The Quantum Theory of the Emission and Absorption of Radiation
Author(s): P. A. M. Dirac
Reviewed work(s):
Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a
Mathematical and Physical Character, Vol. 114, No. 767 (Mar. 1, 1927), pp. 243-265
Published by: The Royal Society
Stable URL: http://www.jstor.org/stable/94746 .
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Emizssionand Absorptionof Radiaction.
243
(2) Observationshave beelnmade of the electricfields anid field changes
associated with 18 distantand 5 near thunderstorms.The suddenchangesof
fielddue to distantlighiningdischarges(> 8 km.) werepredominantly
negative
in sign,those due to near discharges( < 6 km.) predominantlypositive. The
relativefrequenciesof positive and negativechangeswere 1: 5 in the former
case and 43: 1 in the latter. The steadyelectriefields.belowthe 5 nearstorms
wereall stronglynegative.
(3) It is shown that these results indicate that the thundercloudswere
bi-polarin natureand that the polaritywas generally,if niotalways,positive,
the upper pole being positive and the lower pole negative. It is douibtful
if
any active stormsof oppositepolaritywereobservedat all.
(4) The electricmomentsof the chargesremovedby 82 ligltningdischarges
have been measutred.The mean value is 94 coulomb-kilometres.
The Quantum ,Theoryof the Ei'mssion and Absorption of
Radiatton.
and Institutefor
By P. A. M. DIRAC, St. John'sCollege,Cambridge,
Theoretical
Copenhagen.
Physics,
(Commlunicated
byN. Bohr,For.Mern.
R.S.-ReceivedFebruary
2,1927.)
? 1. Introduction
andSu.*mnary.
The new quantumtheory,based on the assumptionthat the dynamical
variablesdo notobeythecommutative
law ofmultiplicationi,
has bynowbeen
to forma fairlycomplete
developed
sufficiently
theoryofdynamics. One can
treatmathematically
the problemof any dynamicalsystemcomposedof a
ofparticles
withinstantaneous
forcesactingbetween
number
it
them,provided
is describable
andonecaninterpret
bya Hamiltonian
themathematics
function,
physically
bya quite definite
generalmethod. On the otherhand, hardly
has beendoneup to thepresenton quantumelectrodynamics.
The
anything
ofa systemn
in whichtheforcesare propaquestionsofthecorrecttreatment
oftheproduLction
of
gatedwiththevelocityoflightinsteadofinstanitaneously,
fieldbya movingelectron,
an electromagnetic
and ofthereactionofthisfield
on theelectronhave not yet been touched. In addition,thereis a serious
the theorysatisfyall the requirements
in miaking
of the restricted
difficulty
244
P. A. M. Dirac.
principleof relativity,since a Hamiltonianfunctioncan no longerbe used.
This relativityquestionis, of course,connectedwiththe previousones, and it
will be impossibleto answerany one questioncompletelywithoutat the same
time answeringthem all. However,it appears to be possible to build up a
fairlysatisfactorytheoryof the emissionof radiation and of the reactionof
the radiationfieldon the emittingsystemon the basis of a kinematicsand
dynamicswhich are not strictlyrelativistic. This is the main object of the
present paper. The theoryis non-relativisticonly on account of the time
beingcountedthroughout
as a c-number,
insteadofbeingtreatedsymmetrically
with the space co-ordinates. The relativityvariationof mass with velocity
is taken into accountwithoutdifficulty.
The underlying
ideas ofthe theoryare verysimple. Consideran atominteracting with a fieldof radiation,whichwe may suppose fordefiniteness
to be
confinedin an enclosureso as to have onlya discreteset of degreesof freedom.
Resolvingthe radiationintoits Fouriercomponents,
we can considerthe energy
and phase of each of the componentsto be dynamicalvariablesdescribingthe
radiationfield. Thus if Er is the energyof a componentlabelled r and Or
is the corresponding
phase (definedas thetimesincethewave was in a standard
phase),we can suppose each Er and ?r to forma pair of canonicallyconjugate
variables. In the absence of any interactionbetweenthe fieldand the atom,
the wholesystemof fieldplus atom will be describableby the Hamiltonian
H = ErEr + Ho
(1)
equal to the total energy,Ho beingthe Hamiltonianforthe atom alone, since
the variablesErn Or obviouslysatisfytheir canonical equations of motion
_
all
allH
= ao = ?
(3r
aEr
Whenthereis interactionbetweenthe fieldand the atom,it could be takeninto
account on the classical theoryby the additionof an interactiontermto the
Hamiltonian(1), whichwould be a functionofthe variablesofthe atom and of
the variablesEr, Orthat describethe field. This interactiontermwould give
the effectoftheradiationon the atom,and also the reactionofthe atom on the
radiationfield.
In orderthat an analogous methodmay be used on the quantum theory,
it is necessaryto assume that the variables Er, Orare q-numberssatisfying
the standard quantum conditions OrEr- ErOr= ,h,etc., whereh is (27)times the usual Planck's constant,like the otherdynamicalvariables of the
problem. This assumption immediatelygives light-quantumpropertiesto
Emissionand Absorptionof Radiation.
245
the radiation." For if vris the frequencyof the componentr, 27VrOr is an
angle variable, so that its canonical conjugate Er/27vrr can only assume a
discreteset of valuesdiflering
by multiplesof h, whichmeansthat Er can
ofthequantum(2wh)'Vr If we nowadd an
changeonlybyintegral
multiples
interaction
term(takenoverfromtheclasicaltheory)to theHamiltonian(1),
to therulesof quantummechanies,
theproblemcan be solvedaccording
and
resultsforthe actionofthe radiation
we wouldexpectto obtainthe correct
andtheatomi
ononeanother. It willbe shownthatweactuallygetthecorrect
ofradiation,
and the correctvaluesfor
laws forthe emissionand absorption
Einstein'sA's and B's. In the author'sprevioustheory,t
wherethe energies
ofradiationweree-numbers,
and phasesofthecomponents
onlytheB's could
of
on
reaction
the
atom
the
radiation
could
notbe taken
and
the
be obtained,
into account.
It will also be shownthat the Hamiltonianwhichdescribesthe interaction
waves can be made identicalwith the
of the atom and the electromagnetic
of
of the atom with an assembly
for
the
interaction
Hamiltonian the problem
of particlesmovingwiththe velocityof lightand satisfyingthe Einstein-Bose
statistics,by a suitablechoiceof the interactionenergyforthe particles. The
numberof particleshavingany specifieddirectionof motionand energy,which
can be used as a dynamicalvariable in the Hamiltonianforthe particles,is
wave in the
equal to the numberof quanta of energyin the corresponding
Hiamiltonianforthe waves. There is thus a completeharmonybetweenthe
wave and liglit-quantumdescriptionsof the interaction. We shall actually
the light-quantumpoint of view, and show that the
build up the theoryfromrl
Hamiltoniantransformsnaturallyinto a formwhich resemblesthat for the
waves.
The mathematicaldevelopmentofthe theoryhas been made possibleby the
author's general transformation
theory of the quantum matrices.: Owing
we are allowedto use thenotion
to thefactthatwe countthetimeas a c-number,
of the value of any dynamicalvariable at any instantof time. This value is
* Similar assumptions have been used by Born and Jordan ['Z. f. Physik,' vol. 34,
p. 886 (1925)] forthe purpose of takingover the classical formulaforthe emission of radiation
by a dipole into the quantum theory,and by Born, Heisenberg and Jordan ['Z. f. Physik,'
vol. 35, p. 606 (1925)] for calculating the energy fluctuations in a field of black-body
radiation.
t ' Roy. Soc. Proc.,' A, vol. 112, p. 661, ? 5 (1926). This is quoted later by, loc. cit., I.
