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DOUBLE-PHOTON EMISSION
is imposed that the laws of conservation of en-ergy and momentum shall hold universally andexactly in reactions between radiation and mat-ter. The problem is attacked by study of themechanical interactions of radiation and matterencountered in the radiation pressure of Maxwell.It is found that the above requirements firstdemand the variation of mass with velocity, andin turn the variations of dimensions and clockrate according to the factor (1 -v2 /c2)I appro-priately placed. The impossibility of detectingthe earth's motion through the radiation-trans-mitting medium follows, as in Lorentz' theory,as a consequence. The nul result of the Michelson-Morley experiment appears not as a postulate(as in the special theory of relativity) but as acorollary of older physical laws of wide gener-
ality, demanding no "new principle.'12 In thesame list of corollaries are the variation of massand clock rate.
Of experimental tests for establishing the re-ality of these contractions, however arrived at,the most crucial is the variation of clock rate;for while the Michelson-Morley experiment hasthe uncertainty that it might be explainable byan entrained ether (Stokes), and the electricalexperiments on variation of mass might be ex-plained by variation of charge with velocity, asis recurrently proposed, no comparable alterna-tives appear to weaken the conclusiveness of theexperiment on clock rate.
12 Newton, Rules of Reasoning in Philosophy, "We areto admit no more causes of natural things than such as areboth true and sufficient to explain their appearance,"Rule 1.
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 37, NUMBER 10 OCTOBER, 1947
Correspondence Principle Analysis of Double-Photon Emission
J OHN A. WHEELERPalmer Physical Laboratory, Princeton University, Princeton, New Jersey
(Received July 21, 1947)
The process of simultaneous emission of two photons (of continuous probability distributionin energy) in the transition from one discrete quantum level to another is here analyzed in thecorrespondence principle limit in which the quantum numbers of the two states are large andthe change in quantum number is small by comparison. In this limit, it proves possible to de-fine a classical dynamical quantity [Eq. (19)], the frequency analysis of whose classical varia-tion with time in a definite orbit determines in the given approximation the absolute transitionprobability and intensity distribution in the continuous spectrum in question.
INTRODUCTION AND SUMMARY
A HYDROGEN atom which arrives at themetastable 2s-state by recombination or
otherwise, will be unable to drop down to thefundamental ls-state by emission of a singlephoton of any polarity whatsoever, providedthat the atom is located in a region of inter-stellar space where it is free from the perturbingelectric fields of nearby ions and from the ex-citing influence of a flux of radiation. Underthese circumstances the system will neverthelessget to the ground state in a mean time of theorder of 1/7 second, according to Breit andTeller,' through a process in which two photons
1 G. Breit and E. Teller, Astrophys. J. 91, 215 (1940).Note added in proof: Eq. (2) predicts a transition prob-
of complementary energy come offtaneously :2
hcw.l+hW2=E(2s) -E(ls).
simul-
(1)
The condition of energy conservation leavesarbitrary the division of the frequency betweenthe pair of quanta. The authors in questioncalculate the intensity distribution in the con-tinuous spectrum, finding a broad maximumcentered at the point of equal photon energies.
The results of Breit and Teller were obtainedability smaller by a factor 27r than that given by Breit andTeller. To date the source of the discrepancy has not beenlocated.
2 Here and below we use co to designate circular fre-quency, in units radians per sec.; and use h to represent thequantum of angular momentum (1.027X10-27 g cm 2 /sec.)as distinct from the quantum of action, h.
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JOHN A. WHEELER
by a standard application of the second-orderperturbation theory of quantum mechanics inwhich-to use a purely mathematical manner ofspeaking-account is taken of the possibility ofvirtual transitions from the 2s-state to all p-states, and from these states down to the s-level. The result of the calculation takes asimple form. It yields the probability per sec-ond, dAKX, of a double-photon transition from theexcited state (labeled b) to the ground state(designated a) in which the photon of smallercircular frequency, coi, comes off in the interval&,, to wl+dw, and the one of higher frequencycomes off in the complementary energy interval,and in which furthermore (to be more detailed)the "virtual oscillator" responsible for the firstphoton is polarized along the K axis (K = x, y, or z)and the other virtual oscillator is polarized alongthe axis (X=x, y, or z):
dA Ki = (4e4 /97r 2 c)coW 23 dw/VIbaK (O) /2 . (2)
Here the composite matrix element for the jumpb-*a is given by the sum over all virtual states,v, (of energy E) for which the relevant ele-mentary matrix elements do not vanish:
MbaK" ((D) = XbvXXvaX/ (E - o,- Ea)
+ bvXXva/ (Ev - i)2 -Ea). (3)
Here xK and xX are the components of the dis-placement of the radiating electron along the K
and X axes. When the polarization is not a matterof concern, the total probability per second, dA,of transition with emission into da is found bysumming dAK1 over all three directions of polari-zation, K, and independently over all three di-rections of polarization, X. From Eq. (3), and itsnatural generalization to many-electron atoms,follow certain obvious selection rules for two-quantum emission, such as the requirement thatthe angular momenta of initial and final statesdiffer either by 0 or by 2h, and that both stateshave the same parity.
