(Advanced book classics) richard p feynman quantum electrodynamics westview press (1998)

Description: This classic work presents the main results and calculational procedures of quantum electrodynamics in a simple and straightforward way. Designed for the student of experimental physics who does not intend to take more advanced graduate courses in theoretical physics, the material consists of notes on the third of a threesemester course given at the California Institute of Technology. Điện động lực học lượng tử (Quantum electrodynamics) Trong vật lý hạt, điện động lực học lượng tử là lý thuyết trường lượng tử tương đối tính của điện động lực học. Về cơ bản, nó miêu tả cách ánh sáng và vật chất tương tác với nhau và là lý thuyết đầu tiên kết hợp được các tính chất của cơ học lượng tử và thuyết tương đối hẹp

ADVANCED BOOK CLASSICS

David Pines, Series Editor

Anderson, P W., Baric Notions of Condensed Matter Physics

Feynman, R., Photon-Hadron Interactions

Feynman, R., Quantum Electrodynamics

Feynman, R., Statistical Mechanics

Feynman, R., The Theory of Ftrndamenral Processes

Norieres, P*, Theory of Interacting Fermi System

Pines, D., The Many-Body Problem

Quigg, C., Gauge Theories of the Strong, Weak, and Electromagnetic Interactions

RICHARD FEYNMAN

Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book

and Perseus Publishing was aware of a trademark claim, the designatians have been

printed in initial capital letters

All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, elecmnic, mechani-

cal, photncopying, recording, or otherwise, without the prior written permission of

the publisher Printed in the United States af America

Cover design by Suzanne Heiser

Editor's Foreword

Addison- Wesley's Frontiers in Physics series has, since 1 96 1, made it possible for leading physicists to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics-without having to devote the time and energy required to prepare a formal review or monograph Indeed, throughout its nearly forty-year existence, the series has emphasized informality in both style and content, as well as pedagogical clari-

ty Over time, it was expected that these informal accounts would be replaced

by more formal counterparts-textbooks or monographs-as the cutting-edge topics they treated gradually became integrated into the body of physics knowl- edge and reader interest dwindled However, this has not proven to be the case for a number of the volumes in the series: Many works have remained in print

on an on-demand basis, while others have such intrinsic value that the physics community has urged us to extend their life span

The Advanced Book Classics series has been designed to meet this demand It will keep in print those volumes in Frmliers in Physics or its sister series, Lecture Notes and Suppkments in Physics, that continue to provide a unique account of

a topic of lasting interest And through a sizable printing, these classics will

be made available at a comparatively modest cost to the reader

These lecture notes on Richard Ft;ynnran8s Caltech course on Quantum Electrodynamics were first published in 1961, as part of the first group of lec- ture notelreprint volumes to be included in the Frontiers in Physics series As is the case with all of the Feynman lecture note volumes, the presentation in this work reflects his deep physical insight, the freshness and originality of his approach to quantum electrodynamics, and the overall pedagogical wizardry of Richard Feynman Taken together with the reprints included here of

vi E D I T O R " S F O R E W Q R D

Feynman's seminal papers on the space-time approach to quantum electro- dynamics and the theon, of positrons, the lecture notes provide beginning students and experienced researchers alike with an invaluable introduction to quantum electrodynamics and to Feynman's highly original approach to the topic

Bavid Pines Idrbana, Elf inois December 2 997

Preface

The text material herein constitutes notes on the third of a three-semester course in quantum mechanics given at the California Institute of Technology

ter were discussed during the preceding semester These are also included, as

the first six lectures The relativistic theory begins in the seventh lecture The aim was to present the main results and calculational procedures of quantum electrodynamics in as simple and straightfaward a way as possible Many of the students working for degrees in experimental physics did not intend to take more advanced graduate courses in theoretical physics The course was designed with their needs in mind It was hoped that they would learn how one obtains the various cross sections for photon processes which are so important in the design of high-energy experiments, such as with the synchrotron at Cal Tech For this reason little attention is given to many aspects of quantum electrodynamics which would be of use for theoretical physicists tackling the more complicated problems of the interaction of pions and nucleons That is, the relations among the many different formulations of quantum electrodynamics, including operator representations of fields, explic-

it discussion of properties of the S matrix, etc., are not included These were available in a more advanced course in quantum field theory Nevertheless, this course is complete in itself, in much the way that a course dealing with Newton's laws can be a complete discussion of mechanics in a physical sense although topics such as least action or Hamilton's equations are omitted The attempt to teach elementary quantum mechanics and quantum elec- trodynamics together in just one year was an experiment It was based on the

viii P R E F A C E

idea that, as new fields of physics are opened up, students must work their way further back, to earlier stages of the educational program The first two terms were the usual quantum mechanical course using Schiff (McGraw-Hill) as a main reference (omitting Chapters X, XII, XIII, and XIV, relating to quantum electrodynamics) However, in order to ease the transition to the latter part of the course, the theory of propagation and potential scattering was developed

in detail in the way outlined in Eqs 15-3 to 15-5 One other unusual point was made, namely, that the nonrelativistic Pauli equation could be written as on page 6 of the notes