I 'Roy. Soc. Proc.,' A, vol. 113, p. 621 (1927). This is quoted later by loc. cit., II. An
essentially equivalent theory has been obtained independently by Jordan ['Z. f. Physik,'
vol. 40, p. 809 (1927)]. See also, F. London, I Z. f. Physik,' vol. 40, p. 193 (1926).
246
P. A. M. Dirac.
" according
i
bya generallsedinatrix
a q-nunber,
capableofbeingrepresenited
matrixschemes,sox e of whichmay have continuous ranges
to manydifferent
to involvecertain.
and may requirethematrixelements
ofrowsand columns,
(ofthetypegiven'bythe a functions*). A matrixschemecan
kindsofinfinities
ofintegration
ofthedynamical
be foundin whichanydesiredset ofconstants
orin whicha setof
bydiagonalmatrices,
arerepresented
thatcommute
system
by matricesthat are diagonalat a
variablesthat commuteare represented
of
values
of a diagonalmatrix
the
elenments
diagonal
The
timie.t
specified
valuesofthatq-number.A
anyq-nuimber
are thecharacteristic
representing
valhes
or
will
haveall characteristic
co-ordinate
momentum
in
general
Cartesian
setofcharactervariablehasonlya discrete
from- o to + oo, whilean actioni
to denotethe
to
use
letters
make
it
unprimed
aJ rule
istic values. (We shall
dynamicalvariablesor q-numbers,and the same lettersprimedor multiply
oreigento denotetheircharacteristic
functions
values. Transformation
primed
of the characteristic
valuesandnotoftheq-numbers
are functions
functions
so theyshouldalwaysbe written
interinsofprimedvariables.)
themselves,
ofthe canonicalvariablesib, rh, thematrixrepreIff(i, ) is anyfunction
schemeinwhichtheLat timetarediagonal
at anytimetinthematrix
sentingf
downwithoutanytrouble,sincethematrices
reprebe written
matricesmnay
at tine t are known,
the L and kcthemselves
senting
naimely,
ik (44
-
i
k 8 WVi),
oftheL and_, we can at
H is givenias a function
ThusiftheHamiltonian
thematrixH(i' i"). We can thenobtainthetransformation
oncewritedowxn
to a nmatrix
scheme(M)in whichthe
(i'/') say, whichtransform-s
function,
as ( '/oe')mustsatisfy
is a diagonalmatrix,
theintegral
equation
Hamiltonian
Wa)*(''
dH
1/R#d>'(4/7,)-A
(3)
levels. Thisequation
valies W(c') are theenergy
ofwhichthe characteristic
waveequationfortheeigenfunetions
('foc'),whichbecomes,
is justSchrddinger's
ofthe
differential
an ordinary
equationwhenH is a simplealgebraicffunction
* Loc. cit. II, ? 2.
One can have a matrix seliem in which a set of variables that commute are at all times
representedby diagonal miatricesif one will sacrificethe condition that the matrices must
fromsuch a scheme to one
satisfythe equations of motion. The transformationftunction
in which the equations of motion are satisfied will involve the time explicitly. See p. 628
in loe. cit., 11.
Jr
Emtissiontand
Absorptionof Radiation.
247
ofthespecialquations(2) forthematrices
L andkoroaccount
representing
Lk and 7.
generalforia
in themllore
Equation (3) maybe Nwritten
JH
(
')d"('i2J-h
a(1)
,(3')
in which
it canbe appliedto systems
forwhich
theHamiltonian
involves
the
timeexplicitly.
Oniemnay
havea dynamical
system
specified
IH which
by a' Hamiltonian
cannot
beexpressed
as an algebraic
ofanysetofcanonical
function
variables,
canallthesameberepresenited
bnlt
which
H(S'ua).Sncha problem
bya matrix
canstillbe solvedbythepresent
sinceonecanstilluseequation
method,
(3)
toobtain
theenergy
levelsandeigenfunctions.
Weshallfind
thattheHamiltonianwhich
describes
theinteraction
ofa light-quantum
andanatomic
is
system
ofthismore
general
type,sothattheinteraction
canbetreated
mathematically,
onecannottalkaboutan in-teraction
although
in theusuLal
energy
potenitial
sense.
It should
beobserved
is a difference
thatthere
a light-wave
between
andthe
de Broglie
orSchrddinger
waveassociated
withthelight-quanta.
the
Firstly,
is alwaysreal,whilethede Broglie
light-wave
waveassociated
witha lightina definite
quantum
directioni
mustbetakentoinvolve
animnaginary
moving
exponential.
A moreimportant
difference
is thattheirintensities
areto be
in different
interpreted
ways. Thenumber
oflight-quanta
perunitvolume
associated
witha monochromatic
light-wave
equalstheenergy
perunitvolume
ofthewavedivided
bytheenergy
ofa singlelight-quantum.
(27rh)v
Onthe
otherhanda monochromatic
de Broglie
waveofamplitude
a (multiplied
into
theimaginary
exponential
factor)mustbe interpreted
a2 lightas representing
quantaperunitvolume
forallfrequencies.
Thisis a specialcaseofthegeneral
rule forinterpreting
the matrixanalysisj accordingto which,if (i'/') or
' of an atomic
t (ik') is theeigenfunction
in thevariables
4 of the state
system
(orsimple
particle),
I (a4')
value L',
[or !ia
(i')
2
I2 iS
theprobability
of each4 having
the
dil' dt' ... is theprobability
of each4 lyingbetween
thevaluesik' andE.k'+ d<'k whenthe k.havecontinuous
ranges
ofcharacteristicvalues]
ontheassumption
thatallphases
ofthesystem
areequally
probable.
Thewavewhoseintensitv
is to be interpreted
in thefirst
ofthesetwoways
appears
inthetheory
onlywhen
oneisdealing
withanassembly
oftheassociated
particles
theEinstein-Bose
satisfying
statistics.Thereis thusno suchwave
associated
withelectrons.
*
Loc. cit.,H. ??6, 7.
P. A. M. Dirae.
248
Systems.
ofIndependent
ofa, A,ssembly
? 2. The Perturbation
We shall now considerthe traisitions producedin aniatomic systemby an
arbitraryperturbation. The iethod wreshall adopt will be that previously
givenby the author,twhichleads in a simpleway to equationswhichdetermine
the probabilityofthe systembeingin any stationarystate ofthe unperturbed
the probablenumnber
systemat any time.i:- This, of course,givesimmirediately
of systemsin that state at that time for an assembly of the systenms
that are independentof one anotherand are all perturbedin the same way.
The object of the presentsectionis to show that the equations forthe rates.
of changeof these probablenumberscan be put in the Hamiltonianformin a
simple m'anner,which will enable furtherdevelopmentsin the theoryto be
made.
systemand V the perturbing
Let H0 be the Hamiltonianforthe unpert-urbed
variablesand may
of
the
can
be
functiorn
dynamical
an
arbitrary
which
energy,
ormaynotinvolvethetimeexplicitly,so thatthe Hamiltonianfortheperturbed
forthe pertuLrbed
must
systemn
systemis H = Ho + V. The eigenfunctions
satisfythe wave equation
i a?at (Ho+ V)D6,
rardr is the solutionof this equation
where(H10-H-V) is an operator. If
that satisfiesthe properinitialconditions,wherethe Q.'s are the eigenfunctions
forthe unperturbedsystem,each associatedwithone stationarystate labelled.
by the suffixr,and the ar's are functionsofthe time only,then Ia 12 is theprobability of the systembeingin the state r at any time. The a,'s must be normalised initially,and will then always remain normalised. The theorywill
applydirectlyto an assemblyof N similarindependentsystemsif we nm-ultiply
each of these ar's by N-!'so as to make X,.Ia7 12 = N. We shall now have that
ar 12 iS the probablenumberof systemsin the state r.
