o z 1 o 3z 1 o + 1 o 4^2

79
6M (1) . 2

s - iue ———

= —c Tr <r T 2 2

6 A (2) 4m c

z o -{ TT (2) + TT *(4)

z z G (1,4) G (2,1)1 o 1 o

+ iyjL'fi G, (1,2) <r G,(2,l) -^—G,(l,2) _<r, G,(2,l)_

9a£T V o - 1 o 9x0i r o+ r b

L 2- 2+ L

6M (1)

s z i|j.e Trcr G (1,2) G (2,1)

6UT

(2)

2mC Z l °X o 6M (1) . 2

s z ip.e

T 2 2

6 A (2) 8m c + o

Tr •{ TT (2) + TT *(4) 0,(1,4) G.(2, 1)

1 o 1 o

- 1

H- £ 2~—Gfl.2) <rG 1 (2 l l)~G1 (l,2) <r G.(2,l) ox 1 o z 1 o 3z 1 o - 1 o 4—2

6M (1) 2

. s z iu.e _ . = —C—-,— Tr

T 2 2

6 A (2) 8m c o v{ TT+

(2) + TT_*(4) G (1,4) G (2,1)

1 o 1 o

^[z-i—G.-U.2) <rG.(2,l]i--^

ox 1 o z 1 o oz G (1,2) cr G (2,1) Tl

1 ° + l °Jj 4-.2

6M (1)

_s z i(ie

T 2 2

5 A (2) 4m c z o

Tr •.{ TT (2) + TT *(4) z z G (1,4) 0,(2,1)

1 o 1 o

+ ijiii -£—0.(1,2) o-G (2,1) --1-0.(1,2)0-0 (2,1) 3x 1 o - 1 o 3x 1 o + 1 o I4—2
The equations (V-50, 51) are given in the "self-consistent

field approximation" (all exchange and higher order correlation

contributions are neglected); the appropriate Green's function to be

used in this approximation will be the solution of equation (V-31) with

the exchange term omitted.

There are a great number of terms in equations (V-50, 51)

involving the Pauli spin matrices. Fortunately, if G is diagonal

a number of the spin terms vanish immediately due to the trace

operations involved in the equations. The spin matrices are given by

(see reference (35), section 10):

'V

and

and thus

?!).-

1 -1

2

(V-52)

'02

The following combinations appear in the equations (V-50, 51) which

vanish identically for diagonal G :

TrG (1,4) o- G (2, 3) = TrGfl.4) cr G, (2, 3)

A O-l O 1 O+l O

= Tro- -1G (1,4)OlG,(2,3) = Tr<rG(l,4) 0,(2,3)

o +1 ol o

81
Tr<r Gfl.4) <r Gf2, 3) = Tro: G/1,4) <r G (2,3)

z 1 o - 1 o z 1 o+l o (V-53)

Tr<r G ,(1,4) cr G, (2, 3) = Tr«r G,(l,4) <r G (2, 3) -1 ozl o +1 ozl o

Trcr Gfl.4) <r G,(2,3) = Tr a G (1. 4) <r G (2, 3) -1 O - 1 O T 1 O+l o

= 0, for diagonal G

Also for diagonal G we have the relations

o- G.(l,4) o- G/2,3) = G.d.4) 0,(2,3) zl ozl o 1 ol o

(V-54)

<r G/1.4) 0,(2,3) = G,(l,4) <r 0,(2,3) . zl ol o 1 ozl o

The use of equations (V-53) and (V-54) with equations (V-50,51)

simplifies and shortens the work of calculation a great deal.

The term TrG,(l, 1 ) appears in the equations (V-50) a

1 + o number of times; this term may be replaced by a constant factor

wherever it appears. The positive charge background is- taken here to

be uniform, thus

n(r ) = constant. (V-55)

If the equilibrium value of the electron number density operator p

(III- 7) is defined as

». - <p>» • < v- 56>

82
we see by comparison of (III-7) with the definition of G (V-l) that

or

n = -iTrG.(l.l ) , e 1 + o

TrG.(l,l) = in , 1 + o e

(V-57)

where n is the equilibrium value of the electron number density.

e Because of the imposed overall charge neutrality of the system one has the result

n = n(r ) = constant.

e (V-58)

83
CHAPTER VI

SOLUTION FOR THE ONE- PARTICLE GREEN'S FUNCTION

Methods of solution for Green's functions have been outlined

in the references (1) through (8). An approach similar to that given

by Ashby (see references (2) and (3)) is utilized here.

A uniform magnetic field is applied to the electron gas both

the equilibrium state and after the action of an electromagnetic per-

-«-T turbation. The magnetic field B (l)(t, ^ t ) is then constant, given

o 1 o by

in

B

T

(1) = V X A

T

(1), o 1 o

(VI- 1) —

—T where A is time- independent. The direction of B is taken along

the z-axis:

B

T

(1) = V; X A*

T

(1) = T B , B 2* . o 1 o zoo (VI-2)

The potential U (1) is taken to be zero for t <t (equilibrium state); o ° T

thus there is no applied electric field. The choice B* = constant

-T T

° (A time- independent) and U (1) = is consistent with Maxwell's

o o equations (Chapter I) and with the equations (V-20) and (V-29) for p and j . The relations

o ' o

(VI- 3)

'_»T T" A , U

o o

= ,

84
can be verified in the "self-consistent field approximation" and thus —

T

T

the choice made for A and U is certainly proper to this order.

o o No attempt is made here to obtain a complete solution of the

equations (V-31) and (V-33); the exchange contributions are neglected.

In order to better visualize the nature of this approximation, we

introduce the function G (1.2) such that

oap

h v i*-j£-- h

T

.(l) }g ..(1.2) = £6 6

4

(1,2)

1 aX ot aX 'oj oXp ap

with (VI- 4)

T 1 2 ^R

a\ o aX 2m 2 zaX aX

and

-f- 6. i*-J hJ

T(l)}G . (2, 1) =-fi6 6

4

(2, 1)

| Xa ot Xa oj op X

' pa

with (VI- 5)

tT 1—2 ^"r

Xa O Xa 2m 2 zXa Xa

and where

o

W-r^ = B mc

(VI- 6)

If both sides of equation (V-31) are pre-multiplied by G (5, 1), then

Ona

summed over the spin index a, and integrated over the set of

co-ordinates (r , t ), one obtains

fd(l)G (5. U-fs . i*-f--hT m 1.G fl,2) = -JfG (5,2

J o^a [ aX Bt a X

x 'of lXpx o o^p )

(VI-7)

+ ie2

Cd(l)d

S

7,-~G (5, 1)G K.t.jrl.t.+j G n J?, t.irl . t, J

. J 3 r o-qa lay 1 1 3 1 o lyp x 3 12 2 o

85
After several integrations by parts and with the use of the assumption

that both G,(l,2) and G (1,2) vani

1 o separations, one obtains the result

that both G (1,2) and G (1,2) vanish for infinite time and spatial

1 o

G. .(5,2) = G (5,2)

lrjp o or\p

2 + ^-Cd(l)d3

r* —G (5, 1)G 1 (T.t :T,t+J G ffl, t j7 , t ) , •h J

3 r ona lay 1 1 3 1 o ly(3 3 1 2 2 o

or (VI-8)

G (5,2) = G (5,2) o o

2 + $-§*M% ^c

o

(5. DOjS;. yT,. y^c^. yy t2

) o

.

Similarly, if one pre-multiplies both sides of equations (V-33) by

G (1,5), sums over the spin index a, and integrates over the set oan

of co-ordinates (r ,t'), one obtains

Cd(l)Grt (1.5)/- 6. i£-~--hJT

(l) }g (2, 1) = #G (2, 5)

J oan {_ Xa at \a oj ip\ o opr^

(VI-9) + ie

2

fd(l)d

S

r*.~-G (7,t ;T t ) G (T t ;T t+)G (1.5) . J i

-* l|3y2231o lva 3 1 1 1 o oa-q

The resulting integral equation is

G (2,5) = G (2,5)

lp-rj o o£Jr|

H-^rddidV -J-Glft (T t ;T_,t ) G, (T,t,;r_ t t+lG (1.5)

'n J 3 r lpy 2 2 3 1 o lva 3 111b oat)

86
or (VI -10)

Gt2,l) = G (2,1)

1 o

. 2

l£-fd(4)dV -i-G.C? .t ;7,t ) G (T ,t ;7 t+)G (4, 1) . -nj 3r,. 12234ol3444 o 34

Equations (VI-8) and (VI- 10) are equivalent; they are integral equations for G in terms of the function G . We see that if the exchange

1 o interactions were absent, then the result would be G, (1,2) = G (1, 2).

1 o o Thus, G is clearly the Green's function for an equilibrium system

o

of electrons interacting only with the applied fields

3A T

(1)

Eo

(1) " Vi Uo^) -7 at = ° o 1 o c at.

B (1) = V X A

T

(1) = i B = constant. o 1 o z o

In the "self-consistent field approximation"

(VI-11)

G.(l,2) & G (1,2) . l o o

(VI-12)

For sake of completeness, the essential ideas in the solution

of equation (VI-4) are now discussed. The treatment employed here

is that of Martin and Schwinger (see reference (1)) and of Ashby

(references (2) and (3)).

We shall use subscripts (f) with quantities and operators

relating only to the interaction of the electrons with the applied field

^TB . The time- independent Hamiltonian leading to the equation (VI-4)

is given by

kt '-^A^i6

^*TM

87
where \\i and ijj are the creation and annihilation operators, respectively, for the electron. The number operator is

N. = fdV ^(1) ^ (1) . J la a (VI-14)

In the same manner as before a new Hamiltonian directly related to

the Grand Canonical Ensemble may be constructed with the origin of

the energy redefined (see equation (III- 6)):

tf

i = H

f

- C N

f

. (VI- 15)

The field operators i[j and \\j corresponding to the Hamiltonian <#

obey the usual equations of motion for Heisenberg field operators:

i-n

9^ (1) a

8*1^(1) a

9t

1

a I

a i

(VI-16)

The Green's functions are also defined as before:

G U, 2) = -ie .(1,2) Spif

oap 1 [_ i a p ])

= -ie^l. 2)/ * (D+l(2)

a P

(VI- 17)

where

and

-1 -P*f

f

f - Z e

-(3c?£

Z

f

h Sp {e

^ .

(VI -18)

88
The use of the equations (VI- 16), together with the definition for G

(VI- 17), leads to the equation of motion (VI-4) for G . A generating functiongy (1,2) is defined as follows:

(T-iph)

Y

x

(1,2) =

Spie fi

( T -i(3n)#

iejd.2) •4i (Dvli' rnoJ (2)i)}

Sp p"-,M*'}

.(VI-19)

where t = t„ - t ^0 is a real parameter with dimensions of time,

f o The generating function «J differs from the function G by the re- placement p—p+— t in the "exponential" operator

The function jSi satisfied the same equation of motion as does G

{

'.>.'« H--»>.}* 1*,""TM"-" "V*"'" (VI-20)

The difference between G and ^ is manifested in different sets of o subsidiary conditions for the two functions; these subsidiary conditions

are to be developed next. Consider first the auxiliary functions

i

(T-ipfi)

SPi e

f ft (T-iph)o^

(1.2) = -i

V

f ^(D^(2:

Sp^e;.

-(T-ipfi)jjf

}

S <

(T-ipfi) Sp-^e

(1.2) = +i-

^(T-iph)<^

f ^<2 >v4

-S-(T-ipR)^

SpJeX

« 1

(VI-21)

89
and

SpJe \ (1)^(2

= -l

sP f^

Spnje r ^(2)^ (1)

G .(1,2)

<-y\

= +i

Spje L

(VI-22)

From the definitions (VI- 17) and (VI- 19) we know that

(T-ifft)

(1,2) =<

%**-*h>H

fi^-Wu.z). t,<t <y\ 1 2

and

G

oV X^' 2 >H

G

>yX {1 ' 2) ' t

l >t

2

G

<V X

(1 ' 2) ' t

l <t

2

(VI-23)

(VI-24)

The Heisenberg operators ij; and \\> corresponding to the Hamiltonian

#€ are given by

TaTa 1 o

ip (1) = e ^ (r t ) e a a 1 o

(VI-25)

90
as may be verified with the aid of the equations (VI- 16). We shall

consider the equations (VI- 25) to be valid for complex times as well

as for real times. Now, from (VI-21) and (VI-25)

SpJe fl e \ (r.,t )e * ^(2)1

I v

( V WV = _1

\ ~ Spf*'^

r " p<*f '-l

T

^f t

_(3^ - Pj*h Sp^e e B \j^(2)e 4, (r^tje I

= -1

l( T -ipfi)#;

SpJef 1 (VI-26)

1

= -1

f --(T-iph^if

SpJe ^ i|^(2)i|i (F^+ ipfiH

in which the cyclic invariance of the trace has been utilized; recall

also that T = t, - t > 0. From (VI-21) one also has

1 o

f -^(T-ipfi)Jif

Sp-U i|;'(2)^ (7 t +i0ft)l

*srpi.v«^v«-

t

x :

' ° !

Thus we see from a comparison of (VI-26) and (VI-27) that

>;;

ipK>

<vvv2

> - <;

ipK,

<v„+i<*:V2'- <***>

91
Similarly

By the same procedures that led to (VI-28, 29) one can obtain subsidiary conditions for the Green's function G , which are

o

>y\ 1 o 2 2 <y\ 1 o 2 2

and

(VI-30)

G^ v (7, -t, ;r" ,t -ipfi)= -G^ . (7 t ;T , t ) : <y\ 11 2 o

r >y\ 1 1 2 o

In the case that £ —0, the generating function^] p

(1,2)-^^/" (1,2)

yA. y\

and the subsidiary conditions (VI-28, 29) reduce to the boundary

conditions

(VI-31)

(T), If the solution of cU . (1, 2) is known, the Green's function G . (1, 2) is

obtained from jQ .(1, 2) by the replacement t — - i(3ft\ ^

(t) The boundary conditions (VI-31) define ir^ , U>2) for the re- y\

stricted intervals of the time t ** t ^t„, and t ^ t <t. however,

o 1 f o 2 f the final results are functions of the difference T = t . - t and by ana-

( \ ±o

lytic continuation one obtains £f (1,2) for unrestricted values of t and t.

2

(T) The function

^J (1,2) depends explicitly upon the difference

(t - t ) in times t and t as can be seen from a substitution of the

92
equation (VI-25) into the equations (VI-21):

n^a.z)

- i( x-^)^

f

+

i(.f

t >#

£

- i<tr

to>*

£ +i(Vy#£ .-5%y*fi

SpJe |

= -1

r-^-W^

and (VI-32)

,^>,i. 2,

=+i-

-l(T-i0n)#+i(t-t)i .l(t_t)^+I(t

r t)^ -I(t-t)^

| -n f -n Z o f t-» ., -fi 2 o f 1i 1 o f —., if] o f| Spje e 4,l(r2

,to)e e ^(F.ye

j.

SpJe L

, 4<T '^>*£ 4<Yy*f tr t

. +I*1-V*£ , - t

.1 SpJe e ^V1 -,' 1 ) e ^( r ,' t )f

|^ X 2 o T

\ 1 oj

by the cyclic property of the trace.

=+i

93
The time dependence of

jfcf (1,2) may be expressed by the

Fourier expansions

iqjn;

I'a-^-y. r'a.2)- Ye" T

' 2 0-.-2

' q

and . (VI-33)

*>

T>' 2)s Z

/fL(tfV

*g

T>r1

' r

2 '-

where the index q represents the set of all integers. The boundary

conditions (VI-31) (also known as "antiperiodicity" conditions) restrict

the range of integers q to the odd integers alone. This is shown as

follows. From equations (VI-23, 33) one has the results

Sfa-vvv -.frSfvvvv

or . . (VI- 34)

V -—<VV M V —IV',) , , I<

T £ 2 £

TM-2

> = l^ { V;>v-2

>. v\ xn 12'// aj >v; q q

and

or (VI- 35)

^

e ^

Yx

(<j:r

r

r2

> = £

e ^<v

x

(q;r r

r 2»-

Since equations (VI-34, 35) are valid for arbitrary t in the range

t < t < t , one may conclude that

o 2 f

$V*, -V - ^r >

c;^^i-T

2

> # <>;7

i -^ (VI - 36 >

94
Since t = t r - t , one has from equations (VI- 31) the results

f o

I

lqjr.

q

^<t,-t,)

-Z

e e

(VI- 37)

I

lqir.

(-"^ T wv&«w •

and

T 1 O o, (t) ,-* —

^>vX(q;T

l' T

2

) = ?(t i"V . w . - - ^<^(q;r

i' r 2

)

= -I

lq-rr. (VI-38)

(t--t ) , x lqtr t 1 o o. (t) . -* — . ^<Y X

(q:r

i' r 2

)

KJTT. . n+i ( T ) — ~ T

(Vv £-««>&h*i-T

a>'

Since t and t are arbitrary in the ranee t < t , t < t, we have

1 2

y B o 1 2 f the result

^>>;T

l' T2

)= «-

1 » q+1

J&<'l<C

' ;T

l' T2»- ,VI - 39 »

Comparison of equations (VI-36) and (VI- 39) shows that the choice of

q even is contradictory, while all the conditions are mutually satisfied

for q odd. This result is related to the fact that the electrons obey

95
Fermi-Dirac statistics. In the case of particles obeying BoseEinstein statistics, the boundary conditions would be periodic and the

choice of the even integers rather than the odd integers would then be

the correct one (see references (2) and (3)). The orthonormality

IHZt . relation for the set e T is

Cf i(q-q'K

i r —t—z — \ e dt = 6 . (odd integers) . tJ q.q

1 (VI -40)

(Recall that, by definition, < |t - t | < t. ) Thus far the choice of the gauge of the vector potential A (1)

has been left arbitrary, but before a solution of equation (VI-20) can —

be obtained, the gauge of A (1) must be specified. Since

\

x i By + i B x x o 1 y o 1 = 0,

one can take

o 2 o 1

i B y, + i B x, x o 1 y o 1 (VI-41)

i B y , + i B x, x o 1 y o 1 + \ i B y + i B x, x o 1 y o 1

where X is any real constant. The representation chosen here is the

"symmetric" gauge (\ = 0):—

1 —

-» Ao U) jB.Wie.,, (VI- 42)

96
The solution to the Schrodinger equation for an electron in the field

=*T

B with the gauge (VT-42) is developed in Mathematical Appendix V,

o together with a number of useful identities. The eigenfunctions are

designated by

r

n, i, k 1

ikz

ikz

u .(p , 9 ) (polar cylindrical co-ordinates)

(VI-437

w (x , y ) (Cartesian co-ordinates) , n, l 1 1

and form a complete orthonormal set. The quantum numbers n and I form a discrete set (all the positive integers 0, 1, 2, 3, . . . co),

while k has a continuous range, - co ^ k < + co. The energy

eigenvalues of the Hamiltonian H, are characterized by the symbol

E , where a is the spin index. The energy eigenvalues of the

a, n, k

Hamiltonian <#, are given by £ , = E , - t; the eigenfunctions

i ^ a, n, k a, n, k

are the same as for H.. The orthonormality relations are

co co dx dyw* (x, y) w (x, y)

J J n i» i

i rV i? CO -co

and

l'~l ~? ~Z

2tt co

= \ d9 \ pdp u* (p, 9) u (p, 9) = 6 6.

co

2

(VI-44)

I

i(k -k )z

dz e

l Z

= 2 it 6(k - k )

•co (T), The generating function rff

*" *(1, 2) may be constructed as an

expansion of the functions (VI-43) and of the set e T

97
,("0

(1.2)

CO CO oo oo 00 dk "dk.