The experiment was unsuccessful The total material was too much for one year, and much of the material in these notes is now given after a full year grad- uate course in quantum mechanics

The notes were originally taken by A R Hibbs They have been edited and corrected by H T Yura and E R Huggins

R R FEWMAN Pasadena, California November 196 1

The publisher wishes to acknowledge the assistance of the American Physical Society in the preparation of this volume, specifically their permission to reprint the three articles from the Physical Review

Contents

Editor's Foreword

Preface

Discussion of Fermi" mehod

Solution of the Maxwell equation in empq space

Relativistic partide mechanics

Ref a t i ~ s t i c Wave Equation

Units

Ktein-Gcrrdon, Pauli, and Dirac equations

Alpbra of the y matrices

kuivalence tramformation

Relativistic invariance

Hamiltonian form of the Dirac equation

Nonrelativistic approximation to the Dirac equation Solution of rhe Dirac huation for a Free Particle

Defirtieion of the spin of a moving elecrron

Norrnalizatian af the wave functions

Methods of obtaining matrix elements

Intepretation of negative energy states

P o t e n ~ a l Problems Itn, Quantum Eleetradynamics

Pair creation and annihilation

Consewation of energy

The propagation kernel

Use of the kernel K, ( 2 , l )

Transition probablility

Scattering af an electron from a coulomb potenrial

Galccllation of the propagation kernel for a free particle Momentum repreenration

C O N T E N T S

Relativistic Treatment of the Interaction of Particles with L i h t

Radiation from atoms

Scattering of gamma rays by atomic electrons

Digression on the density of final states

Cornpton radiation

Two-photm pair annihilation

Positron annihilation from rest

Brernsstrahlung

Pair production

E&cts of screening of the coulomb fieid in atoms

Interacdon of Several Electron

Derivation of the "mules" of quantum electrodynamics

fro-photon exchange between eiectrons andlor positrons

SeXfeenergy of the electron

Method d integration of integrals appearing in

quantum electrodynamics

Self-energy integral with an external potential

Scattering in an ex ternal potential

ResoXut ion of the fictitious "inbred catastrophe"

Anocher vproaclx to the infared di&uXcy

Egect on an atomic electron

Chsed-loop processes, vacuum polarization

Scattering of light by a potential

Padi Principle and the Dirac Equation

Replcints

Summary of Numerical Factors for Transition

Probabilities, Phys Rev,, 84, 123 (1951)

The Theory of Positrons Phys Rev., 76, 749-759 (1949)

S p a c e - x ~ Approach to @anturn Electrodynamics

Phys Rev., 76,169-189 (1949)

This page intentionally left blank

Interaction

The theory of interaction of light with matter i s called wanturn electro-

d y n m i c s The subject i s made to a p m a r more difficult &m it actually i s

by the very many equivalent methods by which it may be formulated, One af the simplest is &at of Fermi We shall take another starting point by just postzzlating for the emission o r absorption of photons In this form it i s most immediately appXicabXe

Suppose a11 the atoms of itbe u d v e r s e a r e in a, box Classically the box may be treated a s haviw natural modes d e ~ c r i b a b l e in terms of a distribu- tion o-f: harmonic oscillatars with coupling between the oscillators and matter

The trarxsfUan to wanturn electrodynmics involves merely the assump- tion that the oscifladars a r e quantum m e c h a ~ e a l instead cif classieal, They then have energies (a + X / 2 ) b , a = 0, 1 ,, with zera-point e n e r w 112fiw

The box is considered to be full of phstom with a~ distribution of energies

&W The interaction of photom with: matter causeha the number of photons of' t y m n t a increase by & l [emission or absorption)

Waves in a box can be repressxrted a s plane e t a d i a g waves, spherical,

waves, o r plane rmning wavtsa exp (iK * X$ One can say there i~ an instan-

t Revs Modern Phys ., 4, 87 (1932)

4 Q U A N T U M E L E C T R O D f x i N A M f C S

tanems Coulomb interaction e 2 / r l j between all charges plus transverse

w v e s only m e n the Coulomb forcea may be put into the Schr6diwer equa- tion directly a e r f o m a l means of e a r e s s i o n a r e M

in Hamiltonian form, field o p r a t o r s , etc

Fermi's technique leads to an infinite self-energy term e 2 / r f i It i s poa- sibie to eliminate this term in s u i a b l e caarr3inate systems but then the trans- verse waves contri bute czn i d t n i ty (interpretaaon more obscure) This mom-