The equationthat determinesthe rate of changeofthe a,'s is?
(4)
ihd7r
=EsrsVas
wherethe V,,'s are the elementsofthe matrixrepresetiungT.
imaginary equation is
z.q -Ssas
-tazr - zq7Ts
ATse
The conjugate
~~(4/)
t Loc. cit. I.
(1926)]
I The theory has recentlybeen extended by Born [' Z. f. Physik,' vol. 40, p. 167
be
that
states
the
may
in
stationary
adiabatic
changes
so as to take into account the
in
not
is
used
This
transitions.
extension
the
as
well
as
the
by
perturbation
produced
the present paper.
? Loc. cit., I, equation (25).
Emissionand Absorptionof Radiation.
249
Ifweregard
conjugates,
equations
(4) and(4') take
a,.andiha,* as canonical
the Hamiltonianformwith the HamiltonianfunctionF1 = rsar*V,T.as,
namely,
da}.
I WIl
da
dt i- i"Faa'
-
* da7*
aF,
ihR
--q
dt
aar
We cantransform
to thecanonical
variables
transNr, jr bythecontact
formation
4
N le-4i4Ah a_* =N
*
aC.
Thistransformation
makesthenewvariables
N,.and0,. real,Nrbeingequal
to aa,.?*=1 1ar2,the probablenumberof systemsin the stater, and Orlh
beingthephaseoftheeigenfunction
thatrepresents
them. TheHamiltonian
F1nowbecomes
()S)h
FJ.= ZrsVr.sN);1Ns2
and the equationsthat determinethe rate at whichtransitionsoccurhave the
canonical form
WI
~~aF,
q
8?>r
aN.r
A mnore
convenientway ofputtingthetransitionequationsin theHamiltonian
formn
may be obtainedwiththe help ofthe quantities
b = a e-w1th
b_-*
a,.*%XYtIh
W,.beingthe energyofthe state r. We have Ib,.12equal to Ia.
numberof systemsin the state r. For brwe find
12,the probable
ih b,.= WrbI.+ ihare- iwrtIl
i (WI-W
- 'Wrbr
W.)tlh
+ Es V,.8bse
t/h,so that v.s is a constaint
withthe help of (4). If we put Vr,
Vres(W
whenV does not involvethe timeexplicitly,thisreducesto
ih br
Wrbr+ Esvrsbs
-ESHI'ibS)
(5)
whereHrS-- WI.01.8
+ v,.,,whichis a matrixelementof the total Hamiltonian
H -ll + V withthe timefactoret(Wr-Ws) t/hreimoved,
so that H,.,is a constant
whenH does not involvethe time explicitly. Equation (5) is ofthe same form
as equation (4), and may be put in the Hamiltonianformin the same way.
It shouldbe noticedthat equation (5) is obtaineddirectlyif one writesdown
the Schrbdinger
equationin a set of variablesthat specifythe stationarystates
of the unperturbedsystem. If these variablesare Rh,and if H(i'4") denotes
VOL. CXIV.-A.
5
P. A. M. Dirac.
250
H in the (4) sIheine,this
a matrixelementof the total Hamiiltonian
equation would be
Schri5dinger
ik4a ( ')lat
> H (g'i")
qg/"),
(6)
fromthe previousequation(5) onlyin the
like equation(3'). This differs
r beingthereusedto denote;
a stationary
state instead
a singlesuffix
notation,
of a setof numeric values i forthe variables k, ud ,beingnsed instead
also equation(5r), canstillbe usedwhen
of:4 ( '). Equation(6), and therefore
whichcai notbe expressedas an
is of themoregeneraltyype
theHamiltonian
of a st of canoniIalvariables,btutcan still be represented
algebraicfuinction
by a inatrix H (g't") or HI,,.
Wenowtak-eb,and ihb to be canonically
conjugatevariablesinsteadof
equation-nvill
(5) and its conjugateimaginary
The equiation
ar and ihaj'
Hamiltoniian
function
form,
with
the
nowtaketheHamiltonian.
F
rsbr
(7)
HrsbXsb
we makethecontacttransformation
as before,
Proceeding
N_'
N eb-.=N_
bi
rcl'e
(8)
to the new canonicalvariablesN., 0,, whereN, is, as before,the probable
F
in the stater,and 0ris a -newphase. The Hamiltonian
iiumberofsystemis
now
become
will
F s=
,., N j Nje2Cic0,-)/h
forthe ra.tesof changeof N, and 0, will take the canonical
aandthe equLations
fornm
Ni
a
Or
aNr.
be written
m-ay
?heHamiitoonian
F
-
?,.WrN,
-F Eq,, S2Nj
Ns2
e'(&0
)/h
(9)
and the
is the total properenergyof the assembly,
The firsttermE,.WrNN
If
dueto theperturbation.
as theadditionalenergy
secondmaybe regarded
ld increaselinearlywith the time,
tih perturbatio is zro, the phases 0, wvo
whilethe previous phases +,. would in this case be cons-tants.
theEinstein-Bose Sttistics.
? 3. The Perturbation
ofan Assemblysatisfying
of a perturbato thepreceding
sectionwe can describe
theeffect
According
ofindependent
systems
tionon an asse-mbly
by ineansof canonicalvariables
ofthetheorywhich
and Hamiltonian
equationsofmotion. The development
EEmission
and Absorptionof Radiation.
251
naturallysuggestsitselfis to make thesecanonicalvariablesq-nunmbers
satisfying the usual quantumconditionsinsteadof c-numbers,
so that their Hamiltonian equationsof motion become truequantuimequations. The Hamiltonian
functionwill. now providea Schr6dinger
wave equation,which.mustbe solved
and interpreted
in the usual manner. The in-terpretation
will give not merely
theprobablenuLmber
of systemsin anystate, but the probabilityof any given
distributionof the svstemsamong the various states, this probabilitybeing,
in fact,equal to the square of the modulusof the normalisedsolutionof the
wave equation that satisfiesthe appropriateinitial conditions. We could, of
course, calculate directlyfromelementaryconsiderationsthe probabilityof
any given distributionwhen the systemsare independent,as we know the
probabilityofeach systembeingin any particularstate. We shallfindthlatthe
probabilitycalculated directlyin this way does not atgreewith that obtained
from.
the wave equationexceptin the specialcase whenthereis onlyone system
in the assembly. In the generalcase it will be shownthat the wave equation
leads to the correctvaltLeforthe probabilityof any given distribut-ion
when
the systemsobey the Einstein-Bosestatisticsinstead of beingindependent.