L L L L Z J 2n J 2t

n=0 n=0 i =0 i =0 q odd -00 -co

(VI-45)

XCa^'h'\'\' iZ'\'q)W

n

l ,il ,^

r

i

) ^

2

,i

z ,^'Z(r )e

If the equation. (VI-45) is substituted into equation (VT-20), one obtains

by suitable integrations and by the use of the orthonormality relations

(VI-40) and (VI-44) the expansion coefficient C : aP

C (n ,i ,k ;n ,i ,k ;q) =

aj3 1 1 12 2 2

S 5 6. , —6(k - k ) a0 nlf n

2

£ ^ 1 2

rtrh -q-E ,+cl t a, n, k

J (VI-46)

and therefore

#i> 2 >

co 00 CO

5 .frv ,( r ,> v* . k

( r

2

) e -fVV

vy y

r dk °apV vn,i,k^i' v

n,i,

Li Li Li J 2ir ^

n=0 i=0 q odd -co f —q-E

|_ t a, n,

(VI- 47) + %

3 — — The delta function 6 (r - r ) in the representation of the

1 Cm functions v (r) (VI- 43) is

n, x, k

co 00

6

3

(T

1 - T)

*** **• CO = y y f-v , (? > v* jr

)

-6 £ J 2tt n, i,k x l' n,I,k* 2'

n=0 1=0 -00

98
00 00 oo

= I I J§^>l'V^>2' e 2

)e

ik(z r

z 2

)

n=0 i=0 -oo (VI-48)

oo oo oo

= / / \ -r-w (x ,y )w* (x , y )e 6 L/ J 2i n, 4 1 1 n, i 2 2

ik^-z )

n=0 i=0 -oo

(t) The expression (Vl-47) for the function £j (1, 2) can be re- duced to a more useful form with the aid of the well-known identities r

-ittt

T Z/ p -, q odd

^ - fi

|

-i-

-iftT

, < t < T

1 +e

-tot

(VI- 49)

+i1 +e

i^T , -r < t <

(See reference (2) for a proof of the identities (VI- 49).) From a comparison of (VI-49) with (VI-47) we see that

,(T)

>a(3 (1,2)

- — £ (t - 1 ] ao oo oo -* — -n a,n, k 1 2' ., V V r<ik

V

n,i,k

(r

i )Vn,i,k ( r

2

)e

n=0 i=0 -oo n a, n, k

4f<>- 2)

oo oo oo

- +i 6 y y r — n' i>k L n ' i>k : 1 ap Z; Z/ J 2it

" + j_ T

1 + e

.— . , ,— . "ST* "n a, n, v<W K 1 2

(*,)v* . ,.( r

2

) e

n=0 i=0 -oo

1 + e

"ft

' d, n, k~]

(VI- 50)

99
where

a, n, k a, n, k

(VI-51)

Equations (VI-50) satisfy the boundary conditions (VI- 31) for any

time t , and any time t ; thus the restriction < |t - t | < t, on the

1

<h (t) times t and t may be dropped. Now, $ (1,2) may be written as

^(1,2) = ^(1,2)^ (1, 2) + ^(1.2)^(1, 2). (VI-52)

where r\ and n are the "step functions"

,2) =1

> t. n+

do, S

< t

and

i) (1.2) ={

°' *1 * *2

(VI- 53)

(t) With the replacement t — -ipfx in the function £j (1,-2) (VI-50, 52) we ^ a|3

obtain the Green's function G : o

G (1.2) = t,,(1,2)G (1,2) + -n (1,2)G^ (1,2)

oaf} '+ >ap - <a|3

with (VI-54)

OO CO CO

V^-i5J:Uf n,i' k

,

- l

-

r* \ a. r* \ ti a, n, k l 2

k 2

P£

n=0 i=0 -co 1 + e

a, n, k'

100
oo oo oo ,-* . — -h a, n, k 1 <£ G

<ap

M = ti8

afi yyr-W^p Lj Lj J

2tt r

p*§

n=0 i=0 -co

a, n, k'

1 + e

The Green's function (VI- 54) satisfies the subsidiary conditions

(VI- 30). There is another boundary, condition not specified by the

conditions (VI-30); we assume the appearance of an exponential con-

-e|t, -t- | vergence factor of the form e

l L

, with e > an "infinitesimal,

in the generating function and Green's function. This is done to

insure that the functions vanish at infinite time separations. Such an

artifice is not necessary in the case of spatial separations since this

is automatically satisfied. The replacement

G (l,2)-~e ' G (1,2), e-0 (VI-55)

oa|3 oa|3

is consistent with the equation of motion (VI-4) for G . In applications

o

of G , the limiting procedure will be the final operation.

The following identities involving the "step functions" r\ and

t) are useful in calculations involving products of G : o

r> (1,2)7! (1,2) =

(VI-56)

T^(1.2) + 7^(1,2) = 1 .

With the aid of the identities (6-1) in Mathematical Appendix VI, one obtains for the Green's function G (1, 2) (Cartesian co-ordinates):

o

imoj

r

., ,. -2h-(X lVX

2yi

) -

n ,. "'IV'Z 1 G

>aP

(1 ' 2) = 6 G

>aP

(1 ' 2)e

(VI-57)

101
lmu.

G M,2) = e

<af5

G

<aP

(1 ' 2)e

where we have used the definitions

V i 12 oo T /Tr . co ik(z -z_)-—

, (t -tj

f n „

mcJ B TyWfdk . * 2 * a ' n>k x *

G

>a P

(1

' 2)= ^ 6

a|3 TrfT 6 Z

"15

J 2^ 1 TF

n=0 -oo Tl + e a ' n' k ^

oo ik(z -z_)--£ v

(t.-tJ

y wrdk e

l 2 * a>n>k

a.0 2tA Z n! J

2tt p £

12 oo T /Tr . m«„ —- ^-^ L (V, _) ~ ,, ,, ,. - B 2 V n 12 (' aK O (l,2)=+i6 _^^-e

<af3

n=0 -oo 1 + e

a, n, k

mw

V =

12 2fi (x

1 -x

2

)2 + ( y r y

2

)21 (VI-58)

and where L (V) is the Laguerre polynomial of order n. With the n

definition

lmu —-— (x y -x y ) -e 1 1 -t I G (1,2) = e ^ L c L G (1,2) e C

, (VI-59)

oa(3 oa(3

one has the result

G «^' 2 ) s ^(1,2)G (1.2) + ti (1,2)G _(1,2). (VI-60)

oa(3 + >af3 - <a(3

It is apparent that the functions G , G , and G depend manifestly

upon the differences (r - r ) in the spatial co-ordinates and the time

difference, (t - t ).

102
CHAPTER VII

THE EQUILIBRIUM ENERGY DENSITY AND THE CHEMICAL

POTENTIAL OF THE ELECTRON GAS

The energy density of the gas is given by the expression

e (1) = <h (1)> (VII- 1)

and in the "self-consistent field approximation" (subscripts f)

t f 1 — 2

^WR 1 h (1) = ^d)U — tt(I) +—-^o- fl liMl) (VII- 2) vf a a£ 2m 2 z a(3J p

(c.f. (Ill- 8) and (VI- 13)). Thus for electrons in an applied field B

=^T

6

vf (1) =

<h

vf

(1) >

uhwB

(1) +

2 zap" <+J(2)+ p

(l»}

= -{

l __ 2

^WR

6 J_T(1)

2 + _B (r a|3 2m v

' 2 zap"

(VII- 3)

G fl (1,2) o£a 2—1

= -i Tr

{

G (1,2) o 2—1

103
The electron number density n , is found from the definition

ef

(III-7) and from (VI- 14) to be

n =

<P*rf>

= <+;<i) *aa» (VII-4)

iG (1,1 ) = -i Tr G (1, 1 ) , oaa + o +

which is analogous to the equations (V-56, 57).

Both e . and n . are equilibrium state values and are

vf ef

therefore independent of the co-ordinates r and t. The solution of

equation (VII- 3) for e . using the Green's function (VI-54) is found to

be

2 co co co

Vf

^v^ r^ /» ii E , v , (r )v* , (r V V V C dk a> n,k n, i,k V 1

; n, I, k

V 1 L L L J 2tt _

~ pi;

a=l n=0 1=0 -co 1 + e

mco.

2ti£

2 oo co Y y f** ».*.

L L j 2tt _ §e

a=l n=0 -co 1 + e

k

a, n, k

(VII-5)

where we have used the identity

co I

1=0

mo) v (r ) v* (r ) n,je,kV l' nj,kl l'

B

2irh

(VII- 6)

(see Mathematical Appendix VI). The number density n . is similarly

given by

104
2 oo oo co

n - y y y r^ia^Lj^l^Lii ef L L L J 2tt _ pe u_ a=l n=0 1=0 -co 1 + e

(VII- 7) men

27ria

CO CO L L J 2tt a=l n=0

pe

1 + e

a, n, k~

From (VI-51) one uses the replacement £ = E - £ in

a, n, k a, n, k

(VII-5) and (VII-7). The energy eigenvalues E , for the two

a, n, k

values of spin index are:

and

1. n, k

'2, n, k

*2 2

(n+l + g)* uB

+ V^2m

*

2v2

/ \* , -n k

(n - g)*u> + —

B 2m

(VII-8)

where 2g = u - 1 > 0, g<<l from (III- 1 ) (see Mathematical Appendix V).

We consider the solutions of the equations (VII-5.) and (VII-7)

first for a non-degenerate gas in the classical limit e << 1 :

CO CO mw

*vf 2T7h"B B£ e^ Z J 2tt n=0 -co

-BEK

E

l,n,k, „

t i e + E, . e

1 , n, k 2, n, k

BE

2, n, k'

B co oo mw

B ^'—hr^ u^B-irv —

- n^w--

e j as^cch-—. n*

n=0

r dk

J 2tt

„2 2 Bj k

2m

co

105
B

sinn

IJ ^

i=0 oo (VII- 9)

H-P^w„ _

r ^ -np-Ku « „ ^2.2

2 cosh B

i=0

J 2t7 -oo

-n k 2m

e 2m }

W

B

C PC

HP*i

2n

2 cosh B

P*»,

• v [ " B

sinn —

p-R<jj. -gi—jl + p^e csch-^J

and

+ ptfw. 1 - |x tanh

M-P^B

_

moo

oo oo n . ca b

e wyrdk

ef ~ 27* Zv J 2tt n=0-oo

j3E_ -PE.

1, n, k

r

2, n, k' e + e

*C 2 1 2

00 „^ °° P^ k

upfioo^.-^ -np-nco__

_^ e 2 cosh r 1 / e [I

n=0

3 M-P^^B

Su er= . v -r cosh

H B /m \2 2

pdk 2m

J2tt

G

-co

(VII- 10)

.PC

-n V

2irp / P^b

sinh

In the solution of (VII-9) and (VII- 10) the following definite integrals

and identities were used:

oo I

dx e

oo

= nTtT

106
I

2 -x 4lT dxx e = ——

(VII- 11)

and

ooI

xR = rb • for ^ x < 1 n=0

ooI

n=0

oo nx = x dx

V n d

Z

x = X ^

dx \ 1 - x n=0

(VII- 12)

(1-x)

, for «= x < 1

Thus, from (VII- 10) one obtains for e n

e K £

3

^UB

„ „2 v"7 n , sinh —-

2-nfi p \2 ef

m p-ncj B

Hf3nw B

cosh —

(VII-13)

and we see that a number of conditions would satisfy the restriction

OY of

e << 1. Equation (VII-13) may be used to eliminate the factor e from equation (VII-9):

|3nc n B

0hCO, evf 7jf{1 + f*"B e csch^+pfi^ 1 - utanh

u(3fico Bn

(VII- 14)

Equations (VII-13) and (VII- 14) represent the expressions for the

107
chemical potential and the energy density of electron gas in a magnetic field as calculated with Maxwell- Boltzmann statistics. In the

limit of no magnetic field (u>._ = 0)

M

n

ef

2

2Trh

3

2 .T 2 ^2

m

r\2 _ef_ /2Trn \

J " 2 UnkTy

and

vf —3 3 n „ = — n , kT

20 ef 2 ef

(VII- 15;

In the case of the fully degenerate gas ((3 — co) with applied

field B a precise formal solution of the equations (VII- 5) and (VII- 7] o

is possible, although quite cumbersome. For convenience, distribution functions W and W , r and V $ are defined such that

1 w 1 £t Vn' k> —iwr-TV) • -

1 + e Ink

, r,(n,k) = 1 - W (n, k)

(VII- 16)

W (nf k) = -— — , r_(n,k) = 1-W (n,k)

2

, P(E2nk" C) 2 2

1 + e

lim W (n, k)

P—co

lim W (n, k)

w

= <

,22

1, if (n+l + g)*iu_ + V1"- ^< °

r> ^m

.2 2

0, if (n+l+g)fiu_ + -^~ - C>

(VII- 17)

2 2

1, if (n- g)*u>_ + 5-^- - t< B 2m

= -< P—co ,22

0, if (n- g)<nu>B

+ -~p - C> .

108
Thus, W (n, k) is non-zero at T = only in the range

- N/S- (n+l+g)&w_ <

-nk

B N/2m

< + nTc- (n+l+g)fio>_ ,

While W (n, k) is non-zero only in the range (VH-18)

hk

- n/£- (n - g)hwB

< ^f^f < + n/C- (n - g)1iW]

as long as the quantities under the radical (square root sign) are nonnegative. We introduce a "cutoff" value of n, defined as follows:

N = "cutoff" value of n, such that

c (N

c

- g)*uB

< Z, < (N

c + 1 - g)iiuB

(VII- 19)

with (1 - 2g) > 0.

Thus, for (N - g)hco < C < (N + g)nu> the energy density is given

c B c B

by

,- .2

N

c"

2 + ^-(n+l + g)ha>.

(2iTh) as

n=0 o

(n+l + g)hoJB+ X

N

c W£-(n-g)n00 n

B

d\

n=0 o

(n-g)nu>B

+ \ 1}

x wB

(2m)- N -2

2 c

(VII- 20)

3 2

(2irfir n=0

+ 1 + g)hoo B«s/£- (n+1 +g)nCj i

N

+ ^ C+ 2(n-g)fic1 > ] n=0

109

•s/S-(n- g yfiw B |

,
and similarly (N + g)-nu_, < £ < (N + 1 - g)"nco_ leads to

C ri C Jb

oo B

(2m)- N -1

2 c

Vf

3(2*n)

2 {In=0

C+ 2(n+l+g)iia) 1

N

n/£- (n+l + g)hV

^ h + 2(n-g)nco

;| n=0

<vn-2i)

N/C- (n-g)no>n

The change of variable X. = i << was made in the equations V 2m

(VII-20, 21). We also calculate the number density n for the same

ef

conditions: For (N - g)noj ** t, < (N + g)fru) c B c B

fl r?m x2 N

c"

2

WC-(n+l+g)h(o

N

c W^-(n-g)nco_

b r v r v p

n

ef

= 2 ( > I dX > \ dX

(2TTh) n=0 o n=0 o }

or (VII- 22)

,2 V2 N

°B (2m)r1 v v

2 ^C-(n+l+g)h(oB

+ 2 ^^-(n-Cv (2*n)

2

n=0 n=0

•and similarly for (N + g)hco_ < C < (N + 1 - g)hoo c B c B

n

co B

(2m)

ef " 2~

(2TTft)

7 N -1

2 c N

| 2

^^-(n+l + g)hcoB

+ ^ N/C"(n-g)KwB |

. n=0 n=0

(VII-23)

The equations (VII- 22) and (VII-23) give n as a function of £,, or vice versa. The equations (VII-20) and (VII-21) give e as a function

110
of t, , or vice versa. Thus, in principle, one has t, as a function of n „, and thereby e , as an implicit function of n „. Both n , and e _ ef vf r ef ef vf

are readily evaluated for small N (large value of u> ) directly from

(VII- 22, 23); for large values of N (small co ) this is more difficult.

The behavior of the function t, versus ^ for a fixed density B

n , is rather interesting. The function £, is continuous for all values

ei

dC

of u> , but the slope B dwB

is discontinuous at each of the "boundary

points" {oj } such that t, = (N - gHiicj } , and at each of the L Bb be L B

J b

"interior connecting points" { w } where t, = (N + g)-h{oj } for all

L B 1 i c

L B i values of N except zero. The "boundary points" as defined form the

c upper limit for go in the range defined by N and the lower limit in

the range defined by N - 1, while the "interior connecting points" occur within the range defined by N . For arbitrary N > the

c c "boundary point" is given by the expression

r W } = —

2-n-n ef

3 B

J b m TT^l N

c c

sj N - u - n + y n/ N - n n=0 n=0

n2

(VII-24)

from (VII- 22). The slope at this point evaluated from below ({w } - ; B

J b

upper limit for the range defined by N ), is, again from (VII- 22)

dC

dco B

= (N

c

- g )h (VII- 25)

The slope from above ({go } +), evaluated using (VII-23) (here re- placement N —N - 1 must be made in the summations since this will

c c be the lower limit of the range defined by N - 1), is found to be

c

111
<H

doj B-+

N -1

c N -2

c N -2

c N

I Li VN - n Z^ n/N - u- n ^ c Z^ c

J

~ n=0 c n=0 c n=0 n=0

N -1

c N -2

c

Li */N - n Zy n/N - ll - n J

n=0 c n=0 c

(VII-26)

The "interior connection point" for arbitrary N is given by

c

Wi = TO

m

2irn

3 ef

N -1

c N

> (VII- 27)

„2

VN - n- 1 + > n/N +ll -n - 1 n=0 n=0

where we have used (VII-23). The slope at this point evaluated from

below ({cj }.-) is given by

doo,. = (N + g)n , c (VII- 2 8)

from (VII-23). From above ({«„}.+) the slope at this point as evalu- B i ated from (VII- 22) is

do; B.

112
N -2 N N -1 N

n=0 C n=0 C n=0 n=0

N -2 N

c c / Y . 1 + y .

l X (VII-29)

\ Lj n/N -n-1 L, N/N +|i-n-l J n=0 c n=0 c

The discontinuities in the slope of t, might be termed

"oscillations" in the function £, and the "spacings" between discontinuities (values of Aco ) might be called the "periods" of the "oscilB

lations. " The "periods" decrease with increasing N , becoming

c "vanishingly small" as N -* co, and the number of "oscillations"

become infinite since the set N = 0, 1 , 2, . . . co is infinite. Since

c for increasing N the quantities {co } and {co }. decrease monotonically, we see that the number of "oscillations" increases without

limit as co goes from a non-zero value to zero, or, as co increases, B B

the number of "oscillations" decreases until the field reaches the

point 1 1&/L 2 \ 3

"B

= m(27Tn

ef) '

after which the "oscillations" cease altogether. The function t, presumably converges to the value otherwise calculated for zero field

(co B

= 0) :

6 - Tlim

( N -g)*{u„k = Hm

(N + g)tf{co } w„—0 N -co c

Bl l B* b N ^co v c

Bl Be ^ B^i

c

1_ tA

2 / 2

v 2TTn m V ef

Itan (N

c

" g)

N -co N -2 N

c c c r V V j

2

3

>N/N-u-n+ >n/N - n

(VII- 30)

n=0 n=0

113
1 lim c -rm / Z

N 2Trn

N —oo N-2 N 2 m V ef

c c c —

\ 4N~rn Tl + / «s/N + u-n- 1 1 n=0 n=0

or

1 1

"eT°

"

This zero field value for £, is obtained from equation (1-95) of

reference (3), in which

G

oii (1 'V = W^V =iI3-

^

<C

(2tt)

m£

2 2 k dk i /2mO = l 2r

"I 3 2 1?

(2tt) 3(2tt) \*T

and (VII- 31)

n. =-iTrG (1,1) = -* /5»£

6f °

+

3( 2lr) 2 U2

Thus the behavior of the chemical potential in the ground state

of the system (p = oo) is marked by an "oscillatory" character superimposed upon an overall decrease in value as u> increases from zero. B

The function £ is also the "Fermi Energy" of the system in the

ground state.

114
In equations (VI- 8) and (VI- 10) the chemical potential t, is the

i both the expressions G

oaf

cases one generally has the result

same in both the expressions G (1.2) and G (1,2) ; in such

oaf} lap o

n * n r

. (VII- 32) e ef

'For purposes of calculation, however, we let

n = n _, (VII- 33) e ef

and the calculated value of t, will be an approximation. This procedure permits a reasonable solution for £ in any approximation of

G with a specified number density n for the gas.