Without justification at this time the "laws of q u m t m ele,ectrdynamics9" will be stated a s follows:

1, The amplitude that m atomic system will absorb a photon d u r i w the

process of transition from one state to another 18 emctly the same a s the

amplitude that the same transition will be made under the influence of a p-

tential equal to that af a classical e l e c t r o m w e t i c wave representing that photon, provided: (a) the classical wave i s normalized t a represent an en- ergy density equal t a b times the probabilty per cubic centinneter of find-

e - hU" and e* ' w t , and only the e- ""tart is kept; md (c) the potential acts only once in mrturbatlon; that is, only terms to first order in the electro-

magnetic fietd strength should be retained

mplacing the word "absorbedM by 'kmit?' in rule X r s q d r e s only that the wave represented by exp (+iut) be kept instead of exp (-iwt)

2 Tbe n u m b r of states a v a i l h i e p r cubic centimeter of a given po2ar- izatlon Is

Note this ia exactly the same a s the number of normal modes per cubic cen- tirneter in classical theory

3 , Photons obey Boee-Einstein atatistlcs, That is, the states of a csllese- tion of identical photons must be symmetric (exchmge photons, add m p l i - tudes) Also the statistical w e i e t of a state of n identical photons i s l in- stead of the elassieal a!

Thus, in general, a photon may be represenbd by a solution of the classi-

Although many forms of expression a r e p s s i b l e i t is most convenient to describe the electromagnetic 8eld in terms of plane waves A plane wave can always be represented by a vector potential only (scalar potential made zlero by suitable gauge transformation) The vector potential representing a real classical wave i s talqen a s

I N T E R A C T I O N Q F L I G H T W I T H M A T T E R

A ZZ a e cos ( u t - f?;*x)

We want the nctrmalizatian of A to correspond to unit probability per eu- bic centinneter of findiw the photon Therefore the average e n e r a density should be 60

Now

IFf = (l/e) (&A/ a t) = (U a/e) s sin (w t - K; * X )

for a plme wave, Therefore the averscge energ;y density is equal to

( 1 / 8 r ) f l ~ 1 ~ + 1 ~ 1 ' ) = (1/4a)(w'a2/c') sin3(wt - K*@

Setting this equal to t-icc: we find that

Thus

-

- er f m p f-ilut - K * xll .c exp [+ilot - K * X ) 1)

to be

For emission the vector poLentiaL is the s m e except for a positive exponen- tial

Example: Suppose an atom i s in an excited s t a b \ti with energy Et and

m h e s a transition, to a final state Jif with energy E r , The probability of

transition p r second i s the same a s the probability of transition m d e r the Influence of a vector potential a8 expf-t-ifwt - K-X)] r e p r e s e n t i ~ g the ernit- ted photon Aecasding to the laws of quantum mechanics (Fermi's golden rule)

Trans prob./sec = 2n/A /f(potential)i l2 * (density of states) Density of states =

6 Q U A N T U M E L E C T R Q D U H A M I C S

The matrix element U f i = /f@otential)f/2 is to be computed from pertur- bation theory This is explained in more detail in the next lectws First, bowever, we shall note that more tjhm one choice for the potential may @v@

the s m e physical results (This is to jusWfy the possibi&t;y of always chws- in@ Q, = 0 for our photon,)

Tjzzz'rd Lecture

The representation of the plme-wave photon by the potantials

i s essentially a choice of 6"gauge." The fact that a freedom of choice edats results from the i w a s i m c e of the Pauli equation to the; q u m t m - m e c h d c a l gauge t r m f o r m

classical, where, if

and if X is my scalar, then the substitutions

leave E and B invariant,

h quantum mechanics the additional transformation of the wave function

Pauli equation i s

I N T E R A C T I O N O F LIGHT WITH M A T T E R

The partial derivative with respect to time introducaes a term

( @ X / B t)Y and this may be included with 4e-jx @ Therefore the sub-

stitutions

The vector potential A as defined for a photon enters the PauB H m i t - tonian a s a perturbation pohntial for a transition from s t a k i to @tale f

rersults in the matrix sbment U,$ given by

This emression indicates that the prturb2ttrion h a the a m 8 eB@ct ae a time-

ergies are, respectively, E ~ and ~ El w As ~ i s well known? the most impor-

ta& contribution will come from the states such that Ef = Ef - wR

Usim the previous results, the probabilly- of' a trmsritioxl per second is

f See, for exmplti3, L D, Lznndau and E M Ufshitz, "@atnhm Meehan- tca; Non-Relativistic Theory," Addison-Wesley, R e A i q , Massachu-

setts, 1958, Sec, 40

Q U A N T U M E L E C T R Q D Y N A M I C S

To determine U f i , write

Because of the rule that f i e potential acts only once, which Is the s m e

a s requiring on& first-order terms to enter, the term in; A * A does not en- ter this problem Making use of A = aa exp [-i ( a t - XC X)] and the two operator relations

where K-@ = 0 (which follows from the choice of gauge and the Mmtvell

This result is exact, It c m be simplified by uai% the ajlo-calked '4dipolet" approximation To derive tMs approximation consicler the term