We assumethe variablesb,.,ihbY
l of ? 2 to be canonicalq-nLmbers
satisfying
the quantumconditions
b.i. ihb2.*-i7bl.*. b2.= h
bqb2- b;. -_ 1,
or
and
- bsbr 0,
brbS
b
-bs
-bs*br
0
- bs-b,,
br*bs,*
(s
4
0,
qr).
now be writtenin the quantumform
The transformation
equations(8) m-wust
b. = (N2.-+ 1)
e-01)1
b_t
-
e- (1o2N,i>1
(Ni.+ 1)U,
Nr]e6i&/h_ 6ioGlh
3
.
also be canonical variables. These equations
in orderthat the Ni., n0may
show that the N. can have only integralcharacteristicvalues not less than
zero,t which provides us with a justificationfor the assumptionthat the
of systems
variablesare q-numbersin the way we have chosen. The nuimbers
statesare now ordinaryquantumniumbers.
in the different
t See ? 8 ofthe author'spaper 'Roy. Soc. Proc.,'A, vol. 111,p. 281 (1926). What are
valuesthata q-numbercan take are heregiventhemoreprecise
therecalledthe c-number
valuesofthatq-number.
nameofthe characteristic
s
2
P. A. M. Dirac.
252
(7) nowbecomes
TheHamiltonian
2rsb7*Hr
bei8z
F =
-
(Ns +
lhHrs
(N, +Er$HrsNri
1-A
1) e- i0s11
(11)
rs)!ei(0-)h
in whichtheHrsarestillc-numbers.We maywritethisF in theformcorreto (9)
sponding
F - ErWrNr+ ErtvrsNr"
(Ns+1
-
ci(Or@)/h
rs)k
(11')
in whichit is again composed of a proper energytermErWrN,and an inter-
term.
actionenergy
The wave equationwrittenin termsofthe variablesNr ist
ik
N2%,Ns'
(N1%,
...) =
Fs (N1', N2%,N3'
),
(12)
to meanih a/aNrt.
in F beinginterpreted
whereF is an operator,each Oroccurring
If we apply the operatore+iG0Ih to any functionf(N1', N2', ... Nr, ...) of the
variables N'I N ', ... the result is
e ? i,hf(Ni',
N2', ... Nr'
-)
eFb/IN,'f(N1', N2',
=f(N1',
... Nrt'..)
N2', ... Nr' T 1, ...
If we use thisrulein equation(12) and use the expression(11) forF we obtain:
ih Tt+ (Nl', N2', N3'
-rbs
HrsNr (Ns' + 1. - 3rs)l (N1,'N2'... Nr' 1,... Ns'+ 1, ...).
2
(13}
We see from the right-handside of this equation that in the matrixrepresenting F, the term in F involving ei(, - 0.)!h will contributeonly to
those matrix elements that refer to transitions in which N, decreases.
by unity and N. increases by unity, i.e., to matrix elementsof the type
F (N', N2' ... N,' ... Ns'; N', N2'... Nr'-1 ... Ns' + 1 ...). If we find,a,
solution6 (N,', N2' ...) of equation (13) that is normalised[i.e., one forwhich
1] and that satisfiesthe proper initial conN2' 1+ (N,', N2 ...)12
of that distribution
in
*w) 2 will be the probability
N2'
ditions,then I 4 (N1%,
in
...
at
state
time.
in
state
1, N2'
any
2,
whichN1' systemsare
Considerfirstthe case whenthereis onlyone systemin the assembly. The
probabilityof its being in the state q is determinedby the eigenfunction
that the label r of the stationarystates takes the
t We are supposingfordefiniteness
values 1, 2, 3, ....
i
When s = r,# (N.,, N2'... Nrt'- 1... N*' + 1) is to be taken to mean # (N1'N2' ... N,'...).
aind Absorptionof Radiation.
lErntsswon
253
in whichall the N"s are pnt equal to zero exceptNq', \vhichis
t! {q}. When it is
substitutedin the left-handside of (13), all the termsin the sumumation
on
the right-handside vaanishexceptthoseforwhichr-q, and we are leftwith
4i(N1' N2', ...)
puLtequal to unity. This eigenfunction
we shall denoteby
ih
k
Hq44}
at
whichis the same equationas (5) with Q {q} playingthe part ofb,. This establishes the fact that the presenttheoryis equivalentto that of the preceding
sectionwhenthereis only olnesystemin the assembly.
Nowtake thegeneralcase ofanlarbitrarynumberofsystemsin the assembly,
and assume that they obey the Einstein-Bosestatistical mechanics. This
requiresthat, in the ordinarytreatmentof the problem,only those eigenfunctionsthat are symmetricalbetweenall the syste:msinust be taken into
beingby themselvessufficient
to give,a complete
account,these eigenfunctions
quantumsolutionof the problem.t We shall now obtain.the equation forthe
rateof chang,e
ofone of thesesymnietricaleigenfunctions,
and show that it is
identicalwith equation (13).
for the
If we label each systenm
with a numnber
n, then the Hamniltonian
a.ssemblywIll be H =
H (n), whereH (n) is the H of ? 2 (equal to Ho + V)
of
expressedin terms the variablesof the nth system. A stationarystate of
the assemblyis definedby the numbersrl, r2... rn... whichare the labels ofthe
stationarystatesin whichthe separatesystemslie. The Schr8dinger
equation
forthe assemblyin a set of variablesthat specifythe stationarystates will be,
of theform(6) [withHA instead of H], and we can write it in the notationof
equation (5) thus:
ihb(r1r2...)
Esl,s...
1A(rlr2
1SS2 *.-)
b(s1s2...),
(14)
whereHA(rlr2 ...; ss ...) is the general matrix
of HA [with the time
elem.ent
factorremoved]. This matrixelementvanisheswhenmorethan onlesI, differs
fromthe corresponding
when sm differsfrom rmand every
rn; equals Hrnl,;,,
other sn equals r ; and equals SH ,. when every s. equals r,
Substituting
these values in (14), we obtain
ihb(r1r2... )
=ms&frJ'rsk
(r1Lr2
+m}1n+1
)
Y} XnHr)trnb(1'rr2...). (15)
We mustnowrestrictb (r1r2...) to be a symmetrical
functionofthe variables
to
r2
order
obtain
the
Einstein-Bose
statistics.
This is permissible
...
in
rl,
since if b (r_r2'
at any time,then equation (15) showsthat
-") is symmetrical
t Loc. cit.,I, ? 3.
P. A. M. Di-ac.
254
at thattime,so that b (rr2 ...) willremain
b(r1r2...) is also symmetrical
symmetrical.
state
inthestater, Thena stationary
ofsystems
LetN,.denotethenumber
maybe specified
eigenfunction
ofthe assemblydescribable
by a symmetrical
by the numbersN1,N2 . N... just as well as by the numbersre, r.2...
and we shall be able to transformequationl(15) to the variablesN1, N2
We cainnotactuallytake the new eigenfunction
b (N1, N2. ) eqaal to the premultiple of the
vious one b (r.1r2...), but must take one to be a numuerical
with respect to its
other in orderthat each may be correctlynoimlnalised
in
fact,
respectiVevariables. We musthave,
2
Sr,,?,...16(rl
...) 12
I
lb(N1,
-
N2
.) 12-
2 fo
2 equal to the suzmof ib (rLr2 .
and hence we must take Ib (N1, N2
r, m2... suchthatthereare N1 ofthemequal to 1,N2
allvaluesofthenumnbers
equal to 2, etco. Thereare iN!/N12!N2! ... termsin thissum,whereN
of systems,and they are all equal, sinee b (rr ...) is a
is the total inumnber
Hence we musthave
of its variablesr1,r2 ....
funictionl
symmetrical
)
b (N1 N2
...)
(N !/N1 ! N2!
b (rr2
...)
...).
left-handside will becomne
If we make this suLbstitutiont
in equation (15), theG
b (r,.r2** rL1 8 ,/in-1
The term
ih (N1 ! N2 ! ... /N!)- b (N1,N2 ...)
* *)
on the right-handside will becolme
in the fist summamtion
!.. .(Ns+ J1)! .. I/N!'H,sb (N,, NN
2 ...
[N ! I (N2;-l)
..