115
CHAPTER VIII

INTERACTION OF TIME-DEPENDENT ELECTROMAGNETIC

FIELDS WITH THE ELECTRON GAS

In this chapter we consider the formal solution of the linear

wave equation (11-29). The "conductance tensor" (11-31) for the

electron gas is discussed in terms of its Fourier transform. Plaae

wave propagation in the direction of the applied magnetic field is con- sidered for a non-degenerate gas. Some discussion is devoted to the

case of complex frequencies. The quantities

6 yn

6AT

(2) m o

are calculated from the equations (V-50) and from the Green's function (VI-54). The quantities

-±2ilL. Mi) , and

6J/U

5UT

(2) 6AT

(2) 6UT

(2) o mo o may be calculated for the electron gas from

6J (1)

I 6AT

(2) m o

through the connecting relations (11-11, 15).

116
Calculation of the Quantities

6AT

(2) m o From the equations (V-50) and (VI- 5 4) one obtains the

following results:

6J (1) e

2n

6 (1,2)

5A>) + o

°°dk "dk °° °° jVJV I I

Q,Vk

1 in2'Vi'V Dw

|n

i' ki

; VVvV -co -oo n =0n=0

00 dk TMdk °° °° tl-ST-Jirl I xivVvVYV^vVvVvV

-co -oo n =0n =0

, 6J M

(1) + 6Aj(2)

+ o

6j + (D

6AT

(2) o

CO CO OO CO dk ZTdk

= \ —-\ —- > ) Q(n , k ;n ,k ;t ,t )D (n ,k ;n ,k ;7,,7)

J

2tt J Zir L L \ 1 2 2 1 Z

1 +- 1 1 2 2 1 z' -oo -co n =0 n =0

1 2

oo,, oo oo oo dk _ dk

+ 1T— Ct^ C^y 7

X(n.,k;n ,k.it ,t )F/n ,k ;n ik ;7.7,; 8x J 2ir J 2ir 6 Z_/ 1 12212+1 1221 2

" -oo -co n =0 n =0

1 2

117
5J M

(1) + 6AT

(2)

6j+

(l)

6AT

(2) z o

00 oo,. oo oo dk n dk

JVJVI I ^vWVVV^vWV^• V l -oo -oo n =0 n =0

6j M( "+

1 T 6A (2) z o

6j_(l)

6Aj(2)

+ o

"dk ?dk °° °° jVjV Z I ^VVWVV^'WVVVr ) 2' -oo -oo n =0 n =0

1 2

oo ,, oo oo oo dk „ dk

3x

2+ I-zT-J-srl I^i-N^Vi'V^i'S^S'7!' r ) 2

; -oo -oo n =0 n =0

1 2

5j M

(1)

-

6 A* (2) + o

6j (1) en

(VUI-1)

c

. T, mc

6 A (2) o

e 4

118
oo oo oo cx> dk _ dk

J-sri'-sr^ I

Q(n

i' krV k

2

;t i'VD-

(n

i' k

r

n

2

' k

2

;T

r

T

2

)

oo -oo n =0 n =0

1 2

°°dk °°dk °° °° + ^J^r$-dI ^

x^VvVvyF>i'VvVVr ) z' -oo -oo n =0 n =0

1 2

6AT

(2)

6j_(D

8AT

(2) z o

°° dk. °°dk °° °°

= \ —- \ —- / / Q(n , k ;n , k ;t , t )D (n , k ;n , k ;T ,T ) J

2tt J2tt L L 1 1 2 2 1 Z

1 -z 1 1 2 2 1 2 -oo -oo n =0 n =0

1 2

M + T 6A (2) z o

6j (1) z 6A^(2)

+ o

°° dk ?dk °° *

= Ct^ Ct^/ / Q(n ,.k :n .k ;t ,t )D (n k ;n k ;7 7 ) J 2ir J 2ir 6 Lj 1 12212 z+1 1221 2 -oo -oo n =0 n =0

1 2

oo ,. oo ,. oo oo

., dk dk

iszJ-*r$*r2 I^vVvVVV^vVvVV'V -oo -oo n =0 n =0

1 2

M z 6aJ(2) + o

119
6j (1) z 6AT

(2)

CO j, CO j, CO CO dk n dk v— Iirl-srZ I^WWYt )D (n ,k ;n ,k ;r ,7_) 2 z- 1 1221 2 -co -co n =0 n =0

CO,, CO,, CO oo dk _ dk

9x

2-

\ -=-> \ "=— / / X(n ,k ;n ,k ;t , t )F (n ,k ;n ,k ;T ,~t ) J

2tt J 2tt A Zy

x 1 1 2 2 1 2

7 zv 1 1 2 2 1 2

• oo -oo n =0 n =0

M z 6AT

(2)

8j (1) e n. —I— = '^rr 6 (1 ' 2)

6AT

(2) z o

oo ,, oo* , oo oo dk _ dk

IttJ'itZ I«n,' k

1 »

2 'V, i'yD

«»

,n

i' k

i :n2 'VvT2

| -oo -oo n =0 n =0

1 2

M Z T 6A (2) z o

with

Q(n

i' krVk

2

; VV

tl^l.2) r (n , k )W (n , k ) + r (n , k )W (n , k ) r l r i

l 2 2

7 2 i r 2 2 2 8iim

2

c <-

120
-il(]Ak*)(t -t )-i(n -nj«_(t -t )-e|t -t | 2m 12 12 12B12 '12' ._ n+ X e , e > 0, e— ,

D (n,k;n,k;r,r)

++

v i r 2 2 r r

oo oo

i=0 1=0

1 2

, 2mnto

L> B .*^V^>),^^V«^^fc f

i)

+ (n +l)v (T W* fir" )v (r )v# fir" ) i l) ^i.vv x n

i+1 'VY * WV 2 vVV x

2 lyl^ 1 nj.^.kj 2 ^-l.!^ 2 n^l.l^ 1

+ <s/(n+l)n v ,(r)v* (rjv ,(rjv* (r ) 1 2 lyi^ 1 n^l.^.k^ n

2

.i

2

.k

2 * n^'W

VVWWV

CD CO

¥ 'I

° ^Vv^Vv^vv^vv^

2 2 n

2

+l,i

2

.k

2

2 n

2

-l.i

2

.k

2

l lyl^l n^,^ 2

+ ^I Vv^Vv^vv^Vv^i

121
D (n , k ;n , k ;T ,T ) + z 1 12 2 1 2

CO CO

=1 / -h\T2mfiGJ

Z^ B

1=0 £=0

1 2

»Jn +1 k v (r)v* (r )v Cr)v* Cr) *l 2 Y"^! 1 Vx*! 2 YW VfV

+ Vn+Tk v , „

. (r)v* . . (r)v „ . (r)v* . , (r ) 1 1 ryl.^.k'l' n^^k^' n

2

,i2> k^2; n^.k^l'

2 2 n

l ,i

1 ,ki

1 ryl^ 2 lyl^ 2 x^-l.l^ 1

+ N/^"kv k P,) V* . k

(T

? )Vn J k

(T

2

)Vn-l* k

(T

l )_l 2 1 n

l ,il> ki

i lyl^ 2 n

2

,i

2

,k

2

2 ^-1,^,^ i

D (n , k ;n , k ;r , r ) -+ V 1 1 2 2 1 2'

CO CO

=1

4 = i=0

1 2

1 2 V'-V1!

1 "VW V^W V2'V

^Vv^vWv^v^

N/n (n +1) v , (r)v* , (r")v , (rjv* . (r)

* -if^^r ^Wltl

^ t ^V. ^

122
(VIII- 2)

D..

(n

i' k

i ;n2' k

2

;r

i' r

2

)

oo oo I

2mfiw

•I B

1=0 i=0

1 2

.*?^V^*^VvP^rV^

+ n

i Vr1 li k^l )Vn-l * k

( S

)V

n i k

(^? )VJ / k^?

l

) 1 n^l.ij.^ 1 ^-l.i^kj 2 n

2

,i

2

,k

2

2 ^.^ 1

2 VV*! * "l'V^ 2 n

2

+1 'V k

2

2 n

2

+1,i2' k

2

!

+ N/n (n +l)v . .{7 )* , . (7 )v . (7 )v* , . , (7 ) J 1« 2 n^.kj l' n-l.I ,k< 2> „

2

,<

2

.k

2

< z' a2+ l,l

2

.k^ l'J

D.z

ln

ik

l ta2- k

i'

XV'z]

OO CD

=1 L B

£ = £=0

1 2

1 2 n-U.k 1 n.i.k 2 n .ivk 2 n .i..k 1 1 11 111 222 222

+ VTkv ,. ,(r)v* i<r ,) v

, fc.ffl)v* (r ) 1 1 nf

l.l

1.k

1

1 nj.ij.kj 2 n

2 ,l2

,k

2

2 n^ 1^

^ 1

+ \/n~Tl k v (r )v* (r )v (r )v* (r ) 2 2 vVk< l' a,.!,.^ 2

IVn

2

.i

2

,k

2

( 2> n2+ l.i

2

,k< l'

-* n + 's/K~fIkv ,(r)v* (r )v iJrJv* i •**,) 2 1 n , I , k 1 n,.i,.k, n,'*, » k, 22 n^,,,i,,k' i^,^ 2 n^+l,i^,k^ 1

7 111^222 n+l,i .„

2 22

123
D (n , k ;n , k ;"r , T ) z + 1 12 2 1 2

oo oo -11*

i.= 1=0

1 2

HO, .^s^^VvJ?^

+ ^

Ik

1 V

»

1 .i1

. kl"l> V

»Vl..1

. k «W

2

. Vkf2>^2

. Vk<'

1

>

+ N/rTk v , (T )v* . . (7 )v , . , (7„ )v* . . (r*

)

+ «sfiT+Tkv k

< r

2

> V

n # k

(r

l )Vn i k

(T

l )Vn+W k

(T

2

} 1 2 n2*V 2

2 n2'V k

2 * VV 1

X

1+l, V 1 2

D (n ,k ;n , k ;7 ,7 ) z- 1 12 2 1 2

oo oo

i=0 1=0

1 2

+ N/n~k v . , (r )v* , , (7 )v , (7 )v# , (7 )

+ 's/iTTI k v m , (r )v* . . (r )v . , (r") v * . , (7 )

+ N/n~k v (7 )v* (7 W (7 W* (7 1 1 2 nj.ij.kj' l ,V

n

1-l,i

1 ,k1

< 2

,V

n

2

.i

2

.k< 2> %, i^'l'

D (n ,k ;n ,k ;r , r ) zz 1 1 2 2 1 2

oo oo -*WZZ;-^^*v^vvP

if i|0

124
and

X(Vk

i ;n2' k

2

; VV

4m c W

1

(n

1,k1 )ri (n

2

.k

2

)-W

2

(n

1 .k1 )r2

(n

2

.k

2

+ 1 (1.2) )U

v -S<k

lk2

,(t

lt2'-i(n1 -2>»B«tlt2>-e l tr

t2

l „ „+ X e , e > 0, e— , Vvvvvvv

oo oo

1=0 a=Q

1 2

^VMr^vV^V^'SfVP

+ ^/r^v . , (T )v* . , (T )v , (T )v* . , (r )

F- (n

i' k

i ;n

2' k

2

;7

l* 7

2

) (VIII- 3)

oo oo =1 1"^5

B

^/n~ V (r W* (T^v (r )v* (r ) ' vurV VrV VA2 VzV

+ <s/n +1 v (r W* (r )v (r )v* (X )

F

2

(n

i' krV k

2

i7i'V

oo oo

= *(k +k ) > Yv . .(r)v* . t (*l)v i v<^) 1 2. v* . v^O- Li Li n,,i . k 1' n. , i. , k, 2 n_,i_,k_ 2 n ,i_,k_ 1' 1=0 |=0 1' 1 1 1' 1 1 2' 2' 2 "2 2 2

125
The variational derivatives of j are given as before by (V-41), and

the associated variational derivatives of M are given by

6M (1)

s + T 6U (2) o 6M (1)

s + T, 6A, (2) + o

6 M (1)

s + T 6A (2) o 6M (1)

s + T,

6 A (2 z o 6M (1)

sT, 6U (2 o 6M (1)

sT 6A L 2) + o 6M (1)

sT

6 A 2) o

6M (1)

sT, , 5 A (2 z o

6M (1) s z T, 6U 2 o

CO ,, CO ,, CO CO dk „ dk

1

- -frS ~^§

^

I^-S'^rVVvW/VS co -co n = n=0

1 2

=

CO,. CO,. CO oo dk ^ dk

£a'm

f\ f\ -oo -co n=0 n=0

i. c*

= o (VIII- 4)

=

CO ,, CO ,, CO oo dk. ^ dk

8z j" ir

-co

j*-cosrl

n = n=0

I ^i^^WWi10

^

°°dk °°dk °° °°

= 2 *ytrJ it I I

^^VzVVWiinz

k

zV^

-co -co n=0 n=0

1 2

OO,, CO., CO CO dk „ dk

= -^JVJttI ^("..S^VVW^^ -co -co n = n=0

1 2

126
V

6M (1)

s z

00 dk °?dk °° °° = 7?jVjVZ I

xVr^VrVV^! 6 A (2 2c V*Vz>

+ 'o -oo -co n=0 n =

1 2

CO CO CO CO 2^2 „dk „dk

i2|j. & 3

c 2+ §-zrS-dl 1

Q^k

1;n2

k

2

; VW-YVzVi''r ) 2

; -co -oo n =0 n =

1 2

6M (1 • „dk_ „dk

s z rrfirj-srl Z

x(n

i' ki

; ^VrVv

3

(n

i' ki

6 A-

(2

* 2c

;n^v 'o -oo -oo n =0 n =

1 2

oo oo oo oo

. , 2^ 2 " dk n dk

12^ ft a 2 8x,

c 2- Iirj" -£1 I Q^VVzV^Wi'Vz7!' r ) 2' -oo -oo n=0 n=0

1 2

oo oo oo oo 6M (1) . ^dk. ^dk

s z

= -r§-zr$^rl l

x

<*t

kPWttFJ*ik?*l¥tri

6 A (2) c z o -oo -oo n=0 n =

with

Y(vWk

2

:t i'V

2 2 _ |a e ^i \_2. ~ _ 2 2 2m c {n'd. 2)r

2

<n

1 .kl )wi (n2

.k

2

) + n*<i. 2)*^.^) r^i^)}

X e , e > , e^O , Z(n

l' k

i ;n

2' k

2

;t l't 2

)

2 2 _ p. e x4i

! , 22 2m c

|r1 ^(l,2)r

i (n

i ,k

i )W2

(n

2

,k

2

) + T1 2 (l,2)W

1 (n

i ,ki )r

2

(ri 2,k

2 )|

X e H^-k2

2

)(t

1 -t2

)-i(n

1 -n2+ll )„B

(t

1 -t2

)-ci«

1 -«2

127

, c > 0, €—0 ,
V

1 (n

1 ,k

1 ;n2

,k

2

;r

i .r2

) (vni-5;

co oo s 1 1 V„t1 ,k I <r

i»

v

SI .*1 ,k<T2>V,r.k,<?

Z>

v

„*,.,k,<5?i'

„ „ 1 1 1 111 2 2 2 2 2 2

i=0 1=0

1 2

V

2

(n

1 ,k

1 ;n2

,k

2

;r

1 ,r2

)

CD CO s l 1

N/2mhw

j i f

o i=o

"s/nT V Z^n^i^k <*!> S^.k^ V^z'^Vz^1

' ^Vv^^vi^.^Vv^^v^.

WWVW

CO CO

£l>mhcoZ^ B

l f

1=0

N/n + 1 v „ , (r,)v* . , (r )v , , . (rjv* „ . (r,)

. 2 V'l'Y ! n ,i .kj* z' n +l.i^k « Z' „^i k » l' ^VV^1' V'Vi^ VW^ VV^.

The above expressions were derived using equations (V-54) and (VI-56).

It is seen from the equations (VIII- 1) through (VIII-5) that all

the quantities

6j„(l)

6AT

(2) m o

depend upon the time difference (t - t ), and therefore the current

density equation (11-25) is applicable here. From the connecting

relations (11-11, 15) and from (11-21) one sees that

128
6p(l)

6UT

(2)

6.(1) . 6j

i

(1 » 6AT

(2) m o

, and

6UT

(2)

also depend upon the time difference (t - t ). Thus the charge density

J- w fluctuation equation (11-25) is also applicable here. The same would be

true for any system with time-independent Hamiltonian because of the

structure of the field creation and annihilation operators (see equations

(VI-25)).

By making use of the identities (6-1) from Mathematical

Appendix VI in the equations (VIII- 1) through (VIII-5), one obtains the

resulting equations for the coefficients D. , F , and V : im m m

D

+» (Vk

l JB2'Wr2

)

.moj v 2 r- 12

= +i*(V I^WZES^ (-*) ±L^

(Xz+. Xi +

)

i(k

1-k

2

)(z

l -z

2

)

L (v )

(n

x + 1) d^; L

n 1+ l(Vl2)

12 1 -i -L (v ) n 12 dv

12 n

2

12 }

D (n , k ;n ,k ;r , r ) z+ 1 12 2 12

-v. mu_2 i— 12 B\ n/tt e

- i*(k

i + V^^b ("ar) ^T-^r (x2-- x

..»

e

i(ki-V|, i-2

»

L (v ) n_ 12 r X

r _2_ " r_£_ , \ (n

x + 1) [dv

12 nj+l^i; n. 12

d L (vj 1

dV

12 n

2

129
>mw 2 " V

12 i(k

i -k

2

)(z

i -z2

) VvVvVV*^ = 2mHi(-2^r ) ^rnrr e

3 L .lA,)

L dv

l2 ni+1 12

. —— L (v ) L dv

12 n

2

12

. + L (v ) L (v ) n +1 12 n 12

n^L (v ) L (v )\

2 n 12 n -1 12

X £0

,mco„ v 3 "12

WWV'l'V2m&,B (-2^-)

^-(X

2+

- X

l +

): e

X W1 3 L 4-1^

L dv

12 n1+ l 12 _ (n2

+l) [dv

12 \+liV12]

L dv

i2 n

i 12

. dv

l2

n

2

12

.