(e/2mc) (p .e e f ' "X), whiclr is Ehci or&r of the velocity of an electron in the atom, o r the current The aponent can be exwaded

K x i s of the order ;ao/ic, where Q = dimension of the atam and h = wave- Xewth, I[f +/A<< I, all terms of higher ordfits than the first in ao/h msby be neglected To complete the dipole a p p r o ~ m a t i o n , it i s also necessary to neglect the last term, This is e ~ i l y done since the last term may be t&en

as the order of fiK/mc) = (2i~c/rnc') * (mv2/2mc2) Although such a term i s negligible even this i s an overestimate, Wore correctly,

I P J T E R A G T X O N O F L I G H T W I T H M A T T E R

(efif/2mc)cr (K x a) e + iK " X v i e x [martsix element of

A g o d appro&xnatian allowa the separation

Then to the accuracy of this apprsximiation the integral is

l$ *W +I Uf*(a = (K x p ) ) ~ I d v o l = o

For the present, the dipole approximation i s to be used Then

where Xfi = l $r * X @$ d vol The total probability i s obtained by inte-

grading Ptf aver din, thus

e2u4

Total prob./sec = j a 2 7 (e lf )' dS2

(2x1

10 Q U A N T U M E L E C T R O D Y N A M I C S

me term s xfi i s resolved by noting ( F i g 3-21

ixti el = lxfil sin 8

FTG 3-1

Substituting for 2,

Total prob./sec = 5 G 3 1 1 x ~ l 2

&eorptioa of lLig.ht, The amplitude to go from state k to s t a b Z in time

T (Fig, 4-1) i s given from perturbation t;heoxty by

&is time dependence m d p r f m m i n g the intt;;grat;fon,

the transition probability i s given by

This is the probability that a photon of frequency w traveliw in direction

(0, $B) will be, absorbed The dependence on the photon direction i s contained

in the matrix element u j k * Far e x m p l e , s e e Eq, (4-1) far the directional dependence in, the dipole approximation

If the incident radiation contains a r a w e of fremencies a d directians, that is, suppose

probabilit;y that a photon i s present with fre- ency w to w -t dw and in solid aqXe dsZ about the direction f@,@ci3)

and the probability of aboorption of any photon travicsling in the (B,@) direc-

tion is desired, it is necesrJary to integrate aver all fr-equencisa, Tbis ah- sorption probability is

when T is large, the faotor a i n ? ~ ~ / z f i ) has an appreciable value only for Ew near El - Ek, ancl P(@, @,Q>) wf ll km substmtially eomtant over the small r a e in o which contributes to the integral s o &at i t may be t&en out af the integral Similarly far ulk, ao that

Trans prob = 2 ~ ( % ) - ' lulr 1' $)dQ

where

12 Q U A N T U M E L E C T R O f ) ' Y N A M P C S

This can also be written in terms of the incident iatensity (energy crossing

a unit area in unit time) by noting that

U s i w the dipole approximdion, in which

the total probability of absorption (per second) i s

It is evident that there is a relation b t w e e n the probability of sgontane-

ous emission, with aceompaying atomic transition from etate 1 to atate k,

Probability of spontaneous

emission/sec

8 t a k k to state l, Eq @-l), although the initial and final states a r e re-

versed since /ulkf = i/uklf This relation may be stated most simply in terms

of the concept of the probability n(u, 8 , (13) that a pmtieular photon state i s

accupied, Since there a r e (2trepo2dw df2 photon stales in frequensy range

diw and solid angle dBZ, the probability that there is some photon vvit-hin this

range is

Ewreessiw the probability of absorption in terms of n(w, 8, $1,

Trans prob./sec = 2 n ( ~ ) * lulk nfw,~,(13)(2nc)-~ wk12 d o

(4-4) This equation may be indewreted a s follows Since a(w,@,cp) is the prob-

ability thiit, a phdon state i s occupied, the r e m a n d e r of the h r m s af the

rfght-hmd side must be the probhility per second that a jphoton in a& s t a b

will be absorbed Comparing Eq, (4-4) with the rate of spontanems emis-

sion s h w ~ &at

I N T E R A C T I O N Q F L I G H T W I T H M A T T E R

h what follows, i t will be shown that Eq (4-4) is correct even when there is

a possibility of more &an one photon per: state provided nfo,8,$) is t&en a s the mean number of photons W r state