- I . Ns+ 1..), (iG)
f r,,,and s fors,. This termmust be s,uLime'dfor
wherewe have Arritten?or
forr takingeach ofthe values
mustthenbe summ-ed
s
and
a11valuesof exceptr,
hus each term(16) gets repeatedby the summationprocessui Ail
i1, r2 T...
it occursa total ofNr times,so that it contributes
Nr [NI 1Ns
. (N -i.) ! ... (Ns+ i)
N-1(s
+ I)` (NL! N
to the :right-handside of (15).
IN!t]-!T-.;36,
(N,, N2..N1.-
-N-
.-I ..N +
,sb(N,
r.)
Finally, the term XE.ff
( n_9
Hrff,binb
ErNHrr* b (rl2 Er2Ns1Hm.,_
(N1,! N2!
...
...+Ns + 1...)
...)
becomes
/N !)- b (N1.,N ...)
Hence equation(15) becoines,witlhthe reuoval of the factor(N,.! N2 !..../N
IN)1
ihb (N1,NS ...) =
N; (Ns+l)i
Ns + 1 ..)
+ E)N7HTb (N,I
2 ..
15. (N1, N2 ... N-1
...
(17)
Fnission a6-nd
A bsorptionof Radiation.
255
whichis ideniticalwith (13) [exceptforthe fact that in (17) the primeshave
been omittedfromthe N's, whichis permissiblewhenwe do not requireto refer
to the N's as q-numbers]. We have thus establishedthat the Hamiltonian
ofa perturbation
(11) describesthe effect
on an assemblysatisfying
theEinisteinBose statistics.
? 4.
TheReaction
onthePerturbing
oftheAssemnbly
System.
Up -tothe presentwe have consideredonly perturbations
that can be represented by a perturbingenergyV added to the Hamiltonianof the perturbed
system,V beinga functiononlyof the dynamicalvariablesof that systemand
perhaps of the time. The theorymay readily be extendedto the case when
the perturbationconsistsof interactionwith a perturbingdynamicalsystemn
the reaction of the perturbedsystem on the perturbingsystem being taken
int-oaccount. (The distinctionbetweenthe perturbingsystemaand the perturbed systenm
is, of course,not real, but it will be kept up forconvenience.)
We now considera perturbingsystem,described,say, by the canonical
variablesJ,, , the J's being its firstintegralswhenit is alone, interacting
ofperturbedsystemswithno mutualinteraction,that satisfy
withan assenmbly
the Einistein-Bosestatistics. Thi-etotal Hamiltonianwill be of the formi
IT
=
Hp(J) + E.1- (n)
whereH. is the lam iltonianof the perturbingsysterm
(a functionof the J's
only)and H (n) is equal to the properenergyHo (n) plus the perturbation
energy
V(n) ofthe nthsystemofthe assembly. H (n) is a functiononlyofthe variables
of the nthsystemofthe assemxbly
and of the J's and wv's,
and does not involve
the time explicitly.
The Schr6dingerequation corresponding
to equation (14) is now
hb
( nlf2...e
EJ sI>2
..
HT(
l
2
;J
SIS2
...*)b(J ,802...)
in whichthe eigenfunction
b involvesthe additionalvariablesJ'. The matrix
elementHT (J', rLr2... ; J", s1s2...) is now always a constant. As before,it
vanishes when more than one sn,differsfromthe corresponding,.
When
fromn
and
differs
other
qrn
every
it
equals
reduces
to
H
(J'r
;
J"tsn),
sM,
s.
r?,,
whichis the (J'r; J"s.n) matrix element(with the time factorremoved)of
H = Ho +- V, the proper energy plus the perturbationeniergyof a single
systemof the assembly; while when every sn equals rf, it has the value
Hr (J') aj j + - H (J'r,; J"rqb). If, as before,we restrictthe eigenfunctions
P. A. M. Dirac.
256
in the variablesr_,r2 *.., we can again transformto the
to be symmnetrical
variables N1,N2 ... whichwill lead, as before,to the result
ihb(JtnNt,
2' *...=Hr
(J) b(J', Y, N2/..
t
+ SJ?,sNi~~2b(JFfl
N/r:N 2
(Y rs) J'r,3s)
. .Ni
..Ns,
^ + I . ) (18)
to the Hamiltonianfunction
corresponding
This is the Schrddinger
equLation
= Hp (J) + rsIfrsNP (Ns + I--s)-l et(0-)/,
(19)
ofthe J's and w's, being such that whenreprein whichH,Js
is now a fuLnction
sented by a m-atrixin the (J) schemeits (J' J") elementis H (J'r; J's). (It
shouldbe noticedthat HIS,still commuteswiththe N's and O's.)
F
Thus the nteraction of a perturbing system and an assemiblysatisfyingthe
Einstein-Bosestatisticscan be describedby a Haamiltonianof the form(19).
to (11') by observingthat the matrix
We can put it in the forn corresponding
H
of
the
element (J'r; J"s) is composed
sumiiof two parts, a part that comes
r and
fromthe properenergyHo, which eqaals W, when Jk"= Jj, and s
energyY,
the j-uteraction
vanishes otherwise,and a part that comes fronm
be denotedby v (J'r; J"s). Thus we shall have
whichmnay
HRIS =
i.
ks
+
V2ss.
wherev,. is that functionof the J's anidw's which is representedby the tiatrix
whose(J'J")elementis v (J'r; J"s),and so (19) becomes
F ci Hp (J) + EIW,N, +
sVsNr
(Ns + 1 - A
j(OO9!b.
(20)
oftheproperenergyofthe perturbinlg
system
The Hamiltonianis thusthe sunm
and the perturbaHp(J), the properenergyof the perturbedsystemsY,WVA7,Nr
)/h
tionl energy E,. ,v;s.AJ (Ns + 1-- &,s)Ae(
? 5. TheoryofTransitions
in a Systemfomn One StatetoOthersoftheSame Energy.
Beforeapplying the results of the precedingsectionsto light-quanta,we
shall considerthe solutionof the problempresentedby a Hamiltonianof the
type (19). The essentialfeatureofthe problemis that it refersto a dynamical
systemwhich can, under the influenceof a perturbationenergywhich does
fromone state to othersof
not involve the time explicitly,make tranlsitions
the same energy. The problemof collisionsbetweenan atomicsystemand an
electron,whichhas beentreatedby Born,*is a specialcase ofthistype. Born's
methodis to finda periodlicsolu-tionof the wave equation whichconsists,in
so far as it involvesthe co-ordinatesof the collidingelectron,of planiewaves,
* Born.'Z. f. Phvsik.'vol 38. D. 803 (1926).
and Absorptionof Rad'ation'.
Er-mission
257
systenm,
whichare
representing
theincident
electron,
approaching
theatomic
or diffracted
in all directions.
ofthe
ThesquareoftheamplituLde
scattered
is thenassumed
byBorn
wavesscattered
in anydirection
withanyfrequency
in thatdirection
with
to be theprobability
oftheelectron
beingscattered
the corresponding
energy.
in any simplemanner
Thismethoddoesnotappearto be capableofextension
from
onestatetoothers
-tothegeneral
problem
ofsystenms
thatmaketransitiors
andcertain
,ofthesameenergy.Alsothereis at present
no verydirect
way
toapplytoa non-periodic
ofinterpreting
ofa waveequation
a periodic
solution
that
physical
phenomenon
suchas a collision.(Themoredefinite
mnethod
it being
will nowbe givenshowsthat Born'sassulmptionis not quiteriglht,
of
a
certain
to
the
by
factor.)
necessary multiply
thesquare
amplitude
is tofind
a non-periodic
ofsolving
Analternative
a collision
method
problem
of
whichconsists
Solultion ofthewaveequation
initially
simply planewaves
overthewholeofspacein thenecessary
direction
withthenecessary
moving
oftimewavesmoving
torepresent
electron.In course
frequency
theincident
mustappearin orderthatthewave equationmayremain
in otherdirections
oftheelectron
inanydirection
beingscattered
satisfied.Theprobability
with
ofthecorresponding
anyenergy
willthenbedetermined
bytherateofgrowth
of thesewaves. The waythe mathematics
is to be
harmonic
component
is by thismethod
interpreted
quitedefinite,
beingthesameas thatofthe
of ?2.
beginning
ofa system
tothegeneral
Weshallapplythismethod
which
makes
problem
ofthesameenergy
undertheactionofa
transitions
from
onestateto others
oftheunperturbed
system
perturbation.