1 (n 1+ l)

Ln

(V

12 ) ^d2

2 >-dv.12

2

'V1 (V

12>]

' #*S^'feVvl}

D

z-

(n

i'

k

i ;n2' k2

;r

i' T

2

) =

— -v

'mo>_ N 2^r- 12

+ i*(k +k W2mfiu> (^r) , , 12 (x, -x,J B\ 2irfi 7 ni' xV 1+ 2+

X £

i(k -k )(z -z ) 1 l' 1 2

L (v ) n, 12 ,- *R* SW - L, (v

n

2 "'[is^']}

-•»-* 2 D (n , k :n , k ;r zz 1 12 2

,mcl)_,2 12 .-~

2 > -* «W(iar) ^ng^i^iz*

l <Wrt

130
D. +

(n

i' k

i

; V k

2

;r

i' T2

)

= 2mfc<o T> f -s-5- ) i (x. " x ) e BV 2-nfi ) n,!n! * 1- 2-

' 12

*{-<^ .s-W (n+D dv „ n + 1 12

ieVv

i2>i L K J

2 ' L 12 2

1

12 "1 "JLdv

12 n

2

12 (n

1+l)Hi

2

(v

i2)

57T)\(vi^

-d2

Ldv

i2 * ^d2

dv" 2

12

nM}}

D->l' k

l

J Vk

2

;r

l

,r

2

) (vin-6)

-V

.mw 2 12 i(k -k )(z -z ) = 2rrrho> B\ e 12 12 f - w ^ —r—? e Bll J n^n^

f^fe\^JfeVtt]+ \^M

n

l L

n

1- 1<"l 2

> L

n

2

<

1'12 »}

D (n , k ;n , k ;r , r ) -z 1 12 2 1 2

/mwr»N 2 .JT

'" V

12 i(k -k )(z -z )

' 12

L

n

(V

12 )

131
— -V

/Hiu 2 i— 12

L (v ) n„ 12 r {

+i

L dv

12 n 1+ l 12

_ + L (v ) n 12 .^'-.V

i(kv -k )(z -z ) 1 2

M

1 2

; e

^_^_^^ ,mu 2 >— 12

L (v ) n. 12 r *^

3

L + i (v

i 7

) L dv

i2 V1 12

- + L (v ) n

2

12 *Tz W

ifr-k )(. -2

)

/tnco 2

" V

12 i(k-kXz-z)

F (n k ;n k ;r ,r ) =tf(k+k)( —£) -rnL (v JL (v )e '*- z 1 12 2 12 i Z\^ 2im 7 n In 1 n 12 n, 12

= *(k1+k

2

)V

1 (n1 ,ki: n2

,k2: r

1 .r2

)

V

2

(n

1 ,ki ;n2

,k

2

;r

1 ,r2

)=F*(n

1 ,k2: n

2

.k

i ;r1 .r

2

) VVVVVVV

= ^''I'VVVVV

"where

moo,

V, „ = ^ (x - x ) (x - x ) 12 2n v 1+ 2+ 1- 2-'

m<o. 2h

(xr

x

2

) +(yr

y

2

)

From the equations (VIII- 1) through (VIII- 6), it is obvious that

the quantities

5J

I (1)

6AT

(2) m o

132
are manifestly dependent upon the differences (r - r ) in the spatial

co-ordinates as well as upon the time differences (t - t ). This was

J. d

to be expected since the system (electron gas) was taken to be

unbounded, and since no restrictions other than an axis of symmetry

(direction of applied magnetic field) are imposed upon the system, it should exhibit translational invariance which, of course, is the precise physical interpretation of a mathematical dependence upon the

difference in spatial co-ordinates. Therefore, if the Fourier transform of 6j (1) is defined by the expression

6J-CD = f£-**£ j(E; u)e

l ^

Wl

. (VIH-7)

J (2tt) *

then from equations (11-19,25, 31) one obtains the result

j,(k,co) = s (£, u)e

T

ar, w) , (VIII- 8)

£ £m m

where

T ic r

bj (i) &&-Z.)-i»(t -t ) l* E«) = -— \d(2) rp e

Z l Z l

. (VIH-9)

ilX1 W J A T /7\

The fact that

6A (2) m o

6AT

(2) m o

depends upon the difference (r - r ) in the spatial co-ordinates as well as upon the difference (t - t J of the time co-ordinates imposes

the following condition upon the Fourier transform of

6AT

(2) m o

133
.4 3.-*

q

i m

( V<V

1y <02

) = (2tt) 6 ^'"S^S'^^m^r "^' ( VIII- 10 >

and from a comparison of the connecting relations (11-11, 15) with

(VIII- 10) one finds that

fflyu ;k*,u ) = (2tt)

4

6 (l^-E^Cu -«a l^,^)

g (k U,£ )U ) = (2TT)

4

6

3

(k -IT )6(u> -a>,)g (ir.w.) (vm-ii) m i 1 Z Z iz l c m l l

Obviously then, all the functional derivatives

6 P (1) 6p(l)

m 9 rp 9 rp 6U (2) 5 A (2) 6U (2) o mo o

Sjjjd) 6^(1)

and

6AT

(2) m o

depend upon the differences of the spatial and temporal co-ordinates in

the case of the uniformly magnetized electron gas in thermal equilibrium. In this case the connecting relations (II- 15) reduce to the

following equations:

icof(k, w) + ik p (k,«) =

icogm

(k, w)+ ik

i q£m (k»co) = —f(k,w) - ik g (k, to) = m m

(VIII- 12)

-—P (k~, co) - ik qfl (k, &) = . eg m im

From equations (VIII-9) and (VIII- 10) one has the result

134

(VIII- 13)
Some Useful Relations for the "Conductance Tensor, " the Current Density, and the Wave Equation

The current density equation (11-25) represents three scalar

equations which can be expressed as £ 6jx

(l) = -cjd(2)

r 6J J1

) T^ 6E x^) +

L 6A (2)

6j x

(1) T

6j x

(1) -X—-6E^(2) + -^

F- 6E ;(2)

5A i (2) y 6A i

(2) y* 'o z 'o ^V1 = - C Jdej V1)

L 6AT

(2) x o

6J„W 6J„(!)

6e'(2)+-^— 6E*(2)+—*=- 6 E ;(2) X 6A i (2) y 6A i (2) Z J y o z o

(VIII- 14)

Wx %M = -c^d(2)

r 6J' (1) 6j (1) 6E (2) +

L T,„, x w r . T

6A (2) x 'o

6A (2)

y

v 'o

6j 2

(l) T Ey(2)+—^— 6E iz

(2) 6A (2) z 'o

in Cartesian co-ordinates, or they may be written in the form

^6j+ (l) = -c[ d (2)_. 6E X 6E (2) + (2) +

6A (2) 6A (2)

"V" 6Ez

(2)

6A i (2) Z

z o

8^-« = -cjd(2)

rSj_(l)

-6AT

(2) +

v 'o

T

6J (!) T6J (1) T " 6E^(2) +

~T

6E_(2)+

"T

6 E *(2)

6A (2) 6A^(2)

z o

(viii- is;

atT

6^)

6J„W 6jJ!) f r

Ja

Vj T

J z w T

= -c\d(2) -4—- 6E '(2)+-Z—-6E i (2) +

6JJ1)

L 6A (2) +

v 'o 6A A (2)

r..

6E

Z

(2 ) 6A (2) z

x 'o

135
where

E*(l) = E*(l) + iE*'(l)

E

T

(1) h E*(l)- ieT(1) . x. y

(VIII- 16)

The form given by (VIII-15) is the most useful for calculations since

the functional derivatives are in a form best suited to operations on

the basis functions (VI-43). Then, from the definitions (V-40), we

find that

6p(l)

6AT

(2) x 'o

"6

P (1)

( 6 P (1) -6A^(2)

o 6A^(2)

o

-J

6p(l) _ 6AT

(2)

y o

i |-6p(l) 6 P (1) Ua^(2)

o 6A^(2)o

-1

\0) 12

r

6j +

(l) 6j.(l)

-J 6UT

(2)„ UuT

(2) 6UT

(2)J

5j

x

(l)

! r6j +

(l) 6j +

(l T ~

~2~

6A i (2) x o

6j (1) Sj (1)

L 6AT

(2) 6AT

(2) 6AT(2) 6AT(2)

%M _ 6AT

(2)

y

v 'o

6j +

(l) 6j+

(l) 5j (1) 6j (1)

L 6AT

(2) 6AT

(2) 6AT(2) 6AT(2) +

v 'o -

v 'o +v 'o - '< %® _ 6AT

(2) z o

r

6j +

(1) -6AT

(2) z 'o

6j.(l)

6AT

(2) J

z

N 'o

(VIII- 17)

136
6j

y

d) i f

6j +

(l) 6j_(l) -

6U

T

(2) o

2 L 6UT

(2) o " 6UT

(2) - o

6j

y

(D i ~2~

-6j

+

(l) 6j +(l) 6j_(l) Sjjl)

i

6A

T

(2) x 'o -6A X (2) +

v 'o 6AT

(2) - o 6AT

(2) +

v 'o 6AT

(2)

- o

6j

y

d) 1 f 6j+

(l) 6j +(l) Bj_(I) 6j_(l)

1

6AT

(2)

y o

2

- 6AT

(2) + 'o 6AT

(2) - o T ' 6A i (2) 6AT

(2) - o

6j

y

d)

i r 6j +W 6j_(l) -

6AT

(2) z o

2

- 6AT

(2) z o 6AT

(2) - z o

6j z d) - 6j z

(l) 6j z d) -I 6AT

(2) x 'o T

6AT

(2) - - o

6J Z W

i

- «.« 6Jz (D n

• 6A

y

(2 >o

- *A>) 6AT

(2) - - o

From (VIII- 13) and (VIII- 17) we have

T — 1 s (k, co) = •? xx v 2

T" .. "p ». p .. p .. "\ s (k, co) + s, (k>u) + s (k, o)) + s (k, co) ++ +- -+ —

8

xy

(k, U) = - T— T -* T— T —

s (k, co) - s (k, co) + s_ +

(k,co) - s__(k, co)

137
T— ifT— T — T — T — ~! s (k, 0)) = -—i s (k, u) + s (k, w) - a (k, co) - s (k, w)

yx 2 [_ ++ + - - +

T — s (k, u) = yy

1 r t — t —

2

s ++

( k'^) " s +

_(k,(o)

T — T — ' s (k, o>) + s__(k,(o)

T -*

s (k, co) = xz ITT— T — - ! s (k, to) + s (k, to) 2 |_ +z -z

(VIII- 18)

T — s (k, to) yz

T — T — s (k, to) - s (k, to) +z -z

T — s (k, to) = zx T -»• T — 1

s , (k, w) + s (k, w) z+ zT -*

s (k, to) = zy

T —

T — s , (k, to) - s (k, to) z+ zFor convenience, we define the Fourier transform of the

-*A

applied current density Aj (1) by

ArA

a, -y

3-~ i k • r , - itot

d kdto— t»/t* . I (k, to)e

(2ir)

(vm-19)

From equations (LI- 12), (VIII-9), (VIII-19), and the wave equation

(11-29), we obtain the Fourier transform of the wave equation

138
k -k —5—1 —r-s (k,<j -kk-i——s (k,u$ -kk-i——s (k.cc) x 2 2 xx xy 2 xy xz 2xz

c c

7 c c

. 4ttu) T -— -r—2 ,2 u . 4ttco T -* . . , 4ttw a-* . kk-i ——s (k,c4 k-k--=--i —=-s (k,u) -kk-i ——s (k,u)

y x 2. yx y

£

2

c 2 yy

x "• y z Z yz

-kk-i —r-s (k,c4 -kk-i —r-s (k,u) k-k--^-i —=-s (k,a} z x 2 zx z y 2 zy z 2 2 zz

c c * c c

e (k.cj) x

T —

e (k.co)

T — e (k.co) z

. 4-itco 1 —rI (3T,C>)

I (k, w) y

I (k, u) z

(vin-20)

It is convenient to use the following definitions:

k +ik

x y

k -ik

x y

I +il

x y

I -il x y

(VIII-21)

+

T

T . T

e + 1 e x y

T . T

e - 1 e x y

139
Then the wave equation (VIH-20) may be written in the form

kk o a 2

+ -2 . 4ttoj T =» . u

c c

+ . 4ttco T=- . 4tto) T /T* ,

c c

- . 4ttw T /T* . 2 2 -+

v ^

c

kk , z - . 4ttco T -*

c

k k -, „ 2

- +^,2 . 4ttw T -* a) . 4tto> T ^ , ~5~ +V1 Ts (k

2 z c -- '^-~Tc -kk-l—

-z 2

-s

-z0^

k k

z + . 4irw T -* __.1 _.s

z _ (k^

c kk-l-—-8 k,<4-^

+ - 2 zz 2

c c

(VIH-22)

4ttco

From equations (VIII-20, 22) it is apparent that for plane wave propagation not in the direction of the applied magnetic field (k £ or k ^ or both), the "plasma oscillations" of the gas are generallycoupled to the "transverse" electromagnetic fields.

Some Calculations of the "Conductance Tensor 1

T — The quantities s (k, u>) (VIII-9) are difficult to calculate in

j?m

general, but there are special cases of interest which are relatively

easy. One of these is for plane wave propagation in the direction of

140
the applied magnetic field (z-axis). In this case, one need evaluate

T — s (k, go) for non-zero valu

Ixn identities (6-2) one obtains

s (k, oj) for non-zero values of k only. With the aid of the

T

s , (k , oj) +- z T

s (k , u>) -+ z

T

s, (k , u) +z z T

s (k ,w)

z+ z (VIII-23)

T

8 (k ,0)) -z z T

s (k , w) z- z = ,

while

2 _ en

T n \ e -iU8 M (K ,U) = ++ z m

e

2

ojl " ^ r r (n,k + X)W,(n+l, \) + r (n, k + X) W,(n+1, \) 'B 4iAi

-co n=0

(n+1)

-n 2

u + ie + w - —- (k + 2k X) B 2m z z

W, (n,k +\)r (n+l,\) + Wfn.k + X)r (n+l,X)

1 z 1 2 z Z oj-ie + oj,, - -z— (k + 2k X)

B 2m z z

2 2 CO CO

\l e «_ , n

^ r T (n, k + X) W (n, X)

B, 2 { ,. \ 2 z 1 k

2 z

8tt m

r d ,v r ^ • . ;

i'

•

J L« + ie + iw -—(k + -co n=0 u«_- -— i* r 2k X) n B 2m z z W (n,k +X)r(n, X)

2 z 1 w - i£ + ^B"2^ (kf

+2k

z V e—

141
2

rp en

-iojs (k , w) = z m

(VIII -24)

2 2 cd oo

e cj r (n, k + X) W (n-1, X) + T (n, k + X) W„ (n-1, X)

2 z 2

tt -ft ^ „ L oo+ie-co -t:— -oo n=0

(k + 2k X)

B 2m z z

Wfn.k +X)ri (n-1, X) + W_(n, k +X)r (n-1, X)

1 z 1 2 z 2

-ft 2 w-ie-w --— (k + 2k X)

B 2m z z

2 2

(i e u

oo oo ^

k

2

2. z

8-rr m KtSr, (n,k + X)W (n, X) z 2

-oo n=0

-ft 2

uu^-. —(k + 2k X) r B 2m z z

2

T e n T e

>icos (k , co) = zz z m

Wfn.k + X)r (n, X)

1 Z

£

-ft 2

oo-ie-uco --— (k + 2k X) J 1 B 2m z z e-~0

2 oo

e co oo —* fdX(k+2X)2

y f-i

l6Tr

2

m J Z ^L -oo n=0

rn (n, k + X) W (n, X) + r (n, k + X) W(n, X) z 1 2 z 2

<o+i£--^-(k

2 + 2k X) 2m z z

W, (n, k + X) r (n, X) + Wo

(n, k + X) r (n, X)

1 z 1 2 z 2

-ft 2

co-ie- =— (k + 2k X) 2m z z

In this case (k = k =0) the "symmetry" of the "conductance tensor"

t —* x y

s. (k, cd) is such that it has rotational invariance for rotations about

Sxn

the z-axis (direction of the applied magnetic field). From equations

(VIII- 18, 23) one obtains the results

142
T T T T

s (k , to) = s (k ,co) = s (k , to) = s (k , co) = 0, xz z zx z yz z zy z

T T 1 s (k , to) = s (k , to) = —

xx z yy z 2

T T

s (k , co) + s (k , co) ++ z -- z

(VIII -2 5)

T

s (k , co) xy z T „ , i s (k ,co) = -r yx z 2

T T

s (k , co) - s (k , co) ++ z -- z

As an aid to the evaluation of equations (VIII- 24), the well

known identity

lim 1 _ /l ,

, r- = P( - )? itt 6(x)

e _0 x± le \ x

(VIII- 26)

may be used (see reference (1), equation (3. 31), reference (46), and

Mathematical Appendix VII, Part 1).

For a non-degenerate gas (e P <<1) one obtains the following

expressions with the aid of the expressions (VII- 12), and equations

(VIH-24):

2

T

G n

-icos (k ,co) £

++ z m

1 phco

1 oo ^B 8hV

A J^ csch 112 fd\

2Trm|3y 2

2m

-oo

-ft 2

L co + ie + co - -— (k + 2k \) B 2m z z

pnco2- £^(k +

- (k X)

2 2m z e

-H _ 2

co - i £ + co_- -r—(k + 2k X) . B 2m z z

143
+ ,^ll^Lch^fdX

Zirmfi J 4m

2 2m

-oo ^^^V^^^V1

2

m e n

-icos (k , co) = -- z m {

2 2m z * n 2 r B 2m z + 2k \)M JJ Z e-0

+

1 \

2 ^]

2Ttmft

(3frco oo csch B fdX

2 2m

(VIII -27)

2 ) — I -6 2 TM Lu + i£-un--—(k + 2k \) -co B 2m z z

- B - g_ +X)2

2 2m z

e £ 2" co - i e - co_,- -— (k + 2k X) J

B 2m z z

- 2 2 2

1 \

2 ^ ** k

z 2Trmft

fif3nco 00

4m sech B

-00

^"b _ |SK

2

\

2

2m

^B"ll (kz +2kz

X)

^^R ft* 22

2 2m z

e w " ie " lxwB"^r (kz +2k

z

X) -€—0

2

T en

-itos (k ,00) S

zz z m {

144
1 2Trm{3

oo S- Cd\(k +2\y 4m J z -oo

2 r e

pn X

2m

L +ie- -^-(k

2 + 2k X) 2m z z

2m z

"^ „ 2 „, u)-i£- T—(k + 2k X) 2m z z lb

In the case of the fully degenerate gas (p—oo) the calculations

are more difficult, since equations (VIII-24) must be evaluated

piecemeal for each value of "cutoff" number N . For k = 0, however, r c the procedure is straightforward and the following results are obtained with the aid of the equations (VII-12, 22, 23) ("cutoff" number

N unrestricted):

c

-ioos (0,oo) =

-ius (0,oo) =

2

e n

i m —; i^cj^, 6(oo + oo_)

oo + oo_ xi rS

2

e n r- e m

B

i^oo^ 6(oo-oo_)

(ji - oj_ a a

(VIII- 28)

-icos (0, oo) = zz

2

e nm

Another special case of interest in the evaluation of the

T — quantities s (k, w) is for "weak spatial dispersion." By series ex- T

Jam pansion s (k~, go) may be evaluated up to a given order in k. If one £m

is concerned only with plane waves of small propagation constant

(|k | ^ 0), or if one is concerned with superpositions (groups) of such

plane waves, then the series expansion will be a good approximation.

145
The identities (5-50), (6-2), and (6-19) from the Mathematical

T

Appendices can be used for calculations of s (k, cj) up to the second

order in k.

Complex Frequencies

For fields which vary harmonically in time with exponential

damping (complex frequencies), one has

—T — —>T — -iftt 6E (r,t) = 6£ (r) e for t >

where (VIII- 29)

Q = co

1 - ico", c*>" > (<o

! and o>" both real).

We are dealing with the linearized expressions for p, j , and the

wave equation here, therefore the use of complex quantities for the —

—

electric field is quite proper. The real parts of 6 E , p, j , and the

wave equation so calculated are the physically observable parts.

Since the current density generally depends upon the entire temporal

history of the fields, it is convenient to make (or postulate) the physically reasonable ansatz

-i£2#t —T — —T — 1 6E (r,t) = 5£ (r) e for t <

where (VIII- 30)

£2* = co* + iu", <o" > (o' and to" both real).

This choice is made so the current density will have the same form of

time dependence as the electric field. There may be other choices

giving the same result, but nevertheless this choice is one suitable for

examination. The complex frequency Q* ("growing fields" for t < 0),

146
T*A -*• the fields, and the applied current density Aj (r, t) are chosen so as

to mutually satisfy the wave equation (11-29) for times t < 0, and

equations (VIII-29, 30) guarantee the continuity of 6 E at t = 0. The

frequency Q is so chosen as to satisfy the wave equation (11-29) for

T*A times t > with Aj = 0. The Fourier transform of the function b£ (r) in equations

(VIII-29, 30) may be defined by

„-*T,—, fdk -*T-* ik-r

6£ (r) = \ £ (k) e J (2ir)

(VIII- 31)

—T — —T —

Then the Fourier transform e (k,u) of 6E (r, t) is given from

equations (H-12), and (VIII-29, 30) by

—T^»

e e») r

T

or> {n^jT) - T(^ir)} (VIII- 32)

Thus, one obtains in this case the result

io)j (k, go) = T •— T —

£m m

,,

* , - .,* n J\, (VIII- 33)

i(co-fi? .(u-n*) i(w-n)o

J

or %»& II

3__

, 3— d kdco d r2 —

r

(2*)

4

X

i(w-n*) i(co-n)

1 1 T ;r~ 8im (k,w)| s/V )e m Z

(VIII- 34)

ik*. (7-7 )-i«t

147
One obtains the same result directly from the current density equation (VIII-7) and from equations (VIII-29, 30).

The integrations over the frequency a> in equation (VIII- 34)

may be performed by use of the Cauchy Integral Theorem (see

Mathematical Appendix VII, Part 2). The results are given by

=< 8t^/1)=

nn> r, ,V m m AT- (7 -7 )-ift*t

JJ 2

(2Tr)

3 1 lm 1- m 2

,t <

JP

3_,

L r y(-in) s (C.n) 6jp- (r* )e

2 . .-5 fm + m 2

(2tt)

fl£ (tyr" J-iQ^

(VIII- 3 5)

,t

x > 0,

where

CO

in* s Rntf

1 im 1

T -* m s (k, n)

jfcm +

T — iws /) __(k,w)^ -i(u)-n*)t e

l 1

, t <

dcu f jm |- 2it | i(w-«*) J

-00

CO

(VIII -3 6)

l(o>- fi)

, t

f> o

-co

For plane wave fields with frequency Q(t > 0) one has

c^rT— -*T ik-r-i^t 0E (r f t) = jf e

•j^.t) = s^or.n)

+ 6ET (r,t).