Lf the initial state consists of tulo photons in the same photon state, i t will not be possible to a s t i n w i s h W1em md the statistical weight of the initial

s t a b will be 1/2 ! However, the m p l l h d e for ab~orption will be Mice that for one photon TaXcjng the statistical weight times the square of the m p U - tude for this groesss, the transition prabrtbility per second i s found to be

twice that for only one photon per photon state m e n there3 a r e Lhree N o -

tons per initial photon state md one is absorbed, the f o l l ~ M n g six proceases

(shown on Fig 4-22) can occur

Any of the *ree incident photons rnw be absorbed and, in addition, there is

the possibility that the photons which a r e not absorbed may be interchanged

The statistical wei&t of the initial state is 113 !, the statistical weight of the final state i s 2/21 , and the amplitude for the process is 6 Thus the t r m s i -

tion probability i s (1/3 1)(1/2 1) (6)' = 3 times that if them were one photon

p r i d t i a t state, In general, the transition probability for n photons per

initial photon state is n times that for a s i w l e photon per photon state, s o

Eq (4-4) is correct if n(cr/,@,$yj) is t&en as the mean n u r n b r of photons per state

14 Q U A N T U M E I L E C T R O D U M A M I C S

A transition that results in the emission of a photon may be induced by incident radiation Such a process (involviw one incident photon) could be indicated diagrammatieztlly, a s in Fig 4-3

One photon i s incident on the atom and two indlstinwiahable photons come

off The statistical weight of the final state I s 1/2f and the amplitude for

the process i s 2, s o the probability of emission for this process is twice that of Eipontanems emission For n incident photons the statistical weight

of the, initial s t a b is f[nI, the statistical weight of the final s t a k i s

l/(n + 1) f , and the a p f i t u d e for the process is (n + 1) f times the amplitude

n + l times the probability of spontaneous emission The n can be said to account for the i d u c e d part of the transition rate, while the 1 i s the spon- taneous part of the transition rats,

Since the gotentials used in computi ng the transition probability have

ztbiiity depends on the: square of the amplimde of the potential, it i s clear

t b t when there a r e n photonsr p r photon state the eorreet transition prob- ability for abearption wodd ba obhia& by aormaliziw the potential8 ta n

pt.lotons per cubie centimeter [amplitude 6 times a s large) This i s the basis for the validity of the so-called semiclassical ~ e o r y of radiation h

potential normalized to the actual energy In the field, that is, to energy nEw

if there a r e n photons, The correct transition pr&&ilily for e r n i ~ s i o n i s not obtained this waly, however, because it is proportional to n + 1 The er-

r o r corresponds to omitting the spontaneous part of the t r a n ~ i t i o n grab- ability In the semiclassical theory of radiation, the spontmeous part of the emission probability i s arrived a t by general arguments, ixzctuding the fact that its inclusion leads to the observed Planck distribution formula Ein- stein first deduced these re lationshipa by sem iclassicsl reasoning

I N T E R A C T I O N O F LIGHT W I T H M A T T E R

%XecLian Rule8 in the f)5pole Appr~~matton In the dipole approamatlon the appropriate matrix element is

Tlxrs components of of X f r a r e xir, Ygr, zjf and

Selection rules a r e detemined by the conditions that cause t N s matrix ele- ment to vanish, For example, if in hydrogen the initial and final shks a r e

S s t a b s [spherically symmetrical), X+ = O and transitions b t w e e n these slates are "forbidden," For transitions from P to S sktes;, however,

X i f f 0 and they a r e 6iallowed p'

Xn general, f o r single electron transitions, the selection rule is

This may be seen from the fact that the coordinates X , y, and z a r e essen- tially the Legendre polynomial PI If the o r b i k l angular momentum of the initial sLak i s n, the wave function contains P,, But

Hence for the matrix element not ta vanish, the angular momentum of the final stale must be n r ~ 1, SO that its wave function will conkin either l?,,

o r P,."$

F o r a complex atam (more than one electron), the &miltonian i s

H = z ( 1 / 2 m ) P, - (e/c)A(x .)lg + Coulomb terms

a

sum i s over all the electrons of the atom As k s b e n shown, (Pafmn i s the same, up to a consant, a s ( x , ) ~ , , and the transition probability is propor- tional to

In particulm, for two electrons; the matrix element is

x 1 + x 2 behaves under rotation, of coordinat-ea similarly to Wlc? wave fmction

of gome "objectY%iGh unit a w l a r momentum If the "object7' and the abm

16 Q U A N T U M 1 E L E : C T R O D Y b J A M I C s

in the initial state do not interact, then the produet (X* + x2) iEi (xI,xZ) can be

formally regarded a s the wave bnction of a system (atom -t otsject) having possible values of J i -c- l, J 1 , and Jz - l for total m ~ l a r momentum There- fore the matrix element i s nonzero only if Jf , the final a.nwlar momentum, has one of the three values JI * 1 o r Ji Hence the general selection rule