Let Ho be theHamiltonian
and
thetimeexplicitly.
V thepertuLirbing
whichmustnotinvolve
If we
energy,
takethecaseofa continuous
bythefirst
rangeofstationary
states,specified
thenmethod
of
integrals,
motion,
then,following
oXksay,oftheunperturbed
? 2, we obtain
ih&(a')
V
V(ooc")d%". a(c'),
(21)
ofthesystem
beingin a state
corresponding
toequation
(4). Theprobability
. doc2'
forwhicheachOCk
at anytimeisIa (Oc)12do-.'
liesbetween
oC'andOCk7'+
daCk'
initialconditions.
theproper
whena (oc')is properly
andsatisfies
normalised
valueofa (o')
the
initial
Ifinitially
take
wemust
thesystem
is inthestateoc0,
a . a (oc'- oc0).We shallkeepa0 arbitrary,
as it wouldbe
to be oftheform
a (oc')inthepresent
case. Fora first
approximation
inconvenient
to normalise
258
P. A. A. Dirac.
fora (o'c)in theright-hand
wemnay
substitute
sideof(21)itsinitialvalue. This
gives
(Mo')eiXW(a')-W(a?)It/h
ih4 (a') = a0V(a'a?) = oc?v
is a constantand W (o') is the energyofthestateoc'. Hence
wherev(M'oJ0)
iha (c.')= a
(o'
-
4o) +
aov
(c,'c() e[W() -W(CI)]t/h-I
(22Y
For valuesof theock'such that W (a') differs
appreciably
fromW (MO),
a (a')
is a periodic
function
ofthetimewhoseamplitude
is smallwhentheperturbing
corresponding
to thesestationary
energy
V is small,so thattheeigenfunctions
statesarenotexcitedto anyappreciable
extent. Ontheotherhand,forvalues
of the cXk'such that W (a') - W (o!2) and OCa'# Oko forsome k, a (oc')increases
withrespectto thetime,so thattheprobability
ofthesystembeing
uniformly
in thestatea' at anytimeincreases
withthesquareofthetime.
proportionally
theprobability
ofthesystem
Physically,
beingin a statewithexactlythesame
being
properenergy
as the initialproperenergyW (ac?)is of no importance,
infinitesimal.
We are interestedonly in the integralof the probability
through
a smallrangeofproperenergy
valuesabouttheinitialproperenergy,
withthe
which,as we shallfind,increases
linearly
withthetime,in agreement
ordinary
probability
laws.
We transform
fromthe variablesrl,,o2 ... oa to a setof variablesthatare
arbitrary
independent
functions
ofthea's suchthatoneofthemis theproper
W,Y -'Y2. yu1. The probability
at anytime
energyW, say,thevariables
ofthesystemlyingina stationary
stateforwhicheachYk lies between
yk'andto
is
now
from
the
(apart
normalising
factor)equal
Yk' + dyk'
d
*dy2' ... d-'l'i
a (')
12
a ( (X4l X2
*
-
(WI, Ti'
X
) - dW'.
Yu-i')
(23)
For a timethatis largecompared
withtheperiodsofthesystemwe shallfind
by valuesof
thatpractically
the wholeof theintegralin (23) is contributed
W' verycloseto WO W (a?). Put
a (a') - a (W', y') and a (cl', OC2'.X -')/a(W', Yl
YU-1') J (W' y )
Thenfortheintegral
in (23) we find,withthehelpof(22) (provided
Yk' ? Y 0
forsomek)
Ja(WI,
J (W', y') dW'
y) 12
a?I2' J
Iv (W, y' ;
-
2
=
WO,yO)12J (W', Y/) [ei(WWO)tIh
12J (W',y')
1a012iv (W,y'; w?,yo)
[1-cos
th_1l dW"
1] [ei- (W'-VO)
-WO)2
~~~~~~~~~(WI
(W'-W?)
t/h]/(W'-WO)2
. dWI
2 ao?12t/h. lv(W?+hx/t,y'; WO,yO)12J(WO+hx/t,y')(1-cosxv)/x2.
dx,
Ermtssionand Absorption(f Radliaction6.
if one makes the substitution(W'-W?)t/h
reducesto
21a?12t/h.
v (WO,y' ;
W?,
-
x. For large values of t this
ye)12J(W0,y')
=27r I ao 12tlh.
25 9
!
v (W?,
(1coS
X)/X2.
?
y' ;W?
dx
2J(
')
The probabilityper unit time of a transitionto a state forwhicheach yk,lies
betweenY,' and yk'+ dyk>'is thus (apart fromthe normalisingfactor)
2
y') dyl'. d2 ' ... dyu-1l,
. I v (WO,y' ; W , yO)12J(WO,
a0 12/h
1ao
(24)
associatedwiththat
elementwhichis proportionalto the square of the nmatrix
energy.
transitionofthe perturbing
To apply this resultto a simplecollisionproblem,we take the ac'sto be.the
componentsof mnomentum
PZ, pg, p, of the collidingelectronand the y's to
be 0 and q4,the angleswhichdetermineits directionof motion. Tf,takingthe
relativitychange of mrasswith velocity into accotint, we let P denote the
equal to (px2 + py2+
resultantmomentumn,
pz2)Y, and E
the energy,
equal to
ofthe electron,in beingits rest-mass,we findforthe Jacobian
(m2C4+P2c2)A2,
jaPi
uPz)-
Psin 0.
a(,,)
E C2
Thus the J (W0,y') ofthe expression(24) has the value
J(W', y') = E'P' sinn0/c2
(2)
whereE' and P' referto that valuefortheenergyofthe scatteredelectroiwxtiich
makesthe total energyequal the initial energyWo (i.e., to that value required
by the conservationof energy).
We must now interpretthe initial value of a (c'), namely,a' 8 (or'--),
whichwe did not normalise. Accordingto ? 2 the wave functionin termsofthe
variables c,.is b (oc')= a (oc')e siW'tohSO thatits initialvalue is
ao 8 (act-_ cc0)e-iWt
a0a(P.'
-
Px0) g (pr - p,Y) g (P'
-
Pz) e-iV't
function*
If we -usethe transformation
(x'lp')
=--(27rh)-3/2eiV3!Xtpx'x'/l7
rule
and the transformation
-
4 (x')
--^ |(x'lp')
+ (p')
dpx' dpy' dpz'
wave functionin the co-ordinatesx, y, z the value
we obtain forthe iniiitial
e-iW'tIh
a0 (22rh)1-2etxp.~P9x'Ih
. The syinbol x is used for brevity to denote x, y. z.
260
P. A. Ml. Dirac.
This correspondsto an initialdistributionof IaQ12(27h)3 electronsper unit
volume. Since theirVeloeityis POC2/Eo,
the numuber
per tmittime strikinga
to theirdirectioni
unit smface at right-angles
ofmotionis Ia 2POc2/(2uh)3 E0.