(VIII- 37)

(VIII- 38)

148
and the wave equation is given by the expression

2 _ "Y lT2 -kk -i^sj gov - im l m 2 ,£m '+ 6,im %2 c c £ = 0. (Vin-39) m

The relations formed from the determinant of the coefficients of £

in (VHI-39),

m

det

c c

, (VIII-40)

give rise to the dispersion relations for propagation.

149
CHAPTER IX

"DIELECTRIC SCREENING" BY THE ELECTRON GAS

The effective ("screened") potential of a static test charge

placed in an electron gas has been considered elsewhere using the

Green's function technique for an isotropic unmagnetized gas (see

reference (3)) and for the non-degenerate magnetized gas (see reference (16)). The essential features of the technique are briefly discussed here.

From the equations (1-9) and (1-12) the total potential is given

by

T A C 3-» p(V 1 c"~' U

X (1). = UA

(1) + \ d r V • GX- l > J 2 r 12

If time retardation is neglected in equation (IX- 1), the first variation

T A

of U (1) with respect to U (3) is given by

6UA

(3)

J 6UA

(3) o o

where (IX- 2)

v(1.2) = ~-6(t - t ) . r

i2

l Z

150
By the "chain rule" for differentiation

-&d3- - f«w 8p

T

(2)

"ft

4*

, ffx-3,

6U (3)

J 6U (4) 6U (3) o o o —

T

where the terms arising from variations with respect to A are

omitted because they are presumably negligible in the "static" limit.

Thus, to this order of approximation

6U (3 >o (IX-4)

6p(2)

= 6

4

(1,3) + ffd(2)d(4)v(l,2) u

ty w Ki

JJ 5U (4) o

From the equations (V-50)

-AdlL = .^L TtC {lt2) G (2>1) , 6UT

(2)

* 1 o 1 o

o

and from the form (VI- 54) for G,(l,2) loosG(l,2) "X-5)

6 P

(1) =~

e

C

' 2 G

o(l,2)Go(2,l),e^0

+

. 6U (2) o

Now' 6p(l)

6UT

(2)o

depends upon the differences (^."^VS" V

o£ aU c °-°rdinate8 » and

151
its Fourier transform is given by

f(k ft0 ) = fd(2)

6

P (1) e

l 2 1 2

(IX-6)

J 6UKIT f?\ (2) o

when we have used (11-13) and (VIII- 11). The function v(l, 2) can also

be expressed by Fourier expansion as

For K(l, 3), we write

K(l,3) =\ —__ofl(k, w) e (IX-8)

J (2TT)

Now, we may solve the integral equation (IX- 4) by Fourier expansion

to obtain the result

<#(£«) = 1 + -^ ffe «)&(£ <o),

which when solved for K(k, co) yields (IX-9)

c/ftk.to) = —T—

—*£, ——»• k - 4irf(k, co)

152
Thus

K(l,2)

k

2

(2tt)4

^lT2-4TrT(k,co)-'

{£• (T-T)-iw(t-t ) ]e 1 ^ • (IX-10)

Then, for the first variation of the effective potential, one has

6UT

(1) = fd( 6U (1) 6UA(2)

J 6U (2)

(IX-11)

= 1

d(2) K(l,2) 6U (2)

—•A

if one ignores the contribution from 6 A ' (valid in the "static" limit).

We take for the potential 6U (2) the Coulomb potential

6UA

(2) =<

0, t-> < to

2

(IX- 12) —*-, t > t r 20

2

°

where r^ = jr.,-^ A.

The potential 6 U (2) may be expressed in

"closed" form with the aid of the integral representation for the step

function (see reference (1), equation (3. 30)):

co

ti +

(1.2)

tl (1.2)

.Pdw e_ Wtf

t )

J 2 it (a) + ie) -co

CO

- if^S i

-Wtf

t 2

)

J 2tt (w- ie)

-CO

1. t,>t

2

1. t <t

(IX- 13)

153
where e is an "infinitesimal" (e > 0). Thus

oo -iw (t -t ) A q fdu e 6U (2) = i-3- \r- -T-

20

-oo J

2tt (w + ie)

oo^ ik ^r 2- r ) - MVt ) (IX- 14)

fd kdu e

= 1 4-n-q \ 4 —2

(2it) k (o> + ie) -co

By substituting the expression (IX-14) into equation (IX-11), one

obtains

ik- (T-7 )-io>(t ,-t

)

6UT

(1) = i4.q p^^^>e — — (2ir)

4

kT

2

(oj + U)

.„3-, • k -«rf

r

oH«*iV

— i

(IX- 15)

4irq \

•

(2tt) (o> +ie)[ k*

2

- 4u f (lT, co) ]

Since we are concerned only with the static potential, we consider

T 6U (1) in the limit (t - t )-»co and make the mild assumption that all

singularities in the lower half to-plane arising from the expression

-iw(t,-tj Xkco) e

vanish ("damp out") in this limit. One can equate the integral

co -iu(t -t ) / j_\ r

-co

154
to a contour integral with the contour closed by a semicircle of

"infinite" radius in the lower half co-plane (see Mathematical Appendix VII). Thus

3- *' {r

i r J

iim 5ut

(d - r^ 4nqe

fW"*00 J (2^ [k 2

- 4irf £ 0)]

(IX- 16)

The solution of equation (IX- 16) from (IX- 6) for the magnetized electron gas was given by Bonch-Bruevich and Mironov (see

reference (16)) for the non-degenerate gas. What they obtained is

essentially a modification of the Debye-Huckel "screened" potential

for a "Coulombic" charge, with the equipotential surfaces "warped"

about an axis of symmetry coinciding with the direction of the applied

magnetic field.

Equation (IX-16) may also be obtained from the "source"

equation (11-30) in a more rigorous fashion. Equation (11-30) is related to the theory of plasma oscillations. Consider first the

relations

6AT

(2) m o From equations (VIII-1) through (VIII-6) one can see by inspection

(the correlations all depend upon the differences of the co-ordinates)

that these relations are symmetrical with respect to the interchange

of co-ordinates and Cartesian indices:

S(1)

6AT

(2) m o saJ(i,o

(IX- 17)

155
The symmetry condition (IX- 17) is reflected in the Fourier transforms by

*kn*'** = W'k»"w) ' (IX- 18)

Now, one must remember that q. is dependent upon the applied

magnetic field. If one designates

cj = 1 Q) B z B

(IX- 19)

then the tensor "a. may be expressed most generally in the following £m

form (for Cartesian co-ordinates only):

V^ = 5

Jem q

i (k

"2

' w2) +

(IX-20)

3

-»-2 2 V + iuq.,(£" ,<o ) / e. (w_) . 3 l_, imn B n

n=l

The quantity e is given by

j2mn

£mn

1 if JL, m, and n are a "cyclic"

permutation of positive integers

-1 if I, m, n are an "anticyclic"

permutation of positive integers

otherwise .

(IX-21)

Equation (IX-20) satisfies both the condition (IX- 18) and the condition

for rotational invariance in form for rotations about the direction of

156
the magnetic field. The combinations 6. , k k , and £m I m

pimn^n

n=l

exhaust the possibilities satisfying all the required conditions for the

tensor q. . £m

The Fourier transform of the expression

6p(D

6AT

(2) m o

is given from (11-13) and (VHI-11) by

.. #,

,

~ -,3-*. ik-(r-r )-ico(t-t ) Sp(D f d **<» = nT i 12' 1 2' ,TV „» S = \ t— g (k, w) e . (IX-22)

6AT

(2) J (2tt)

4 m

m o

The Fourier transform of the "source" equation (11-30) is given from

(11-25) and from the definition

a ^jV", a ik»r -iwt,

Ap (1) = \ J- p (k fW)e (IX-23)

J (2ir) as T -» _ _» T - —A — uk e (k, M) = -4ircg (k, u) e (k, Q) -i 4ttu> p

(k, u). (IX-24) m m m m r

With the aid of the relations (VIII- 12), formula (IX- 20), and the

definitions

157
• „(k.*) - —p|

- j^i

^"e (IT, w)

(IX-25) —T —

e , (k, cj) =

'-*T -*

e (k, oj) I*

expressions (IX-24) may be reduced to the form

—» T —» = —

w | k* | e

.J (k, o> ) = -4ircg (k, w)

• T - t k

m

ei^1 . A -A=r , e. (k, oj )+ -;- -i4ttwP (k, oj) lm |k |

I

(IX-26)

a = tV x T l 4 ^f(k,oj) T . -A^ . = -4Trcg (k, w)e , + — e„ - i4irup (k,oj) m ±m r£* I II

_ T _ T _ 4* k [p (k,u)-i-gm

(k, u)e Jm(k, )]

i |lT | e (k, co) = ik • e (k, oo) = r —

11 [k -4Tr7(k, w)]

—2 —A -* c = -* T —•• 4irk [p (k, w)-i-7- k

i q

jem

(k.co)elrn

(k,oo)]

[ k* 2

- 4tt f (k, to)]

(IX-27)

4tt1T

2

[ p A

(k, to) + - q,(k

2

, co 2 )^- (£xTj (£, «))]

CO J -D -L

[ir

2

- 4TrT(k; („)i

—»2 —A -* -*2 2 — -*T -* 4wk [p (k, w) + q

3

(k , w )oo B

' b (k, w)]

[ k

2

- 4*7(k*. w)]

158
In terms of the potentials one obtains from equations (IX-24, 27) and

from the Lorentz condition (1-3) the results:

ilT-T (k, w) = ik"-[-iku (k, <o) + —T (k*. w)]

= fk"

2 -^ uT

(k, w) c

2

. —2 —A -» -*2 2 — —- -»T -» 4wk [ p

(k, w)+iq

3

(k , w ) u -(kXa (k.co))]

[k"

2

- 4TTf(k, W)]

and

iir-T

T

(k,a)) = (k"

2 --^ uT

(k )W) c / (IX-28)

= -i—— g (k*. co)[-ik u (k, w) + — a (T, u)] +4tt P

(k, u)

cj m m cm

=— T— 4tt = — T— —A — = 4Trf(k, u) u (k, w) + — kq (k, w)a (k, co)+4tt p

(k, co)

to i im m r

or

(k

2

- ~ - 4irffc u)) uT

(k, u) = 4tt[ p A

(k, u ) + -i ij (k*, w) a T

(£ u)] ^ go £m m c

-A —

k

i = — T -*

4tt[ p (k, to) + — q.(k, to) a (k.oj)] u (k, co) =

? 2

[ir -~- 4ufk co)] c

159
The charge density of the test charge is

i °- V ° Ap--(l) ~j

t ;>

(IX-29)

1

where q is the charge. The Fourier transform of A p (1) is given

by

p A

(k, W) = ~*

, z^0

+ (IX-30)

i(o) + 1 e)

By combining (IX-30) with (IX-27) or (IX-28), one obtains more accurate expressions for equations (lX-15, 16). Equations (IX-27, 28) are

the field relations for "plasma oscillations, " and it is evident from

these that transverse oscillations may be coupled to the longitudinal

oscillations.

160
CHAPTER X

SPIN MAGNETIZATION OF THE ELECTRON GAS

In Chapter III an operator M is identified as the "spin magnetization" operator for the electron gas. The "magnetic polarization"

or "magnetization" given by the expectation value M of M is

s s expressed by equation (V-22) in the form

M (1) = ^Tr^Gfl.l). (X-l)

s 2mc 1 +

In the present chapter we derive the equation of motion for

M , that is, we obtain an expression for

s 8M

s at

Following this discussion, we calculate the value of M (1) for the

8

initial equilibrium state of the system.

From (X-l), one obtains the equation

8M (1)

s = ue 2mc

7 4 8t

i "cU3 1

= ue 2mc Tr ~v

G

1Pa

(1>2) + i*XV(1 ' 2)}

: 2 2—1

+ (X-2)

Tr 7jitf -£- G^l, 2) + i^ -i- GjU, 2)j

Z -\

161
By substitution of equations (V-31, 33) into (X-2), we obtain the

expression

9 M (1)

s _

(j.e 3t " 2mc Tr 7(J- f*7

T

(l)

2

- 7

T

*(2)

2

1 G. (1, 2) [2m J 1 ^BT

(l). 2mc 7 a (1.2) - g^1.2)711

J 2—1.

(X-3)

• 3 n*%

2mc I^h^l'S^'S^^l^l'Y^] . . !^'1

! 17!'^ '

From the identities (j, k, 8. represent Cartesian co-ordinates)

and

ar - a

'-k = "^v^- = ior

# U» k, i in cyclic order)

V

2

= I

J

for the Pauli spin matrices, we obtain the relations

(X-4)

v[bT(i>-v •j^TM*1 [v*""m1

and (X-5)

°P

B

T -

(1 >-v] "

6 xP

B

'T(1,+i

[

7x6

xffT(I)

l

Thus, because of relations (X-5), equation (X-3) may be rewritten in

the following form:

162
8 M (1)

8 9t.

2mc {t^-'W}* 1 2mc —

T

B (1)

+ "^— Tr 2mc 4-\ 2m \

1*T 2 -*T* 2"!

ir (1) -ir (2f ^(1,2)1

2—1.

2mc Jru

6(t

3

_t

l+) + 6(V

t l"

) G (1, 3)G,(3, 1 ) 1 1 ++

(X-6)

^t^— Tr-r—-I tt (1) + tt (2) • 2mc s ''2mc""2ml —T — irll) 7

T

tiJG

1(i.2i

2—1.

. 3

^^^^K-^^^-vO ^ 13 L

1 ' 3 ^^3 ' 1^

Equation (X-6) takes a familiar form in the first term on the right

hand side of the equation. The second term on the right arises from

the motions of the electrons; this expression is not so readily reduced

as was the first term on the right, although its significance can be

shown quite easily. Consider the definition

r e e e \

where

Pe ») = -e (+'(1)4- (1)> = +ieTrG(l,l> (X-7) 1 a a ' 1 +

163
and

T(l) s^-Tr e 'v 2m {

ir(l) + ir (2) Gjd.2)

}

2—1.

The quantity p is the electron charge density while j (1) is that

part of the electron current density due to the motion of the electrons

alone. Thus, the "average" electron velocity V is given by

j (1) . v (i) = e v

= -L.

e

K

' p (1) 2m

Tr

{

TT (1)+TT (2) Gjd.2)

-1+ ..(X-8)

TrGjd.y

From (X-l), one obtains the result

V M (1) = --^-Tr

1 s 2mc {

"h„ h

x 1 i 2 ^(1,2)^1

2—1.

(X-9)

„ e rr-T —T* 1 - -zkTr {r (1) "- (2) a^i.z)^}

2—1

For convenience, we make the definition

(1) = i^-Tr-2- 2mc 2m {f

TT (1)+TT (2) "^ U)-~ (2) G^l.2)

2—1

+ (X-10)

If we compare 2 (1) to the combination -[ V (1)'V ]M (1) using the

s els

equations (X-8, 9), we find that the two terms are similar in appearance to each other except for the fact that 2 (1) is a trace over

s products, while -[V (1)« V ]M (1) is a product of traces.

6 IS

164
The value of M (1) , in the "self- consistent field approximas o tion" is given from (X-l) by

M (1) = It^TrTG (1.1).

s o 2m c o + (X-ll)

Since G is diagonal (see (VI-54)), the terms Tr cr G (1,1) and

o x o + Tr a G (1,1) both vanish, and hence

y o +

where

M (1) = i M so z so

If = ^Tr, G (1,1)

so 2mc z o +

(X-12)

From (VI-54), G (1, 1 ) = G (1.1) and therefore

oap + <a,8

M

i|ane so 2mc a

<i,«' "-<><««» »>

CO CO CO une

2mc I I js IV.ki-Wjfc.k,"

jxew

8

n= i= - oo

co oo

n,i,k 1 n, I, k 1

—£ ^ Cdk Tw

2

(n,k) - W^n.k)

n=0 -co

(X-13,

34 For the non-degenerate gas, e <<1, and

so 2

Bit c

co oo I I* {

BE , -BE

2, n, k 1, n, k

e -e n=0 -co }

(X-l 4)

165
pjio) B

n=0 -oo

_, oo oo B*V —r— e sinh —- | > e \ dk e

4tt c

so with the aid of the equations (VII- 11, 12, 13), one obtains

ixefin

M » T—

so Zmc

.phco

tanh (X-15)

For the case of the fully degenerate gas in the ground state (j3-*oo), a treatment similar to that for (V1I-22, 23) results in

r

M

_, un (2m) uhe B

so 2mc (2-nfi)^

N N-2

c

| ^-(n-gyfi^ -^WC-(n+l+g)Ku>B

n=0 n=0

N

c

for (N -g>K<o_ ^ ^ < (N +g)fiu_.

C B C B

N-l

c

(X-16)

) N/C-(n-g)*io>B -^^-(n+l+ g)KWB

n=0 n=0

for (N +g>«<»>1, < £ < (N + l-g)ftu_

C ' B C -D

or

166
N

c N-2

c ^^-(n-g)iia)B

-^N/;-(n+l+g)ficoB

n=0 n=0

N N-2

c Y»v/;-{n-g)RUB +^N/C-(n+l+g)fico

;i n=0 n=0

jtfie n Mso 2mc

for (N -g#co_ ^ C < (N +g>tf<o_,

c B c B

N N-l

c c 2^C-(n-g)fiWB -2^C-(n+l+g)«uB

n=0 n = N N-l

c Y^-di-g^c^ +2^C-(n+l+g>Ku

B

n=0 n=0

(X-17)

for (N +g)6a)T2 ^ t, < (N +l-g)^co„

c B c B

167
CHAPTER XI

DIAGRAMMATIC TECHNIQUES

In recent years the use of diagrams in connection with perturbation theories and in the solution of integral equations by iterative

expansions has become rather popular. To some extent the description of many body problems in terms of Green's functions circumvents

the necessity of such an approach; however, the exchange and correlation contributions to the Green's functions themselves may be described

in terms of diagrams.

Two examples of the diagrammatic technique are considered

here, not for the purpose of obtaining solutions, but rather to illustrate the topological structure of the diagrams.

The first example will be the integral equation (IX-4):

K(l, 2) = 6

4

(1, 2) + CCd(3)d(4) v(l, 3)-^^— K(4, 2).