A J = a 1, O,

Parity, P w i t g is the prowrty of a wave function referring to i t s behaviar

upon reflection of all coordinates, That is, if

parity is even; o r if

parity 18 odd

If" in the matrix elements involved in the dipole approximation one makes the change of variable of integration x = X\ the result i s

If the pwidy of %cf i s the s m e m &at of -$!l, it follows that

Hence tlze rule that p a ~ t y must chawe in allowed transitions For a one- electron atom, L determines f&e gmity; tiherefore, L1 L = O would be forbid-

by algebraic, not vector, sum of individual electron m p l a r momfsnta), s o

hL = O transition8 can occur The a- 0 transktions a r e always forbidden, however, since a photon always c a r r i e s a m u d t of mlf;ular momentum All wave functions have ei&er even or d d parity This can be seen from

is invariant under the parity operation Then, if EIJlr(x) = E\k(x), i t i s also true &at H@(-X) = f t ; J i ( - ~ ) Therefore, if the state is nondegeneirab, it bllows that e i a e r ?P f -X) = Jir (X) o r Q (-X) = -S [X) If the state is degen- erate, i t is possible that ?fr (-X) ;c {X) But then a complete solution would

be one of the linear combinations

4? (X) + % ( - X ) even parity

% (X) - @(-X) odd parity

Forbzddelz Llirtcls, Forbidden spectral lines mzty a p p a r in gases if they

a r e sufficiently rarefied, That is, forbiddenneess is not ablsolute in all cases

It may simply me= thzt& the lifetime of the state is much longer &an if it

I N T E R A C T I O N O F L I G H T W I T H M A T T E R ,

were allowed, but not infinite, Thus, if the colliaion rate i s small enough (collisions of the second kind ordbnarify cause de-excitation in forbidden

cases), the forbidden trmsition may have sufficient time t s occur

the dipole approximation replaces emix' at by 1 If Ws vanishes, the transi- tion i s forbidden, as described in the foregoing, The next higher o r quadru- pole approximation would then be to replace e-jK' X by 1 - i/K* X , giving the matrix element

For light; moving in the z direction and p l a r i z e d in the x direction, &is

~ G O I Z L B B

met the trsrnsition probabiljity is proportional to

whereas in the dipole approximation i t was proportional to

Therefore the transition prob&ility in the qua&upole approxhation is at least of the order of (Ka)' = a2/%, smaller than in the dipole approxima- tion, where a i s of the order of the size of the atom, and h the wavelengtLh emitled,

and consequently that

Note that p,z can be written as the sum

From the preceding prciblern, the f i r s t part of p,z ie seen to be equivalent,

up to a constant, to xz, which behaves similarly to a wave function for

18 Q U A N T U M E L E C T R O D Y M A M X C S

2, even pmity, The second p r l i s the

a r e Been to be A J = * 2, +l, O with na parity chawe, This type of radiation

i s called electric quadrupole, The selection rules far the scseond part of p,z

a r e A J = rt I* OS no parity clzawe, md the e o r r e a p n d i q radiation i s eaUed magnetic dipole Hots that unless A J & 2, the two t y p s of radiation cmnot

be d i a t i w i s h e d by the change in m ~ l m momentum or pwity If A J = *l, 0,

Lfiey eirn only bbe d i s t i n ~ i s h e d By the polarizatia~ of thr; raaation Bo& t m a

may occur simultmeously, producing interference

In the case of electric qudrupole radiation, it is impticit in Lhs rules that 1/2 - 112 and O - X trmasitiona are forbidden (siren though &l' may be

& l), since the required chawe of 2 for the vector m m l a r nclomentm i s im- possible in these caaes,

Continuing to higher approximations, i t is possible by a b i l w reasodng

the photon, and the selection rules for parity chmge and cbmge of total m-

w l a r momentum AJ associated wit;h the varicrus multipole order8 (Table 5-11,

Electric m p t i c EL~trie &wetic Ekectric lMultipole dipole dipoh quadmpolt.: qwdrupole octupale

Actually all the implicit selection rules for A J , which become numerous for the higher multipole orders, can be expressed explicitly by writing the selection rule a s

where 2' is the multipole order o r 1 i s the vector change in angular mo-

m e n t ~ ,

XNTIERACTllOrJ O F L I G H T W I T H M A T T E R If)

Xt h r n s out that in so-eal led parity-favored trmsitions, wherein the p r d - uet of the initial and final parities is ( - l ) l t - and the lowest possible mul- tipole order i s J f - J i , the transition probabilities for multipole typea con- tained witEtin the dashed vertical gnes in Table 5-1 a r e r o w h l equa1.t In !r

parity-unfavored transitions, where the parity product i s (-1) t -ji and

the lowest mullipob order 163 I Jf - Jy 1 + L, this may not be true

ber of atoms p r cubic centimekr in two ~statee, say 1 and k, i s giiven by

the system i s in eqtliubrim, the number of atoms goiw from s t a b k to l per anit time by absorption of photons b must e q u d the number goi% from

l to k by emission If n, photons of frequency w are present per cubic cen-

a m e t e r , then polaabili.t;les of absorption are proportional to n, mci proba- Bility of emission is proportional to n, -t 1 Thus