Dividingthisintothe expression(24) we obtain,witlhthe help of (25),
P
(p'; p) 2
(26)
sin 0' dO'db'.
4n2 (2nh)2 E'
This is the efectivearea that in-st be hit by an electronin orderthatit shall
d withthe energyF. This result
be scatteredin the solid angle sin 0dO' d+'
differsby the factor(27rh)2/2mE'. P'/POfromBorn's.* The necessityforthe
factorP'/P' in (26) could have beei predictedfrointhe prin ipie of detailed
betwecenthe directand
balancing,as the factor jv (p'; pO) 12 iS symnmetrica-l
Teverseprocess s.t
? 6. Applicationto Liyht-Quancota.
We shall now apply the theoryof ? 4 to the case whet the systemsof the
assembly are light-quanta,the theorybeing applicable to this case since lightquanta obey the Einstein-Bjosestatisticsand have no muttualitteraction. A
is in a stationarystate whenit is movingwithconstantmomenlight-quantu-m
tum in a straightline. Thus a stationarystate r is fixedby tthethreecomponentsof momentumof the iqliht-quantum
and a variable that specifiesits
state of polarisationv.We shall workon the ass-umption
that thereare a finite
n mberof thesestationarystates,lyingveryclose to one another,as it would
be inconvenientto use continuousranges. The interactionofthe light-quanta
with an atomic systemwill be describedby a Hamiltonianof the form(20),
in which Hp (J) is the Hamiltonian for the atomic system alone, and the
coefficients
v., are forthe presentunknown. We shall show that this fort
forthe Hamiltonian,with the v,, arbitrary,leads to Einstein's laws forthe
emissionand absorptionofradiation. The light-quant-iium
has the pecuLliaritythat it apparenLtly
ceases to exist
its
wheniit is in one of its stationarystates,narnely,the zero state, in wvhich
momentum,and therefore also its energy,are zero. WVen , light-quantumn
is absorbedit can be consideredto jumnpinto this zero state, and wvhenone is
the zero state to one -inwlich it is
emittedit can be consideredto j'umpfrom-L
* In a more recent paper ('Nachr. Gesell. d. Wiss.,' Gottingen,p. 146 (1926)) Born has
obtained a result in agreementwith that of the prese it paper fornon-relativitymechanics,
'by using an interpretation of the analysis based on the conservation theorems. I amn
indebted to Prof. N. Bohr for seeing an advance copy of this work.
t See Klei and Rosseland, 'Z. f. Physik,' vol. 4, .p 46, equation (4) (192l1).
Emissionand Absorptionof Radiation.
261
inevidence,
so thatit appears
tohavebeencreated.Sincethere
physically
is.
oflight-quanta
nolimittothenumber
thatmaybecreated
inthisway,wemust
thattherearean infinite
oflight-quanta
number
inthezerostate,so,
suppose
thattheN0 oftheHamiltonian
(20)is infinite.
We mastnowhave0G,the
toN., a constant,
variable
canonically
conjugate
since
00= aF/aNo
WWo+ termsinlvolving
No0 or (No + Ir!
and W0is zero. In orderthattheHamiltonian
(20) mayremain
finite
it is,
forthe coefficients
necessary
to be infinitely
small. We shallSeuppose
vo0.
v,.O,
thattheyareinfinitely
smallin sucha way as to makevToNo!
andvorNo
inorder
thatthetransition
finite,
probability
coefficients
maybefinite.Thus.
weput
vrO (No +
1)Ar6 -i
yr vo.NOeio/Il -
12*..
where
vrandvr*arefinite
and conjugate
imaginaries.
Wemayconsider
the
onlyof theJ's and w's of theatomicsystem,
since
v, and v* to be functions
e-ioo'h
and N0oeiOoI
arepractically
theirfactors
therate
(N0 - 1)-2constants,
ofchangeofNobeingverysmallcompared
withNo. TheHamiltonian
(20),
nowbecomes
F
(J)+ ErWrN+ rEr o[v,NA 4/+ v,*(N.1)r-H e-i&/l
HPp
+ Er -0'Es o vrsNr(Ns + 1 -8 rs) et -Gi/h (27);
The probabilityof a transitionin which a light-quantumin the state r is.
absorbedis proportional
to the square ofthe modulusofthat matrixelementof
the Hamiltonianwhich refersto this transition. This matrix elementmustl
come fromthe term v7NIe"OIhin the Hamiltonian, and must thereforebe
proportionalto Nrt'whereN,' is the numberof light-quantain state r before.
the process. The probabilityof the absorptionprocess is thus proportional
to Nr'. In the same way the probabilityof a light-quantumin state r being
emittedis proportionalto (Nr'+ 1), and the probabilityof a light-quantum
in)
state r being scatteredinto state s is proportionalto Nr'(N8'+ 1). Radiative
processesof the more generaltype consideredby Einstein and Ehrenfest,tin
whichmorethan one light-quantum
take part sininltaneously,
are not allowed
on the presenttheory.
To establisha connectionbetweenthe numberoflight-quantaper stationary
state and the intensityof the radiation, we consideran enclosureof finite
volume, A say, containingthe radiation. The number of stationarystates
for light-quantaof a given type of polarisationw:hosefrequencylies in the
t 'Z. f. Physik,' vol. 19, p. 301 (1923).
262
P. A. M. i)irac.
range v. to v,- dVranidwhosedirectionof notion lies in the solid anlgledcx.
The energy
aboutthe directionof motionforstate r willnowbe Avrl2dv.do,/c3.
of the light-quantain these stationarystatesis tlhusN' 2nhv,..Av2dv,dCw)r/c3.
whereIJ is the intensityper unit frequency
This inust equal AcmJ,dvrdor,
about the state r. Hence
ra:ngeof the radia-tioni
1
- Nr (27h)v7/C2,
(28)
to I. and (N' --?1) is proportionalto T2+ (2Ah)v 3/c2.
so t'llmt
NJ. is proportional
to
We thus obtaiinthat the probabilityof an absorptionprocessis proportional
of
an emission
-er unit trequency
range, and that
It.,thtencident intermsity
process is proportionalto 1,+ (2nh)v3/C2, which are just Einstein's laws.
a light-quantutm
ofa processin wNThich
is scattered
In thesamewaytheprobability
a state 'rto a state s is proportio1alto 1, [LI+ (27ch)V73/c2], whichis Pauli's
fro:m
ofradiationby a meiectron.t'
law forthe sea-ttering
for EiiiissionantdAbsorption.
? 7. The Probability
Coeficietlds
fromnthewave
We,shallnowconsidertheinteractionofan atomand radiationi
and
point of view. We resolve the radiatior into its Fourier comiponents,
is
f-inite.
Let
each
niumber
yery
but
be
their
component
large
suLpposethat
labelledby a suffixr, and supposethereare a. componentsassociated witlhthe
type of polarisation per unit solid ai-gle per -unitfreradiation of a definiite
quencyrangeabolt the componentr. Each componentr can be describedby
a vector potential 1c, chosemso as to inake the scalar potential zero. The
termto be added to the flamiltonianwill now be, accordingto
perturbation
the classicaltheorywithneglectof relativitymechanics,C-41X. r. X,rwhereXT
is the componentof the total polarisationof the atom in the directionof K2,
whichis the directionof the electricvectorof the componentr.
nical
We can, as explainedin ? 1, supposethe fieldto be describedbythe ca mo
varialblesNr, 02,of whichN, is the numberofquanta of energyof the componentir,anad0, is its canonicallyconijugatephase, equLalto 27chv,timesthe
0r of ? 1. We shall now have c, ar cos 0,/h,where ar iS tne amplitudeof
97 whichcan be connectedwithN, as follows:-The fow of energyper uinit
KIC
areta per unit time forthe componentr is 17rc1 a 2Vr2. Hence the intensity
-
Llie ratio of stimtulatedto spontaneous emission in the present theoryis just twice its
value in Einstein's. This is because in the present theory either polarised component of
the incident radiation can stimulate only radiation polarised in the same way, while in
applies also
Einstein's the two polarised components are treated together. This remnark
to the scatteringprocess.
t Pauli, 'Z. f. Physik,' vol. 18, p. 272 (1923).