JJ 6U i {4) v o

(XI- 1)

By iteration K(l, 2) may be expressed in terms of the infinite series

K(l, 2) = 6

4

(1, 2) + Cd(3) v(l, 3) 5

f[.(3)

J 6U X (2) o

+ ^d(3) d(4) v(l, 3, r-

(XI- 2)

6p(3) 6p(5)

6UT

(4)

' 6UT(2)

o o

+ . . . . Each of the integrals in (XI-2) may be expressed in terms of equivalent diagrams. We associate with each of the quantities v(l, 2) and

168
an "elementary" diagram. The interaction term v(l , 2) may

6U T

(2)'o be represented by a "wavy" line:

v(l, 2) <*=

I 2 (XI-3)

and the quantity

Sp(l)

5U T

(2)

, which we designate as a "propagator" (or

"generator") may be represented by a "bubble":

6p(l)

6UT

(2)

o

(XI- 4)

For convenience, we also define the diagrams

• i K(l, 2) <==> xx and

4 r 6 (1, 2) <===> :• 2

(XI- 5)

A "chain" diagram of "elementary" diagrams joined at the "end" points

(or "vertices") represents a product of the associated quantities, along

with an integration over the space-time co-ordinates of the joined

"vertices. " For example

6p(2)

1 ^wvwvk

<z 2)

6UT

(3)

2 <=

3

f d(2) v(l,

r 6p(i)

\ d(2) P „r,Jt 6UT

(2) i (XI-6)

Ml) c (2, 3)

P

$TM i».«

Observe that in the last diagram of (XI-6) the integration over tne

delta function may be performed, and thus

• 3 6U (3)„

o: = o

(XI-7)

• 3

169
Therefore, we may represent equation (XI- 1) by the series of diagrams

obtained by iteration

K(l, 2) <^

i i

• 2

• 2 ^14 +

• 2

(XI-8)

• I 3

' 3 1 5

• 2 Y3

• 2

From (XI-7), we see that the diagrams in (XI-8) reduce to the form

K(l, 2) < . 1 >

•2 . • 2 (XI-9)

4- •••

The series of diagrams in (XI-9) represents the series (XI-2) exactly.

The solution (IX- 10) of equation (XI- 1) is precisely the solution which

should be obtained for the diagram equation (XI-9) if it could be evaluated. The interesting thing here in the structure of the diagrams

(XI-9) is that they bear a curious resemblance to the "ring diagrams"

for the Grand Partition Function. As a matter of fact, there is indeed

a direct relationship between the two sets of diagrams; both correspond

170
to the "self- consistent field approximation" (sometimes known as the

"random-phase approximation") see reference (42)).

The second example of the diagrammatic technique will be the

integral equation (VI-8) for the Green's function:

G.(l. 2) = G (1, 2) + f d(3) d(4) V(3, 4) G (1, 3) G. (3, 4) G. (4, 2) , 1 ° J oloio

where (XI-10)

V(l, 2) =

le

6(t +e-t ) —-, £-0+

C 12

Here we designate G (1, 2) as the "propagator" (or "generator"), and

the representative diagram will be

G (1, 2) <* o

(XI- il)

The interaction term V(l, 2) will be represented by a "wavy" line in

this case also:

V(l, 2) <=> (XI-12)

The Green's function G,(l, 2) will be represented by the notation

1 o

G.(l, 2) <£==> a

1 o

(XI- 13)

• 2 As before, when the "elementary" diagrams are joined at some "vertex

point, " this represents a product of the related quantities and an integration over the co-ordinates of the vertex point. There is always an "incoming" and an "outgoing" propagator joined with an interaction at

each vertex point in the expansion of (XI-10);

i '

2l *wvi* <==> f d{2 ) g (1, 2) G (2, 3) V(2, 4) . (XI-14)

4 J

o o

171
With the "elementary" diagrams (XI- 11, 12, 13), the equation (XI- 10) is

represented by the iterated series

G(l, 2) < > a =

1 o ;,

i

2 3

3 <*U

4

rsT

2 •

* 2 P

(XI-15)

i 2

"Zeroth

order"

diagrams

—

'I st order

diagrams

2nd order diagrams

172
5

1 2

-»-•••

_

" ^

3 rd order" diogroms

One of the nice features of this diagrammatic expansion of G.(XI-15) is

that a particular form of diagram appears once and only once. A classification of these diagrams is possible on the basis of the number of

"propagators" in any one diagram; we observe that there is always an

odd number of "propagators" in any one diagram, thus we designate an

nth order diagram as one with (2n + l) "propagators. " The rule for

constructing all nth order diagrams is a simple one; all "joined" vertices must contain two propagators and an interaction (XI- 14), and all

interactions must join to two different vertices without any two

173
interactions "crossing through" each other or joining at the same vertex. That is, there are no diagrams of the types

(XI- 16)

Moreover, since all diagrams must comprise of continuously "linked"

chains of propagators; there can be no diagrams of the type

(XI- 1?)

We emphasize that the directions indicated for the propagators by the

arrows is important since the propagators are matrices; the propagators are arra'nged from right to left in the order of the directed diagrams. For example

=> G (1, 2) G (2, 3) G (3, 4) o o o

(XI- 18)

* 4

Diagrams of the types

i

i 2

!

(i 3

(XI- 19)

where the arrows point in the opposite senses relative to a vertex are

meaningless.

174
We consider two approximate "linear" equations for G. to illustrate the classes of diagrams belonging to each:

G.(l, 2) s G (1, 2)

1 oJ A o

I

+ \ d(3) d(4) V(3, 4) G (1, 3) G (3, 4)

and

G.(4, 2) , o.j A

(XI-20)

G.(l. 2)

1 o. = G (1, 2)

B

°

+ f d(3) d(4) V(3, 4) G (1, 3) G (3, 4)

OJB ° G (4, 2)

Now G, (1 , 2) is represented by the series of diagrams

1 OJA

G,(l, 2)

1 oJA

,

i

* 2

* 2

* z

10

* 2

+ • • •

(XI-21)

175
while G,(l, 2) is represented by the following series of diagrams:

1 Ojg & &

G

l"'

2

»oJb

=>

* 2

+ • ••

(xi-22;

The n zeroth" and "first" order diagrams are the same in both approximations, but the higher order diagrams differ. Each approximation

exhibits a definite symmetry or topology, and we have the "peculiar"

result that the "zeroth" and "first" order diagrams go equally well

with both cases. The well defined symmetries of the two approximations

(XL-20, 21, 22} are occasioned directly by the fact that these equations

are "linear. " If the solutions of (XL-20) were known, they could be

used as generating functions along with G to obtain solutions corresponding to the "non-linear" diagrams.

176
CHAPTER XII

SUMMARY AND SUGGESTED LINES OF FURTHER STUDY

The functional series expansions developed in Chapter II for

charge and current density from fundamental principles of electromagnetic theory apply generally. The series may be extended to nonlinear orders in the field by the method outlined. This result amountsto a formal theory of the electromagnetic properties of physical

systems in terms of charge transport and of charge density fluctuations.

It was shown how the use of a Darwin Hamiltonian in "second

quantized" form leads to expressions for the charge and current

densities of the electron gas, and to expressions for temperature

dependent Green's functions, all of which depend upon the applied

electromagnetic fields. It was further shown how these expressions

are "renormalized" into functionals of the total electromagnetic fields,

relating the charge and current densities both explicitly and implicitly

to the fields.

In order to facilitate calculations, the functional series for p and j obtained in Chapter II were used to express the charge and

current densities of the electron gas explicitly to an order linear in

the perturbing electric field. Calculations were obtained in the "selfconsistent field" approximation in terms of a "conductance tensor".

The nine components of the "conductance tensor" may be expressed as Fourier transforms generally covering all ranges of temperatures in

the non-relativistic gas, although actual calculations are quite difficult except in a few special cases, i. e. , for plane wave propagation

in the direction of the applied magnetic field and for the "weak spatial

177
dispersion" limit, at zero temperature and at the classical high temperatures. The results at classical high temperatures correspond to

quantities obtained through Maxwell- Boltzmann statistics. The results

obtained for zero temperatures (degenerate gas) very definitely show

quantum effects. It is interesting that these latter results permit

calculation of the current density and other quantities quite readily at

very high applied magnetic fields, but the procedure becomes more

difficult at low values of magnetic field. Of course, all of the more

difficult calculations could be programmed for computation and

numerical tabulation by a high speed computing machine.

Although a considerable amount of effort is required to bring

about the formulation of the expressions for the charge and current

densities and to give them in a form showing explicit dependence upon

the perturbing fields, the results given here should facilitate examination of the exchange effects and other higher order quantum contributions. In this case, the solution of the appropriate non-linear integral

equations is required. Moreover, the treatment is readily extended to

include effects of non-linear orders in the field.

178
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182
MATHEMATICAL APPENDICES

183
MATHEMATICAL APPENDIX I A brief discussion is given here regarding the definition of a

functional and of a functional series expansion. To simplify matters,

we consider functions of only one variable.

Suppose that one has an arbitrary function U dependent upon the

variable x: U = U(x) ; (1-1)

and let us further suppose that we have another function Y such that Y

depends upon the variable x explicitly, and also upon the function U

explicitly:

Y = Y[x, U(x)] .. (1-2)

The function Y is designated by the terminology

Y = functional of U . (1-3)

Let us assume that Y is known for some particular function

U = U (x): o Y = Yfx, U (x)l known . (1-4) o

L o

' Then, let us designate any other function U and the associated Y by the

notation

U(x) = U

q

(x) + 6U(x) ,

and (1-5)

AY[x, 6U] = Y[x, U(x)] - Y[x, U (x)] .

184
Assume that it is possible to express AY [x, 6U] in terms of an infinite

series of the type

AY[x, 6U] = Cdx, Afx, U (x); x, 1 6U(x ) L J 1 1 o 1 1

+ "^jTdx

i dx

2 A

2^ x' u

o

(x): xrx

2^

6u(x

i

) 6u(x

2

} (1 " 6)

+ yr (

> rrdx

i dx,, dx

3

A

3 [x, Uo

(x); *it x2> x

3

] SU^) 6U(x

2

> 6U(x

3

)

+ . . .,

where the integrations range over the region for which .the function Y is

defined, and where A , A , A , etc. , are coefficients dependent only

upon the variables x, x , x , . . . , and upon U . If one takes for Y the

1 Z o function U, then obviously in this case Ajx, U

q

(x); Xl]=6(x- x

x

) (1-7)

A=A=A=. . . =0.

2 3 4

We define a linear variational operation upon any functional Yfx, U(x)]

of U(x) such that

6Y[x, U(x)]

B the variational derivative of

6U(Xj) Y[x, U(x)] with respect to U(x.) ;

and (1-8)

6Y[x, Uo

(x)]

6U(Xj ) = ,

6Uo(x)

_ Q

6U(x.)

6U(x) -i- = 6(x - x.) . 6U(xj) J

185
Higher order "variational derivatives" of Y[x, U(x)] may be defined with

nomenclature similar to that of ordinary derivatives. In general

6 Y[x, U(x)] the nth order variational

6U(x ) 6U(x

2

) . . . 6U(x

n

) "derivative of Y[x, U(x)] ,

with (1-9)

6 nY[x, U

o

(x)]

6U(x

x

) 6U(x

2

) . . . 6U(x ) n

6 nU (x) o 6U(x

1

) 6U(x

2

) . . . 6U(x ) n

6 n

U(x)

o ,

o ,

r6(x - x

1

) J n = 1 6U(x ) 6U(x ) . . . 6U(x ) \0 , n > 1 .

Thus (1-10)

5 nY[x, U(x)] 6 nAY[x, 6U(x)]

6U(xJ 6U(x.J . . . 6U(x ) 6U(x ) 6U(x ) . . . 6U(x ) ,

and from (1-6, 8, 10) one obtains (1-11)

6Y[x, ' U(x)] r

. = \dx A [x, U (x); x] 6(x - x.)

6U(x.) J 1 1

L o

J 1 j

J

{6(xlj- x.) 6U(xJ + 6(x_ - x.) 6U(Xl )} 2 2 j 1

J

+ . . . .

It is easily seen that (1-12)

lim 6Y[x, U(x)] 6Y(x)

(6U=0) 6U(x7"

=

6U(x.)

S A

l£

X' U

o

(x)l Xjl *

J jo

186
Similarly,

2

SYfx, U(x)1

6U(x.)

(1-13)

5 Y[x, U(x)] j>

6U(xk

) 6U(x.)

= 6U(x

k

)

and from (1-11) we obtain the result

lim 6

2

Y[ X> U(x)] m

6

2

Y(x) _ A [x, U (x); x., xj . (1-14)

(6U=0) 6U(x, ) 6U(x.) 6U(x ) 6U(x.) 2 ° J k

k j k jo

(In general, we assume A Tx, U (x); x , x » . . . , x 1 to be "symn

L o 1 2 n metric" in the co-ordinates x,,12x. . . . , x . That is, A is inde- n n pendent of the order in which x12, x , . . . , x occur.)

n Generally

lim 6 nY[x, U(x)] 6 nY(x)

(6U = 0) 6U(x,) 6U(xJ . . . 6U(x ) ~ 6U(xJ 6U(xJ . . . 6U(x ) 1 d. n 12 n

(1-15)

= A [x, U (x); x , x , .... x 1 . n

L o 1 2 n

J

It should be clear by now that the operations which have been defined

are analogous to ordinary differentiation operations with the designation of the functions U as the "variables", and U as "constants. " o Thus, the series for Y may be expressed in the form

Y[x. U(x)] = Y[x, U

q

(x)] + AY[x, 6U(x)J ,

with (1-16)

2 3

AY[x, 6U(x)] = 6Y[x, 6U] + 6 Y[x, 6U] + 6 Y[x, 6U] + . . . , and

6Y[x. 6U] = JdXj -Jffi mXl) . 1 o

187
6

2 Y[X, „] . j, ^dXi dx

2 6u(x

6

2>

Y W

Xi)o

6U|^J 6U(x

2

) .

etc.

The symbol 6 ny[x, 6U] is called the nth variation of Y with respect to

U, and the quantities (1-15) are called functional derivatives, or variational derivatives.

All the usual rules of ordinary differentiation apply here, as for

instance, the "chain rule" of differentiation for changes of variables:

6g[x , u(x )] r6 g[ x i» "(x )] 6u(x ) i

1 « \ dx, . , .. 1 r-r-.. (1-17)

Sv(x

2

) J 3 6u(x

3

) 6v(x

2

)

The results of this discussion may be generalized to the case

of functionals of more than one variable, and the same philosophy

applies throughout.

More exact and complete discussions concerning functionals

can be found in a number of sources; see for instance, the reference

28.

188
MATHEMATICAL APPENDIX II

The equation of continuity (1-7) and the gauge transformation

(1-5, 6) lead to connecting relationships between the functional deriva-

* -* T T

tives of p and j with respect to the field potentials A and U . The

first order variational derivatives of the equation of continuity are given

by

_^_ 1pJlL + _^ T(i) __ 0>

5U (2) 8t 6UX (Z) o i o

9 6p(D +v . 5jjl)

a Qf

8t 6UT

(2) * 6UT

(2)

1 o o and (2-1)

6 3p(l) 5 * —

n

^ + r

V • j (1) =0

6A (2) 8t 6A (2)

q o 1 q o or

S£J13 + v . 5i{1) - T... 1 „.T,

at, 5A (2) 6A (2)

1 q o q o One may obtain similar results for higher order functional derivates.

The same results (2-1) may be obtained in another way. As a functional series, the equation of continuity becomes

189
= 8p

o

(l)

8t

l + v

-*• Jo'1 '

y*uD {

a L 8tl 5U (2) o

5 7(1) 1 6UT

(2) - o

6U

T

(2)

+

8

- 9t

5p(D +v . T 1 6A (2)

q o

sT (i)

5A

T

(2) - q o

5A

T

(2) . q

(2-2)

Now, p and j represent the charge and current densities for the

O O rp rp system in some initially "unperturbed" state (8U = 5A = 0), and thus

at.

+ v • j (i) = o

1 o (2-3)

T T

The functions 5U and SA are here considered "independent" and

q

arbitrary, thus each of their coefficients in (2-2) must vanish

separately, again resulting in the equation (2-1).

Further connecting relations are found from the requirement of

gauge invariance. The densities p and j are invariant under the

change of gauge

5uI»5UT

+ — -|-6AT

cot

(2-4)

6A-».5A - V5A , T

where &A is an arbitrary function satisfying the equation

2

-V2

6A

T

+ i

2 2

c at

6A = . (2-5)

190
The functional series for the charge density p and the current density

—

j are given by

p (1) = p (1) 4- fd(2) [

6 PU) 5U

T

(2) + _^ill 5A

T

(2)1+

*- L ATT l?\ KA f>\ ^ 5U (2) SA (2)

q o and (2-6)

J(D = J (D +

\

d( Z ) id(Z)

f" ,

5„t(1) 5ljT(2) +JY

ST 6Al

(Z)

L 5U (2), 8A

1 (2) q J q °

"t" • • • •

If we apply the gauge transformation (2-5) to the equations (2-7), we

obtain

p(l) = p (1) +\d(2) C< J>£J!)_ 6UT

(2) + _5£ilL 6AT

(2)

- L6U1 (2) 5A (2) q - o q o

+ . . .

and

J

+ \d(2) «|<!Li-JL SA^.-iEjav^A^....

6U (2) x c at,

'o 2

6AA (2)

q o

(2-7)

7(1) =T (D +^d(2) S^1

) 51^(2) -4iiil-6AT

( 2) T

L6U (2) 6AA (2) q

q o

+ .

"J

+ UU) LHIL^JL jaV-^-v saTzJ

L6U (2) C 3t

o 2

6AX (2)

q o

191
Now, if we assume that the functions Sp (1) 6p (1) T ' T SU (2) 6A (2)

6 f(1) sT(D

6UT

(2) 6AT

(2)

q o

, etc., all vanish identically at the

boundaries |r | = co and t = co for finite |r | and t , then by integraLa C* X X tion by parts we obtain the integrals

s

d(2)

'

i a 6P(D , v

6p(l)

L c Bt

2

6U

T

(2)

2q 6AT

(2) J o q o

6A(2) +

(2-8)

°=id(2) 5j.(D + V

L c 8t_ 6U (2) 2 o

5j (1) *

6A

T

(2)

q o

SA(2) + . . .

from a comparison of the two sets of equations (2-6) and (2-7).

T

Since the function 6A is "independent", the coefficients of each of the

combinations of terms SA(2), 6A(2)X 6Al(3), etc., in the integrals

must vanish individually. Thus

6p(l)

+ V

c 8t_ 6U (2) 2

x 'o

1 8 6j(l)

c 8t, 6UT

(2) 2 o

5p (1) 2q 6AT

(2)

=

(2-9)

+ V

6jd)

2q 6A T

(2)

q °

= o

and the higher order equations are similarly formed.

The equations (2-1) and (2-9) are precisely the connecting

relationships desired (equations 11-11 ).

192
MATHEMATICAL APPENDIX EI

The Hamiltonian given in. the equations (HI- 2) includes the non

time retarded form of the "magnetic" interactions. It may not be

readily apparent or immediately obvious how this form of Hamiltonian

is obtained and we consider here its derivation.

We begin with the ordinary Hamiltonian operator for two electrons. The "spin" is an unnecessary complication in the derivation,

however, and v/e ignore all spin contributions until we have obtained

the "correct" prescription for constructing the Hamiltonian.

From the equations (39. 14) of reference 35 one has for the nonrelativistic Hamiltonian of two electrons the expression

Hnr

1 2m

Jl

POy't) 1 2m P (?., t)] - e U

A

(T. , t) - e tfV,. t) + -£— , 2

J

l Z r i2

where (3-1)

P(7,t) = £* + | A

A

(7.,t).

1 lie 1

The interaction of the electrons with the applied fields are

included along with the "electrostatic" interaction between the electrons. Now the part of the time retarded "magnetic" interactions

(Darwin terms; see references (31) and (35)) due to the motions of the

electrons alone is

H = - m ,22 2m c r-.{yff".t|.