Thia i s thds Planek black-bdy distribution law

photon being s c a t b r e d by an atom into a new direction ( a d possibly e n e r e )

(see Fig, 6-11 This may be considered a s the absorpt;ion of the incoming photon and the ernisrsion of a new lpfroton by the atom The WO photons t&ng part in the phenomenon a r e represented by the vector potentials

The number to !X determined i s the probability that an atom initially in state

k will be left in state 1 by the action of thds pert;urba#on LP r=: At + AS in the

obscure reason the magnetic radiaiLion predominates for each order of muI- tipole

Q U A N T U M E L E C T R O D Y N A M I C S

time T This probability can be computed just aa any transition probability with the use af Alk, where

where r pins are neglected,

In each i n b g r a l defining AXkr each of the two vector potentials must ap-

p a r once and only once Thus, in the f i r s t integral the term p e h of fJ will

not appear in Ulk The p r d u e t A A = (A1 + A2) - (Ai + Az) will conbibale only its c r a s s - p r d u c t term 2A1& The second in&gral will have no con-

t r i h t i o n from A A , but will be efie sum of two terms The f i r s t term con- tains a U1, based on p * Az and a Unk based an p At The second h w Ul,

based on p AI and Unk cm p *AZ The Lime sequerzcrzs r e s u f t i w in these two terms can ba represented schematically a s shown in Fig, 6-2

lai l

Then the resulting integral i s

where L& = (EI +&W - Ek - i), and the phase angle c# i s independent of n

A term wi& the denominator given by (E, - b 1 - Ek)(EI + b 2 - E,) has been

El + m Ek + tiwl are important The final result e m be written

where iM1 I s determined from Alk by integraaxlg over wz and a v e r q i n g

over e2 Then the complete expresssion Ear the cro8s section cr is

22 Q U A N T U M E L E C T R Q D Y N A L M I C S

The first term under the summation comes from tlxe "first termm 're- viously referred to and t b second from the ""secand term." The last term

If l k, the scattering is incoherent, m d the result is called the '"man effect,$8 If l = k, the s c a t b r i n g is coherent,

Further, note that if all the atoms are in, the ground state and l r k, f;hen

the energy of the atom can only i n c r e a ~ e and the frequency of the Ught ~3 can only decrease, This gives r i s e to "Stokes lines ." The oppsaite e f h c t gives @%ti-Stokes Unes .$'

Suppose wg = (coherent s c a l k r i w ) but further Ewl i s very nearly equal

to Ek -E,, where E, i s some possible energy level of the a b m , Then one

b r m in the sum over n b c o m e s extremely large and d o m i m b s the remain- der The result is called "msomnce s c a t k r i w " B' @ is plo-d agaimt o,

then a t such vdctes of w the c r o s s section hais a s b w maximum (see

Fig, 6-31

FIG 6-3

The @'indexps oaf refraction of a gas e m be obtained by our scattering for- mula, ft can be obtained, a,@ for other t y of scatterilae;, by consideriw the ~ ~lf&t s c a t b r a d in the forward direction

bJslf-Enem, h o t h e r phenomenon &at must be considered in q u m

e l s c t s o d p a l c s i s t h ~ pos~ibilft.Y of m atom emitting a phobn and r e a b ~ o r b - Ing the earne photon This affects tfie diagonal element Akk Its @Beat is

quivalent; to a sMft of energy of the level h e find&

where e is the direction of polarization This integral dive%@@ A more exact relativistic calcuIation also gives a divergent integral This means that our fomulialian of ebctromametic effecter if; not really a completely satisfactory theory The modifications r e q ~ r e d to a v d d this difficulty of the infinite self-energy will be &scusaed later The net result i s a very

@mall sMft bE in position of energy levels mfs shift h m been observed

by L a b and &&erford

e h i ~ velocity The p i n c i p l e has been verified exwrimentally Newton's laws satisfy this principle; for they a r e urrebmged when subject ta a Gall- lean transformation,

because they i nvolve only second derivatives , The e H q a a t i o n s a r e changed, h w e v s r , when subjected to this trainsfornnation, and early workers

in this fieid attempted to m&e an absoluta determination of velocity of the

any effects of this type ultimately led to Einstein" postulate that the Max-

well equations a r e of the @%me form in any coordinate system; and, in par- ticular, that the velocity of light i s the same in all coordinate s y s k m s The transformation beween coordinate system@ wkich leaves the MmweXl equa- tions invariant is th@ Lorentz transf.armatiarr:

where ta& u = v/c Henceforth we shall use time u d t s s o that the speed of light c i s unity The latter form is written to demonstrate the analam with rotation of a e s ,