Emissionand Absorptionof Radiationt.
263
pet unit frequfency
range of the radiationin the neighbourhoodof the coma2 2.
Coinparingthis with equation (28), we obtain
2 (hvr/cajr)`Nj)and hence
a,,
ponent r is T2227r-1
K2.=
2 (hvr/5cc)` N37Cos Or/ht.
The T:amiltonian
forthe wholesystemnof atom plus radiationwould now be,
accordingto the classical tlleory,
F Hp (J) -Lr (27ihVr)N, + 2c1 , (hv,/Cn) XrNr!cos 0,/h,
(29)
whereH, (J) is the Halmiltonianforthe atomlalone. On the quanturm
theory
we must make the variablesNr anid 01 canionicalq- iumberslike the variables
We mnst nlow replace the N7, cos 0,/hin (29)
Jk, w1kthat describethe atoi.
by the real q-nuimber
6i rl + 6---i0rjhN
rlh
N- t
(.4 -1- Ci9rII2}
~iNfiNj e'Io+~zrl2Nj-1~{
e%f
e
(N2.
,sothat the liamiltonian(29) becomnes
XF
t1P (2
) +
+ hA
C
ES (2J;71hVr) N2.X
>2;r
(V)r/(Tr)~
2'X {Nq.gi8 I + (Nr+
v. = v.*-h (h2/Gr)'
cvr
i!t}.
(30)
This is of the form(27), -with
and
)~ C
Xr
(31)
(r, s X0).
0
The wave pointofviewis thusconsistentwiththe light-quantunm
point ofview
and gives values for the unkniowninteractioncoefficientvrs in the lightquantum theory. These val-uesare not such as would enable one to express
the interactionenergyas an algebraic functionof canonicalvariables. Since
the wave theorygives vr.s. 0 forr,as 0, it wouldseemto showthat thereare
no directscatteringprocesses,b-utthis may be due to an incompletenessin
the presentwave theory.
We shall nowshowthat the Hamiltonian(30) leads to the correctexpressions
forEinsteirn's A's and B's. We mnust
firstmodifyslightlythe anal is of ?5
so as to applyto thecase whenithesystemhas a largenumberofdiscretestationary states instead of a continuousrange. instead of equation (21) we shall
now have
ih a (Vc) -V (oa'oc")a
If the systemis initiallyin the state oo,we musttake the initialvalue of a (')
to be 8
whichis now correctlynormalised. This gives fora firstapproximnation
whichleads to
ih
c
(z)=V
ih a (x')-A
ieLW('>)
(cx'ot0)
_--
v(c4'x?
+ v (o'0?.) ; W
eiL
'-W(?
-
w(ol?)]
t|
1L
-
P. A. II. Dirac.
264
to the variablesW, y,
to (22). If, as before,we transform
corresponding
yO)
Y2
Yff-1bwe obtain(wheny'
a (W'y') = v (W', y'; w
_WO).
TO) L[-Ce(W'-W0)t/h]/(Wf
The probabilityof the system being in a state forwhich each yk equals Ykt
t
is Yw, Ia (W' y')12. If the stationarystates lie closetogetherand ifthe timYle
is nottoo great,we can replacethissumbytheintegral(/AW)-ij' a (W'y') 2 dW',
whereA.Wis theseparationbetweenthe energylevels. Evaluatingthisintegral
as beforewe obtainforthe probabilityper unit timeof a transitionto a state
forwhicheach YkT Yk'
(32)
. V (WO,y'; W0, 0) 12.
2n/hAW'I
In applyingthis resulltwe can take the ys to be any set of variablesthat are
independentof the total properenergyW and that togetherwith W define
a stationarystate.
We now returnto the problemdefinedby the Hamiltonian(30) and considler
an absorptionprocessin whichthe atom jumps fromthe state JOto the state
fromstate r. We talkethe variables
JXwiththe absorptionofa light-quantum
y' to be the variables J' of the atom togetherwith variablesthat definethe
directionof imotion and state of polarisationof the absorbed quantuml,burt
not its energy. The matrixelemeictv (W0, y'; W0, TO) is now
h1/C
C:
/,g) /2Xv(J?J')Nro?
( Mr
whereX)r(JJX)is theordinary(J?J')matrixelementofX.
probabilityper unit time of the absorptionprocessis
Hence from(32) the.
27vc hvlk (J0Xr ('JoJ)
1 2N0
z,7w
jcr
comesfrom
To obtain the probabilityforthe processwheiithe light-quantnm
anydirectionin a solidangledc, we musti.aultiplythisexpressionbythenusmber
of possible directionsforthe ligbt-quantumin the solid angle d4, which is
dt) grAW/27rh.This gives
Tcl
a
X (J2J)
Nr0
2
1
-
X (J?J') 12I
for the absorption
with the help of (28). Hence the probabilitycoefficient
value forEiThin
the
usual
X
with
is
agreement
12,
Xr(J0J')
process 1/272cvr2.
A for
in the matrix mechanics. The agreemel
stein's absorptioncoefficient
manner.
verified
the
same
in
be
coefficients
may
emission
the
Emission and Absorptionof Radciation.
265
The presenttheory,since it gives a properaccountof spontaneousemission,
must presumablygive the effectof radiationreactionon the emittingsystem,
and enable one to calculate the natural breadthsof spectrallines, if one can
overcomethe mathematicaldifficulties
involvedin the generalsolutionof the
wave problemcorresponding
to the Hamiltonian(30). Also the theoryenables
one to understandhowit comesaboutthatthereis no violationofthelaw ofthe
conservationof energywhen, say, a photo-electron
is emit;tedfroman atom
underthe action of extremelyweak incidentradiation. The energyof interactionofthe atom and theradiationis a q-numiber
that does not commutewith
the firstintegralsof the motionof the atom alone or withthe intensityof the
radiation. Thus one cannot specifythis energyby a c-numberat the same
timethat one specifiesthe stationarystate ofthe atom and the intensityofthe
radiation by c-numbers. In particular,one cannot say that the interaction
energytends to zero as the intensityof the incidentradiation tends to zero.
There is thus always .an unspecifiableamount of interactionenergywhich
can supplythe energyforthe photo-electron.
I wouldlike to expressmy thanksto Prof.Niels Bohr forhis interestin this
workand formuch friendlydiscussionabout it.
Summary.
The problemis treated of an assembly of similar systemssatisfyingthe
Einstein-Bose statistical mechanics, which interact with another different
system,a iHamiltonianfunctionbeing obtainedto describethe motion. The
theoryis applied to the interactionof an assemblyof light-quantawith an
ordinaryatom, and it is shownthat it gives Einstein's laws forthe emission
and absorptionofradiation.
waves is then considered,
.The interactionof an atom with electromagnetic
and it is shownthat if one takes the energiesand phases of the waves to be
q-numberssatisfyingthe proper quantum conditionsinstead of c-numbers,
treatthe Hamiltonianfunctiontakes the same formas in the light-quantum
ment. The theoryleads to the correctexpressionsforEinstein'sA's and B's.
VOL. CXIV.-A.
T