12 I- P (r . t) +

12

12

T -ZF.t) P(r

2

.t)

(3-2)

193
The operator H (3-2) is Hermitian as is also H (3-1); i.e., both

m nr H and H satisfy the condition

m nr

rd

3 71 d3 r

>

2

0*(H l}>) = f^d3 ^ ^ (H^)* , (3-3)

where <p and ip are arbitrary functions of the time and the two sets

(r ) and (r ) of spatial co-ordinates [see reference (43)] . Moreover,

a combination Hamiltonian

H = K + H

c rnr m

(3-4)

results in an "equation of continuity" of the form —R(« r . t) + Vj • yr,. V

t] *V, • J

2

(^.r

2

,t) = ,

where (3-5)

R (r^ r2

, t) = ijj* (r

1#

x^ t) i|i (r^ r

£

t) ,

and ij; is some eigenfunction of H. From the Schro dinger equation

and

ih— ^=H +

(3-6)

-ih—q; = (Hl|i)* ,

,

194
we find J1 and J to be given by

_ _* _, P(?.t) P*(£.t)

J

i

tri'V tJ ~ ^*~ ^ + ^ —"*~ ^* 2m 2m

2

i|i*P(r„t)i|; + i|j P*(r\ t) ij,* 2mc; r12

'2

12 , -* *"*"»-, 12 r — -**-* + i|,* r [ r

12

• P(r

2

,t)il,] + vp —L__[^

2

. p (^,t)+*]|

(3-7)

r r

12 12

-, - -> P(r2' t) P {r

z'

t]

J

2

(r

i ,r

2'

t

^ S ** ^

+ ^ ^* 2m 2m

2 . *

i|i*P(r. t t) ijj + i|;P (r.t) l|l*

2 m c / r

r + **—^[^-P^.tjqi] -f^-iL-f^. pVr

t)^*]| . r

21 r

21

The non-time retarded equivalent of H (3-2) must also be a m

Hermitian operator (3-3), and it must correspond to the classical

limit. These requirements are sufficient to give an expression for

the "correct" Harniltonian in the non-time retarded case. If we

define operators A. and A such that

195
^i-V" s "7^7

'21 "i'V" * "

1— -*•-•• -*• * 1 P(r. ,t) + P(r t) —

r. _ 2 2 r

L 12

2m c

L

r

21

P^.tJ + P^.t)-

12

1

(3-8)

12 J

then we may write the non-time retarded Hamiltonian H^ for the

"magnetic" interactions in the form

H

mn

P(r t) e _ e _

• — A + — A

2 m c 12 " c 12

P(r

x .t)

2m

P(r

? ,t) e ... e _ P(r ,t) —*

A,, + —-A„ r^—

2 m c 21 c 21 2 m

(3-9)

-(2SF) f'V1

'

• r P(r.,t) + P(r t) • P(r ,t) —

i2

2 l Z ri2 —r P(r,t) • P(r ,t) + P(r ,t) — i2

2 : 2 rl2

P(r

i' t}

}

The operator H satisfies the Hermiticity condition (3-3) and it corres«

• - mn

ponds to the classical limit. This latter condition we can see by

expressing the wave function in the form (see reference 39, section 15)

iAfiif

\\i = ae , -* -> where a and & are both real functions of r , r , and t. With the

expression (3-10), and with the total Hamiltonian defined as

(3-10)

H = H + H

nr mn

(3-11)

the Schrodinger equation (3-6) goes over into a form in which the real

and imaginary quantities become manifestly distinct from each other.

196
The real and imaginary parts of the Schrodinger equation can therefore

be written in the form of two separate equations:

3S 1 <Tt

+ 2mv

1 s + fAA

(71 ,t) 2m V,S + ;AA(7.t)

2 c 2

-eU (r

1 ,t)

..uAg

?

, t)+ Z.^2

_i.fv1 s + frA

(T

1 .t)"

1 2 r 12

^mc;r

12

[l c 1

| VS+-rA

(r t) 2 c 2

2 2

(3 - 12)

•ft 2 V

l a *

2 V /eKx2

1 Ti'V l/eK^ _ 1 „ „ 1 2m a 2m a V"10 /

r 12 a 2 \mc / L

1 21 3

= ,

and

lTT + T" Vf

S + — V'Vi S + ^

Vr A A

(7. . t) + -I-aV,. t)

. V a

^ at 2m 1 mil 2mc 1 1 mc 1 1

+ ^-v.2 s +— Va- VS + -^-V.AA

(7_,t) +-£- a

A

(7 ,t).Va 2m 2 m 2 2 2mc 2 2 mc 2 2

^ i (3-13)

mc / r 12

'^s+frV^t] V,a + — V

2 2 2 r,„

12

V mc V_S+-A A

(7,,t)

2 c 2

a„ 1 "1 V.a + -V

- 1 2 lr

l 2 J

- a JLA_L V . VS} = o mV r l2

l 2 J

The equation (3-1^) is precisely the classical Hamilton- Jacobi equation

in the limit 'd.—Q where the identifications

197
55 = E

9t

V

X S = p^ (3-14)'

V = P2

are made (see reference (32)). In the non-time retarded limit, the

Lagrangian is given by

mV;2 mVl2

. . Z . 2. %•%

+ eUA

(7

1 ,t) + eUA

(T

2

,t) --£-

12

+^ r l2

(3-15)

This Lagrangian corresponds to the Hamilton- Jac obi equation (3-12) if we assume that — 8L -r 3L

i af 2 aT

1 2

^^-vj^Vi (3 - 16)

*2 - IF"

^ m Vc A (r 2' t}

• 2

(See the procedure outlined in reference (32). )

198
The "probability current densities" T and T as derived

from the Hamiltonian (3-11) are given by

P(r ,t) P*(r ,t)

2r

12

Vmc i|;*P(r ,t)v|i + ijiP*(r ,t)i|;*

(3-17)

T

2 Pi,72

.t) = *> 2m * + * P *(r

2

, t)

Zm l|;#

1 / e 2r

12

Vmc i|;*P(T , t)i|i+ vJiP*(T ,t)vjj*

The results which have been obtained for the two electron

system can be generalized to the case of an N electron system (N5= 2) as follows:

N

H = I[h*^

_^._^ 2 A— x ,t) - eU (r.,t)

1 + —

2 /

j*l

jl

2 2mc Z^

j*l

P(7.,t)-A (r.,r ,t) + A (7.,r t)-p(7..t)

J jl J I Jl J I J

where (3-18)

A. (r., r , t) ji

V j' I' 2mc P (7 . t) + P (7., t) n

L Jl Jl

Since relativistic and time retardation corrections of order [ — ) are vv missing from (3-18), one must always remember this equation to be

incomplete.

199
From equations (3-8, 9) we see that the "prescription" for the

"magnetic" interactions part of the Hamiltonian is such that the

operator A._ for two electrons is incorporated into the Hamiltonian

-~A

in a manner similar to that for the vector potential A for a single

particle. Since the operator A takes the form of a vector potential,

J. c we must add to it all contributions from "magnetic dipoles, " i. e. , electron "spins. " This is done in the manner indicated by Slater (see

reference (36)):

X.fr.,~T.t) = - ji j I 2mc

r

— p (7. t) + p (7, t) J- + yk? x-M

a h " r.

t

(3-19)

where xt operates on the £th particle only. The interaction between

spin and magnetic field is given by

N

h = J~. )7 . [v x AA

(7„,t) +i > v xa .(7,7., t) s 2mc L I \_l I 2. L, 6. £j & j

4=1 j±l

(3-20)

In a straightforward manner, the Hamiltonian (3-18, 19, 20) may

be converted into a "second quantized" operator (see reference (29),

Chapter 6) resulting in equation (III-2). We observe that the "second

quantized" Hamiltonian (III-2) is "symmetrical" in all its components.

The fact that an equation of continuity of the form (3-5) does

exist for the Hamiltonian (3-18) (as a generalization of (3-11)) indicates

that one can define a current density for the system in the "second

quantization" formalism.

200
MATHEMATICAL, APPENDIX IV

The derivative of a "step discontinuity" can be represented bymeans of the Dirac delta function. Suppose we have a function f(x)

such that

g(x) , x< x

f(x) = J , . . x

'

| h(x), x> x

^ o

(4-1)

where g(x ) # h(x ) and both g and h are continuous functions in the

neighborhood of x . We can define the "derivative" —

9x

the expression

from

x=x

Im

= e-0+ i

K. +€

O

8f

to <

i - t

= h(x ) - g(x ) O

(4-2)

where t > 0. The equation (4-2) shows that one can uniquely define

or

8x

as x=x

8f

'

3x

= 6(x - x ) o

f(x ) - f(x ) = 6(x - x ) 0+ o- o h(x ) -g(x ) o o x=x

(4-3)

201
MATHEMATICAL APPENDIX V

Solutions have been obtained elsewhere (see reference (39) and

(44)) for the Schrbdinger equation of an electron in a constant mag-

—T — netic field B = i B : o z o i*— = H$

where

~ ufhco H = —Mr.tf + -r-^o- 2m 2 z

(5-1)

$ = Ue i

Et

U =

f(r)

g(7)

We outline the solution of equation (5-1) here and obtain a number of

-T

useful identities. The vector potential A is taken to be

o —T 1 —T — A = ^ B X r • o 2 o

(5-2)

In the gauge (5-2) the operator ir is given by

-JL

i 8x

mco.

IT 4f_8_

i 8y

m

"b

(5-3)

202
1_L

i 3z '

or by

TT

TT

z& a

i 3x + 1 2

X

+

2$. a

i 3x

- 1 2

X

# a

i 3z

(5-4)

From (5-4) we obtain the relations

TT IT + - V i J 3x 3x

(5-5)

*-*+ = [t) 3x^r + ^

uB^-x.ar)+ (-^) x.V^-

Now

2 2 1

TT + TT = —

x y 2

TT TT + TT TT + - - +

(5-6)

_/2*? 3

2

^i y 3x

+

3x_ + ^

UB (

X

+ 3x7 " X

- 8x7)

+ W^) X+XThus

it it = tt + tt - rrmui + - x y B

tt tt = tt + tt + rrmcon

- + x y 1 203

(5-7)
or

2 2

itxy+ it = tt tt + mfico = TT TT - rrrfiGj +- B-+ B

(5-8)

The Hamiltonian may be written in the forms

H = -i-12(tt + tt2-2+ tt ^B ) + a 2m x y z 2 z

1 • 2

= -

(tt tt +rrmco +TT ) +

H»o>,

2m + - B z 2 z (5-9)

(tt tt - mhw_ + TT ) +

II&O. 2m - + B z 2 z

From (5-9) we obtain the identity

TT TT - TT TT = 2lYm<j)

- + + - B

(5-10)

We shall find that it has the properties of a creation operator, while

tt corresponds to an annihilation operator.

From (5-1) we obtain the equations

r i -2

^^

Ej f (r ) = TT + —

2m 2

f(r)

r~. n -2 ^B-

(5-11)

E

£

g(r) = 2m g(r)

The equations (5-11) infer that the eigenfunctions of (5-1) are

U = (. ) v(r) with energy eigenvalue of spin index a = 1, and

U

2

= ( ? )V(T> (5-12)

204
with energy eigenvalue of spin index a = 2, and where

—2

IT 2m v(r) = jA^n

'(a=l) 2

v(r) = ^B-

'(a=2) 2

v(r). (5-13)

The solution v(r) is separable for the z-direction, in which case we

assume a plane wave solution

v(r) cc e ikz

(5-14)

Then, from (5-9, 12, 13, and 14) we obtain

TT IT ——v (r) = 2m v

' u^B

^

B ^2k

2.

'(a=l) 2 2m -(T)

E

(a=2)+2

u^oj^ r ^w„ ^2, 2_ B B -h k

2m (r*)

and (5-15)

TT TT 2m

f

v(T) = ixhw^ ^oj^ ^2, 2_,

E - B

+ B - k

(a=l) 2 2 2m v(T)

(a=2) 2

Lchco„ ^<J„ v?2, 2. _ B B -K k

2m v(r)

By inspection of the equations (5-10, 15) we find the eigenvalue of the

operator ir ir to be less than the eigenvalue of the operator t i by

2rrrfia)„# thus we infer that

+ \ 2rrma>_,

205
is a creation operator and

D

is an annihilation operator. By analogy with the harmonic oscillator

problem, we seek some function v (r) such that

it v (r) = . (5-16)

- o

The function

mu>B

v (T) = o

/

m

"B " 4n

si 2rrn

6

XX

e (5-17)

satisfies the condition (5-1 6) and it satisfies the boundary conditions

v (r)-~0, for !x -co

o

or for | y J -»co ,

and finally v satisfies the normalization condition o

co co 2

(5-18)

Cdxfdylv^T)! = l . (5-19)

-co -oo

The normalization condition (5-19) is easily proved by a change of

co-ordinate system to polar cylindrical co-ordinates:

x = p cos 9

y = p sin 9 (5-20)

z = z .

206
Thus

OO CO

•CO -OO

OO CO

C r

m

"B "

OO -OO

2tt 00 mco R r r

2.hW Pdp

mw

B 2 —2lT p e o o

(5-21)

= 1

Finally, the function (5-17) satisfies the equations (5-15) with

•2 2

E = E . (l + g)lia) + —

(a=l) 1, 0, k

*' B 2m

(5-22) A2

E

(a=2) ^ E

2,0.k

= -^wB +_

2msince p. = 1 + 2g. The function v (r) corresponds to the ground

states of the electron (two spin states). Now, the function

vOr(r) = F(x ) v (r) , (5-23)

- o

where F(x ) is an arbitrary function of x , satisfies the condition

(5-16) equally well. The angular momentum operator (z-component)

L

z

= r(x

87 y ^)

= <^-*^ <5 - 24)

i 89

207
commutes with the Hamiltonian H, and we can remove the arbitrariness of the function F by allowing U to be an eigenfunction of L as well as of H. Thus, if we take

L

z Vo

(7)F

= ^o^'f

where I is any positive integer (5-25)

I = 0,1, 2, 3, ••-,«,

then we obtain "the solution

F(xJ = x* = (x-iy/ = p

l

e"

li9

(5-26)

','"

which satisfies boundary conditions. Thus, the function v (?) is o generalized to an eigenfunction of L with angular momentum quanz turn number i and energy quantum number k:

T75 V 2*

£j± m"B 2 2

(-vr) ( *- iyf

tit v~^"y

-**•! mt0

B 2

W^2

I -i|6 W9 pee ikz

e

where v , is normalized

o, i, k

208
oo oo 2ir oo Mdy|v

o.i.k(7)l -Wp,dp|vo.tk^i =i - -OO -OO O o

It is easy to see from the polar co-ordinate form that v , is

o,£,k

orthonormal in I :

oo oo Wd"ti1

.'*

1 ff>T*i2 .»a fl

-OO -00

(5-28)

2tT OO , V r r

i(k -k )z = M-p dpv. vk ravo, Vk2

<r) = ^e

O O

The functions v (r) exhibit a many-fold degeneracy in the energy

o,£, k

eigenvalues, i. e. , for a given value of k the eigenvalues (5-22) are

the same for all values of I. As noted before, the problem is analogous to that of the

simple-harmonic oscillator (see reference (39), section 21) and

therefore we introduce the energy quantum numbers n with the

functions v . , (r) such that

n, I, k

TT TT

2rnno) n,i, k ' n, i, k

TT TT +

^—• v . i ( r ) = n v . , (r) , 2miuo_ n, I, k ' n, £, k '

TT v . ,.(T) = •Jn+T.v.J.. , . (r)

(5-29)

\/2mTiZ n.i.k n+l,£,k

TT

, " • v . . (r) = sfn v , .

. (rj

N/2mfto,B

n,i,kx

' n-l,i,k v'

209
where the numbers n are positive integers, n = 0, 1, 2, * * •

, oo. Using the equations (5-29), we find that

1

1 \

2

ir. v . , (r) = n/Tv, . . (r) 2mfiuJ + 0,i,k* ' l.i.k

_1_ 2

T-Z ^ ^ v • i @) ~ N^r2 v . . (r) (5-30) 2mfew_y + o,je,k 2,i,k. '

n .

1 \

2

n -

o-T2 ) it . v f (r) = n/2v - (r) , 2rrnico Ry + o, *, k n, JL, k

and thus n

2

(?) = JL/C-i-JN ^ v . . (r) . (5-31)

Now, one can easily prove the identity

mcoB

mcoB

-x x ——r— x x

45 + - 4K "+ - 26 8 „ ,,,

and repeated use of the identity (5-32) results in the "more general"

identity

ma>

B mwB

n ~4n~

X

+X

- ~4n"

X

+X

- /2fi 9 \

n %

e = e ( T"^7'

Thus, if we apply (5-33) to (5-31) we obtain

n moo.;. ma)

-R V

n, JU k^ = -^ (Sg e (ST) e VC £, k^'

(5-34)

210
and from (5-27) then one has the result

l+£-n moo, mo)B

8x

, .,n ,mco v 2 ., —r?^-x x , ~ v

11 X X

.-* (-1 / B\ ikz 4h +-/9\ * 2ft + v . . (r) = . ( ~7-l e e f -—

H

x ) e

Now, from the easily proven commutation relations

(5-35)

(*, >i/3Nn v \dx j v 8x ;

x +

(x ) 8x. J \ 8x<+;

(x r = o

(5-36)

we may write (5-35) in the form

V

n,i.k

(r)

l+£-n* mco

, moj 2 ., y—x x n-l

,

-£)

***** f x%t; _ (^ /

*7~Trn!ii \

n mco B

x x (x^xj e

* ""3" + -

From the definitions (5-20) we obtain the identities

(5-37)

x = x + iy = pe

ie

x = x - iy = XX = X X

+ - - +

pe

2

ie

= p

•ie ax

_a_ . j_ _a_ 8p

_1

p ae

(5-38)

211
ie

8x

= e JL • I JL'

9 P +1

P ae

1 8

x 9x

_1_

2p 8p

+ l p 89

1 8

x 8x.

J_

2 P

_8_ ._1__8_

9p "% 86

Combining (5-37) and (5-38) results in

n,i , k

l+l~n mcoB 2

(-if /""bN 2

ikz 4n K n-£ i(n-!)0 f 1 / 8 .18

e e p e —— f

• sjimljly 2i^

mco B 2

2p \8p p 89

n

2i

-~^-p

p e

Since the operator

(5-39)

"_L/1L -IJLY

2p \,8p p 89y

in (5-39) operates upon the function

21 2tf P e

B 2

P

which is independent of 9, we may make the replacement

mco

2p \^8p p 99 )

B 2

P

mco U 26. P e

1 9 T 2£

"^Sp" 1

B 2

2*T P

(5-40)

212
and since

V

_L _L

2p 3p (5-41)

we obtain the result

v , (r*)

,n jri^ 2

n-i £ _ (i) / B\ ikz i(n-i)9 t 2 2/ /£( <— l ->vr ' I e e 5 e c\ (£)

\fcn\ll \ 2£ J

* ^n,£ vw

where (5-42)

I =

mcoB\ 2 Z^M-.fe«(i),-V

(see reference (44)). The orthonormality of the functions v (r) n,4,k

co CO

\dx \dy v* (rjv

-co -00

k

2 n

i' n2 V'z

2tt oo

i(k

2

-kl)z

(5-43)

= \ d0 \pdp v* (r)v a

. (r) WS Wk

2 o o

can be verified with reasonable simplicity if one uses the polar

co-ordinate form of v (r) (5-42). The expressions (5-35,42) do

n, £., k

in fact satisfy the creation and annihilation relations (5-29) as one can

easily verify by mathematical induction. For convenience, one may

write

V

n.l,k

(r

» '<

llcz e w (x, y) in Cartesian co-ordinates,

n, £.

ilcz e u Ap, 0) in polar co-ordinates,

n, I 213

(5-44)
where

W

n,J

(X' y)

1+l- n m"B 2 2m"B 2 2

(-if /"V 2 "4^-(X+y) /9 . .8>!\ • /

-^"(X +Y }

6 wim-*'

( 5 _ 45:

r+i^l (x-iyf

and

u (p,9)

n, i

Clearly, the functions w and u have the following orthonormaln, I n, £ ity conditions:

CO CO

fdxfdy w* (x,y)w (x, y) = 6 6 ,

.) J n v

Jt 1 n

2

,JL 2 n

i' n

2 V *2 -co -co and , (5-47)

2tt co CdeCpdpu* (p. e)u ( P , e) =6 8

J J ^.ij n

2

,i

2 n r

n

2 ^,^ . o o

The energy eigenvalues for (5-13) are given in general by

2 2

vK 1 E, M = E, , = (n+l + g)tfw_ +

'( Q=l) l,n,k "' B 2m

(a=2) 2, n, k B cm

(5-48;

214
The designation of the energy eigenvalue is

E - E (5-49)

(a) a, n, k

where the index a is the "spin" index, n is the "harmonic oscillator"

quantum number, and k refers to the wave number in the z-direction.