X' = x cos 8 i- y sin 6

y' = -X sin B + y cos 6

Successive transforrnatians vl and v, o r ul and u2 add in the sense that a single tran~farrnation va o r u3 will give the same final system if

Einstein postulated [theory of special relativity) that the Newton laws must

be modified in such a way that they, too, a r e unchawed in form under a Lorentz transformation

An interestin%; consequence of the Lorentz transformation i s that clocks appear to run slower in moving systems; that is called time dilation In transforming from one coordinate system to another it i s convenient to use tensor analysis, To this end, a four-vector will be defined a s a set of four

quantities that transforms in the same way a s x,y,z and ct The subscript

p will be used to desimate which of the four components is being considered; for e x m p l e ,

The fallowing wantities a r e f our-vectors:

a ~ " y y ~ v t

A,, $ 8 A,* V (A,) vector (and scalar) potential

-$ The energy E, here, is the total energy including the r e s t energy me2

S P E C I A L R E L A T X V f T U 2 5

An invariant i s a quantity that does not change under a brexrtz transforma- tion If a u and bp a r e two four-vectors, the "product"

i s an invariant To avoid writing the summation symbol, the following sum-

mation convention will be used, When the same index w c u r s twice, s~

over it, placing minus in front of first, second, and third compments The

h r e n t z invariance of the continuity equation i s easily demonstrated by writ- ing i t a s a ' 4product" of four-vectors V, and j, :

Conservation of chmge in all systems if it Is conserved in one system i s a consequence of the invaiance of &is 'product,' ' t h e four-dimensional di-

vergence V e j , Another invariant is

p,p, = p e p = ~2 - p X Z - pY2 - p,' = E' - = m'

(E = total energy, m = r e s t mass, mc2 = r e s t energy, p = momentum.) Thus,

It i s also inlteresting to note that the phase of a free particle wave function

~sxp f f-i/tr)fEt - p - fo] i s invariant since

The invariance of p,p, can be used to facilitate converting laboratory en- ergies to eenter-of-mass energies (Fig, 6-4) in the following way (consider identical particles, far simplicity):

Center-of-mass system hboratarjr system

Q U A N T U M E L E C T R O D Y N A M I C S

but

and

The equations of e l e c t r & m a m i e ~ B = V X A and E = (i/c)(@A/@t) - V Q,

a r e easily written in temor notation,

where use i a made of the fact that (p is the fourth component of the four-

vector potential A p e From the foregoing it can be seen that B x , B,, B,, E,,

E,, and E, a r e the components of a secoad-ra& tensor:

This tensor i s antisymmetric (F = - F,@ ) and the diagonal terms (Ir = v )

t" f"

a r e zero; thus there are only six ~ndewndent components ( t h e e components

of E and three components of B) instead of sixteen

The Mmwell equations V x B = 4n J -t- (a E / B L) and V - E =: 4np a r e wrltbn

Q U A N T U M ELE:CTRODVEJAMXCS

h empty space the plane wave solution of &e wave equation

where el, and k, a r e constant vectors, and k, i s subject to the condition that

This may be seen from the fact that V operating on e-lk' X has the effect

of multiplying by ik, (V, does not operate on er since the coordinates a r e

rectangular) Thus,

Note that in these operations P, AI, actually forme a second-rank tensor,

V, (V, A, f a &ird-rank tensor, and then contraction on the index v yields a firat-r& tensor o r vector

The k p i s the propagation vector with components

and the condition k * k = 0 meam

Problem: Show that lthe LLOmntz codition

implies that k e = 0

m e n w o r M q in three dimensions it is customary to t a b the poiariza- tion vector 63 such that; K * a = O and to let the @calm potential cp = 0 But

this is not a unique condition; that is, i t is not relativistically invariant and will, be true only in a one-coordinate system This would seem to b a para- dox attaching some uniqueness to the system in which K s = 0, a situation incompatible with relativity a e o r g The "paradox, however, is resolved

by the fact tbat one can always make a so-called gauge transformation, which leaves the field FP, unaltered but which does change e Therefore, choosing; fiC* c? =- O in a particular system amounts to selecting the certain gauge

The gauge transformation, Eq (1 -31, is

where X i s a scalar But V A = 0, the h r e n t z condition, Eq, (7-41, will still hold if

This equation has a solution X = cremik * X , So

where a i s an arbitrary constant, Therefore,

i s the new polarization vector" obtained by gauge transformation h ordinary notation

m u s , no matter what coordinate system, is used,

can be made to v a d s h by choice of the constant a

Clearly the field i s left unchmged by a gauge transformation for