From equations (5-4, 29, 35), one can easily prove the

following identities:

2* a mw

B

TT. = T— -T + 1 + i 3x 2 +

TT =

2-n a . wb

i 9x

+

"

l 2

X

-

TT =

Z * a

i 3z

2* a . mw

B

i 3x " l 2 x

2* a mu

B

l 3x 2 +

* a

TT* = - — -—

z i 3z

tt * = - tt - imcj X

tt* = - tt + imu> X

+ B +

TT* = - TT z z

215
+ n,i,k 1 i 3x Z +) n,£,k B n+l,l,k

/2tf a . mw

B \ r^—. jW

- n, i,k ^ i 9x 2 -y n, £, k B n-l.£,k

TT V = V , = ^lk V . , z n, £, k i 3z n, £, k n, £, k

tt* v* = f i —r—x ]v* . , = N/ZmSoiZ N/n+ 1 v* . . . + n,£,k ^ i 3x

+

2 -^ n, £, k B n+l,£,k

- n, i,k ^ i 3x_ 2 +y* n,£,k WB n-l,£,k

^3 ^i . xr* v* = -r— v* . , = Tik V* . , z n, £, k i 3z n, £, k p, £, k

mw.

(5-50)

2fi 3 . B

tt* v = [- — i —-—x )v , = -W~2rrrfiu„ ^+lv ,, . + n, £,k i 3x, 2 -J n, £, k B n,|+l, k

+

:v „t = [--^5

+i—^—>O v , v

= W2nrf£w„ *7I v , , n, £, k V i 9x 2 + 7 n, £, k B n, £-l,K

* 9 rft TT* V = - — T~ V . , = -Ilk V . , z n, i, k i 9z n, I, k n, £, k

tt v * = ( +i x ] v* = W2nrfia)Z" N/£ + 1 v*

+ n, l,k V i 8x 2 +J n, £, k B n, £+l,k

- f-r--s i—^—x ]v* , = -W2miiu_ n/I"v* , , - n, i,k V

i 9x

+

2 -/ n,i,k B n,i-l,k

w v * = * JLv* , = -tfkv* . . z n, £, k i Bz n, £, k n, £, k

216
2-ii 3

i 8x^ n, £, k

n/tTv . . - + Wi+ 1 v

n-l,£, k n, £+l,k

2j 8

i 8x n, £, k

\T2mhcoB

" Vn+T vn+l,£,k n, £-1, k I

lmojB

•x v

2 + n, £, k

imu

B

-x v

2 - n, i,

k

2-K a

N/2rrrfico

B

""

2

\/2rrrn*eo ,_

N/2mKojB

n+l,£, k n, i-l,k

->Jn v , + W£ + 1 v n-l,!,k n,£+l,k

I

i 8x n, £, k ^n+ 1 v* , . .+ W* v*

n+l,£, k n, £-l,k I

a£ d

v*

i 8x n, je, k

»v/n v* . . , - i^FTI v* , . , 2 n-l,£, k n,£+l,k I

moo

• B

-l —-—x v*

2 + n, £, k

\IZmZu B

-'v/n v * - W£ + 1 v*

n-l,£,k n, £+l,k

2 - n, £, k

N/2mft'("b"^n+Tv* - i\£ v* n+l,|, k n, £-l,k

Thus we see that the operator

ITT' v/ImSu

P

behaves as a creation operator for the function v

n>j?>k

terms of the

angular momentum states, while the operator

-iTT*

2irrHto

B

serves as an annihilation operator.

217
MATHEMATICAL APPENDIX VI

We give here a proof of the following identities:

e £

V

n.i.k(r

i yva.l.k( *2) = 2^nT

i =

x y - x y,) —

X e e L (v ) , n i c

I,

Vn+l,i,klr

i )Vn,i,k tr 2

; ~ „(*+!)• ^T\\ &-

imc

°B

" V

12 ^f z

z» — ^"Yi1 T d

T

. X e e 5v^ L

n+l (Vi

" V 1 T / • • "ON. I ^i.k'A+u.W s w(n + i). ^TTI VT5*") (6_1)

lk(2 rZ

2

} IT" (x

l y2" X

27l

} T d

T

, , X e e -d^^n+l^iz' '

v 2 2 Z,->. # -> 2+1+ / R

v (r ,)v (r ) = ==—- ( —

i = Q n + 2,i,k 1 n,i,k v Z' ff (n + 2)! \/(n + l)(n + 2) ^

Zfi

imu

B

" v

12

ik(Z l" Z 2

} ""2T~

(X

i y2- X

2y

i

} T d

2

T

,

. Xe e 71- L

n + 2

(V

12 ) ' dV

12

218
(X, -X, ) > v (r W (r ) = —

( — \ L n,i,k v 1' n + 2,i.k v 2

; ff ( fl T2)^(a + i)(a + 2 ) V

Z* /

i =

x

lk(2 i- 2

2

} -ar-

(x

i y2" W t" d

2

X e e e — L ± ,(v„) , dv

2 n + 2* 12'

12

and

00 V - - mw

i > v , (r )v (r ) =

-ft ' i =

i = o

**> CO £

V

fl+ U,k( V\lk(r

i )= I Vu^n.U*,/^ ' = x - (6-2)

oo cc I

V

n + 2,I,k(r

i )Va,l,k(r

i )= £

Vn,i.k(r

i )V

n + 2,Ik(r

i

) = '

SL = i =

where the v (r) are the functions developed in Appendix V

n, x , k

(5-35, 42),

mu

B

12 2fi

2 2

(X, " xj + (y, - y)

2 «- 1 2' w l '2'

(6-3)

mcoB

(X, , - X, J (x, - x ) , 2h x 1+ 2+' * 1- 2-

219
and L, (v) is the Laguerre polynomial

v d , -v n. L % v) = e —— e v ) n dv a

(6-4)

= nl J ±

Li ( n

l)

k

n! vk

2 ' (n-k)! (k!)

k = which satisfies the differential equation

d

2

L (v) dL (v) v

T

) + (1 - v) - J

1 + n L (v) = ,

• dv dv n

and the recurience relation

/- L (v) - L (v) = —i~ 4- L ^.(v) . (6-5) dv n a ' (n+1) qv n + 1

'

(See reference (45). ) From equation (5-35) we obtain the relation

1 +1 - n

mco . ik(a - z ) 'n.i.k^l^ii.i.k^z' irnli! ^ 2fi ^

v_ a Jslv. : AxA rrf-zP)

(6-6)

mw rnto -ifi-.

(X

l+

X

I- + X

2-

X

2+V_9i_\

n

, .i " "4^ <X

1+

X

1- + X

2-

X

2+

* e i T~ K I x, x^ ) e

220
Thus

I V./i^i./z1 = ^r("d^)

mwi_fv ik(z - z ) e

X = mw_

( x, , x, + x-, xo.) , ->* Q

2

6fi v 1+ 1- 2- 2+'/_2^_ 8

B 1- 2+

mco —-— (x X - X, X, - x„ x ) 2ii

V 1- 2+ 1+ 1- 2- 2+'

X e

since

and

co o V x X

1 TT =

e

' i =

TM . mw v ^^ x, x„

B \ 2tf 1- 2+

x, x^ ) = e m-k 2-n "1- 2+

1=0

Now, the following identity will be proven by the method of mathematical induction:

-mw

"ST (X

1-

X

2 +

" X

l +X

l-" W/ 2* 8

2

e '/2fi_ 3 \

( muB

3X

l. B 1-

3 V2+'

' ( X, X„" *,.,X , " X. XoJ ( 6 " 8 > v 2£ x 1- 2+ 1+ 1- 2- 2+

x e

L

n

" mW

B

221
The expression (6-8) certainly holds true for n = since L (v) = 1 from (6-4). Now, if we assume (6-8) for arbitrary positive integer

n, then

B / „ , n+1

-— (x X - X, X, - X^ X ) ^ i. 2-K 1- 2 + 1+ 1- 2- 2+ V 2^

e rB

ax

i8x2y

ma> — B , » zr- (x, x - x, x, - x„ x„ ) 2ft

v 1- 2+ 1+ 1- 2- 2+'

X e

-mco -*r (x

i- x

2+

- x

i +xi-- x

2- x

2+) /

,: = e mw d

mojB / \ v •2T- (x

iX2+

- X

l+Xl--

X

2.X

2+

) , X , , e VV

12)

(6-9)

I ,

,

._,> , , nl2 . . nli

= {

(1 ~ V

12 } VV12} + (X 1-- V 3 V

+ ^2+

" X

l +

} -T^7

3L_(v 10 ) aL_(v,

2

)

2^

+

a

2

L (v,j

n 12' "\

1-

8X

2+ I

U-v.J L (v ) +

12 n 12

moj 3 x

2* ^12 , /

V

12

+ (x, " * ) mco 9x 3x 1- 2- 3x B 1- 2+ J.

+ (x, , - x,J

8V -12 - ^u'

2+ 1+ 8x. dv. 2+ J 12

+ -2L III2

. Hi! d2L

n

(v

12 ) 1 mwB

9X

1-

9X

2+ ' dv 2

J

222
where

mu

B

(x. - x., ) (x, - x,J .

Now, from (6-3)

12 2-6

x 1- 2- x 1+ 2+

( x n , " x-> ,) » 8V

12

mc

B

8 *1- 2tf

8V

12 B

8X

2 + 2«

,2

9 V

12

mW

B

aX

l9X2 +

2ft

2fi M+ 2 +

(x, - x, )

'

and thus the expression (6-9) becomes

-moo

(6-10)

(x. x -xx - x x ) n + 1 2<n

v ~l- 2+ 1+1- 2-2+'/ 2*

(

mco 9x, 3x

B 1- 2 +

mooB

X e 2* tXlX

2+

- X

l+

X

l-"

X

2-

X

2+) (6-11)

dL.(v,

2

) /(1-v.J L (v ) + (2v -1) --g-i

12 nl2 12 ^v.

d L(v,J

12 dv2

J 12

Using the series expression (6-4) for L (v), one obtains

n dL (v) d

2

L (v)

(l-v)L (v) + (2v-l) —p v n dv , \c. dv

n .k. ..2 k k+1 . 2 k- 1 = V (-D (n!) l(l+2k)v -v + k v 1 (6 _ 12)

k = (n-k)J (k!)

I

(-l)Vl)! 2 vk

k =

(n + 1 -k)I (ki)

2 = L

n + l (V >'

223
and thus equation (6-11) reduces to the form

-mu)

e / 2h 8 \ K ax

i8xzJ

(6-13)

mco —->— (xx -XX -XX ) „ 2fi v 1- 2+ 1+ 1- 2- 2+' , . n + 1 12

Thus, since the result (6-13) is equivalent to (6-8), one may conclude

that the expression (6-8) has been proven valid for arbitrary positive

integer n. Therefore, equation (6-7) may be given as

e 2/

Vn,i,k(r

i )Vn,i,k(r2

) ~ 2Trhn!

i = mu

B

4h

(2x, x., - x, x, - x^ x_ ) 1- 2+ 1+ 1- 2- 2+

X 6 VV

12'

moo mco ik(z, - z\ —-r? (x^ x, - x, x„ ) B a

v 1 2' 4h v 2+ 1- 1+ 2-

e e 2irhn!

" v

r2

X e 2 L (v ) (6-14) n ic. imw

mc

B

ik (Zi

- Z2 ) -gff-CVz- VlJ e e

2Tr-Kn!

V

12

Xe 2

Vl2>

Equation (6-14) is the first of the identities (6-11).

224
From (5-35) one has the relation

mcj^ x ikjz, - z ) Y v (TW * (7) = -^- (

^ e

X

/_, ri + l,i,kl r n.i.k* 2

; -s/aTi ^ZirSnl^

i = (6-15)

mu

B

(x. x + x x ) n X e 4n x 1+ 1- 2- 2+; / / 2-n 3 \ / 2n 3

mW

B

8xl-AmwB

9X

1-

8X

2 + mco

X e

- (x, X - X, X, - X X ) 2n v 1- 2+ 1+ 1- 2- 2+'

by a procedure similar to that for (6-7).

By comparison with (6-8), we see that (6-15) may be written in the

form

mcj ik(z - z ) V -*. * -

l_j

Vn + l,i,k

(r

i )Vn,i,k

(r

-l

i -

2> = ^TTT ^*nlB\ ~~" % 1 V e

mW

B'

X e

(X

1+

X

1- " X

2-

X

2+} / 2-n 3 mcoB 3Xli

mo.

B

(x X - X X - X X ) M 2tf

v ~l- 2+ ~1+ 1- ~2- 2+'

X T 6 L

n

(V

12 ) (6-16)

"UXT , " Xl ,) J"00 TM V ^(Z 1 " Z

7

) e

_

x 2+ 1+' / B_\

" tt ni Vn+1 V

2

'H /

mto

X e —v— (2x, x^ - x, x, - x„ x_, ) 4ii v 1- 2+ 1+ 1- 2- 2+' VV

12>

1 9V

12

dL

n

(V

12 ) 2-n 1 moo (x., - x, ) 3x, B

v 2+ 1+' !

dv

12

225
2

i(x„ - x, ) m^v iM z , v " Z -J 2 + 1+' / B\ v 1 2' ir(n + l)! \ln + 1 2ft

imW

B

" V

12 IT^z-Yi1 T dL

n4-:

(v

iz } X e e ; , av,

12

from (6-5) and (6-10). Equation (6-16) is precisely the second of the

identities (6-1). The third identity in (6-1) is found from the complex

conjugate of (6-16), with the subscripts 1 and 2 interchanged.

The fourth of the expressions (6-1) is given from (5-35) as

) v (r )v (r ) = ( £

l_j n + 2,i,k 1 n, i,k 2 -rra! *V(n + l) (n + 2) V

2^

i = mu

v

ik(2

i - V if {x

i +x

i- * W/aT 8

<" / « a 2

X e e

// 2ft 9 \ / 2ft 3 \

(m^ 8xJ^B

6xu 8xJ

ma>3 . . 2ft (X1-

X2+" X

l+

X

l-~

X

2-

X

2+' (6-17) X e

*

mu . ik(z, - z\ -r (x, x, + x x ) 2

; 4K V 1 + 1- 2- 2+' = 1__ e /"W

B\ ^1

irn! N/(n + l)(n-r2) V

2* /

*Fv axi J

mco 2 B . -2l" (X

lX

2+

- X

l+

Xi^ X

2-X

2+i T , , VV

12>

226
mco

tt n! N/(n+l)(n + 2) \ 2& ^

mco ik(z -z ) B

(2x x -x x, -x x ) B\ 1 2' 4# v 1- 2+ 1+ 1- 2- 2+'

e e

9* / 9V 1A2 d L <V 19> wbV 9x i_; dv

2

2

2*

moo

a v

\ 3x*_ /

3v, dL (v„)

+ 2(x_, -x, ) 12

B \3x 2+ l + '3x1-J

n 12'

dv

12

-Xr{X2+- Xl+ ] L

n

(V

12 }}

mco

<*2+-*i+

> /^bn2 ik(z r

z

2

} -4^2x

i,

x

2+

-x

i+*i--

x2-*2+) e e

it nl -V(n+l)(n + 2) V

2K

{dv

1;

- dL

n

(v

12 ' dv

" (X

2 +

- X

1 +

12

.2

L (v ) n 12

r dL (v ) n 12

dv

12

- L (v ) n 12 I

mco

B,

Tr(n+2)W(n+l)(n + 2)V 2*

/^B\2 ^Vz1 i ^T(x

l y2- X

2yl )d L

n+ 2

(V

12)

dv

12

X e

2

by the same type of argument as was used for (6-7). Equation (6-17)

is the form of the fourth identity (6-1). The fifth identity in (6-1) is

obtained from the complex conjugate of (6-17) with the subscripts 1 and 2 interchanged.

The identities (6-2) are easily obtained from the expressions

(6-1). The second of the equations (6-4) leads to the following equalities:

lim LT I \ . (v) = n.» v—O n lim —d

-L(v)= -nJ n

v-~Q dv n

(6-18)

227
v— , 2 n 2

dv

Thus, in the case r = r" - r, the equations (6-1) reduce to the

relations (6-2).

The identities

3

3

V i— .—. —. / B\ + Z^V

n,i,k(r)V

n,£.l,k«ri H^~7~

i=0

3

^ ,mu 2 x Z^1 ^^.^ VtkP» = (ysr) v

£=0

3

• V _ / m

"-R\2 X Z^ V

n,£,k<T)V

n,i+ l.k<

T> = (-ST) T

(6-19)

co „ 2

V __ / mWR\2 X

> N/£(FTj v* (r) v (r) = ( —£) — Li n, £-2,k n, I, k' ^ 2h y u

co _ 2

) sTKiTT) v* (?) v _ = (?) = ( _E\ -t

Zv n,£,k n, i-2,k ^ 2fi i it

co 2

r-^ .rnco .2 x ) MU+DU + Z) v* (?) v (?) = r _£A -i

Z_/ n, £+2, k n, £, k I 2ft / tt £=0

co 2

) V(|+l)(|+2) v* (?) v (?) =

f -55T ) — £_, n, £, k n, £+2, k V 2ft J

tt £=0

are obtained from (5-50) and (6-2), by taking simple combinations of

terms.

228
MATHEMATICAL APPENDIX VII

Part 1: We give here a proof of the identity

(7-1)

where e is an "infinitesimal. " Given some function f(x) which is continuous in the neighborhood of x = 0, we consider the following integral:

oo _oo oo f dx ^_ f(x) = r _^i_ dx±i£ c ffi

J-oo x Tl£ J.co(x2

+e2) J.oo.(x2 +

. (7-2)

E ) Now, by definition

, c °°^i dX » iim - c

°° x f(x) dx un\rf~

G f< :<

> - r°° f( x)^

loo * ^--O^T^^-e-O+Ll^ —ax.^—dx

"IOO dx f( X) P( £ )

.

With the change of variable

we obtain the result

x = ez .

lim C e f(x) lim f

f(e:ez) dz

dz

1 +

= it f(0) .

(7-3)

(7-4)

(7-5)

229
Therefore

f(x) = P

00dx^-± iTT f(0) , • 00 and we see that

xtu Pl£) ± itr 6(x), e—

+ V

(7-6)

(7-7)

Part 2 : The Cauchy Integral Theorem may be utilized in many integrations of importance in physical theory. We are particularly concerned

here with integrals containing "complex exponentials" in the integrand.

From complex variable theory, we have (see reference (41))

^\ dco f(oj) 1 ^ du> f(oo)

Now integrals of the form

U) lh4 a> - z

I

oo -icot

dco g(oo) e

(7-8)

(7-9)

(<o - z)

whore g(o) is analytic in the corr.pl ex w plane, can be evaluated via (7-£5-

For t > 0, we choose an integration contour in the lower half <*> plane of

the form of a semicircle;

f, -iwt

f du g(<j) e g

2 srr^r

lim r

R -iwt

dto g(o))e

(u> - z) (7-10)

1

• r- 7r • lim ( -\ Re d6 g(R e ) e (R e" ie

- Z)

, t > ,

+ oo

230
The exponential in the integrand (7-10) causes a vanishing contribution

to the integral about the "infinite" semicircle, thus

iiiit icot

r d« p(M) e = r du g( u) e t>0

J.OO (w - z)

^

(w - z)

(7-11)

- -izt

-l 2tt g(z) e , z in the lower half plane

, z in the upper half plane

from (7-9). Similarly, for t<0 an integration in the upper half plane

results in

£

dco g(co) e («.- z)

icot = +

dco g(oo) e

(co - z)

-ICJt

-, t<0

(7-12)

{

i 2tt g(z) e~

izt

, z in the upper hall plane

, z in the lower half plane.

231 • US. GOVUaMEBT I>»I«TIM& OFFICE : i**4 0-'2*-63)