Apart from these simple deductions from thequantum formulae above, it is difficult to formfor a general atomic or molecular system in aparticular level of excitation a qualitative ideawhether double photon emission is favored ornearly forbidden (as double-photon process goes!) .
To translate this question of rate of the two-
quantum process from the realm of quantummechanics to that of classical mechanics, viathe correspondence principle, is the purpose ofthis note. It is well known that the quantumtreatment of single-photon emission has a closerelation to the classical radiation theory. Like-wise, one has to expect that the above formulaefor rate of double-photon emission can be setinto correspondence with a formula which refersonly to quantities which are well defined for asingle classical orbit, in such a way that the classi-cal and quantum expressions will come intoasymptotic agreement with each other in thelimit of high quantum numbers.' Specifically, itmust be possible, in principle, to find a classicaldynamic quantity-in the same sense that co-ordinates and momenta are such quantities-thefrequency analysis of whose classical variationin time for a given orbit will determine theprobability for two-quantum emission with anaccuracy which is the higher, the closer one is tothe correspondence principle limit of largequantum numbers.
We are successful in our goal to the extentthat we are able to define a classical quantity ofmotion which determines the emission- prob-ability in the given approximation. We are un-successful in the present note in the larger aim ofreadily visualizing double-photon radiation inthe sense that the classical dynamical quantityin question has a not easily grasped relation tosuch usual features of the motion as displace-ment and velocity. However, a little insight intothe features of the motion responsible for two-quantum emission is acquired by applying thegeneral correspondence principle treatment totwo idealized cases of one-dimensional motion(although these particular results could be de-rived directly from the quantum formulae (2)and (3)). In the case of the harmonic oscillator,double-photon emission is found to have zeroprobability for all choices of initial and final
I For a discussion of the correspondence principle in thelight of wave mechanics see, for example, H. A. Kramers,Quantentheorie des Elektrons und der Strahlung (Akadem-ische verlagsgesellschaft, Leipzig, 1938), Vol. II, Section 85,also appears in Vol. I, Part II of Hand- und Jahrbuchder Chemischen Physik. For a correspondence principletreatment of double-photon processes before the periodof wave mechanics, see H. A. Kramers and W. Heisenberg,Zeits. f. Physik 31, 681 (1925). For a wave-mechanicalanalysis, see M. Goeppert Mayer, Ann. d. Physik 9, 237(1931).
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DOUBLE-PHOTON EMISSION
state. In the case of an otherwise free particlewhich moves back and forth between two rigidwalls, the probability of simultaneous emissionof two quanta in a continuous spectrum is foundin the correspondence principle approximationto be independent of separation of the walls,provided the frequency of collisions with thewalls is fixed; and this probability vanishes ex-actly unless the quantum numbers of the upperand lower levels differ by an even integer. Moregenerally, in the case of a one-dimensional mo-tion, only when the potential is asymmetric withrespect to a reversal of the sign of displacementswill double-photon jumps be allowed in whichthe quantum number changes by a single unit.
The remainder of this note is devoted to thederivation of the correspondence principle in-tensity formula and the examination of the twoexamples just mentioned.
DETAILS
The starting point for a classical treatment ofthe radiation from a dynamical system in agiven state of motion is usually taken to be aFourier analysis of the intensity-determiningquantities of the motion in terms of the funda-mental frequency of the motion, c, and itshigher harmonics, kw, thus:
displacement =x(t)= Xk exp(ikcot), (4)k=-Xo
(where the reality of x(t) requires that
X-k = Xk*) and
acceleration = -k 2 co2Xk exp(ikwt). (5)k=-co
For instance, to take a well-known example, theclassical expression for rate of loss of energy byradiation,
(2e2 /3c3) (acceleration) 2 average, (6)
takes in the Fourier representation the form
Z (4e 2 /3c3) (kW)4 Xk 12, (7)k=1
from which follows, in the approximation of thecorrespondence principle, the probability persecond, Ak, of a jump from the given state to a
state lower by k quantum units:
Ak = (4e2/3hc3) (kw)3 Xk 2 (8)
in direct relation to the accurate non-relativisticquantum formula for dipole radiation:
Ab b_- = (4e2/ 3hC3 ) (VWb, b-k) 3 Xb, b-k 2 * (9)
As this example recalls, we translate from classi-cal theory to quantum theory and conversely bythe approximate correspondences:
for frequencies: kC>Ctb, b-k,
for energies: khwoEb-E
for displacements: Xk->Xb,b-k,
b-7,
(lOa)
(lob)
or,
T-1 fs dt exp(-ikwt)x(t)->flb*X1b-kdx. (10c)0
Here T is the period 2r/w associated with thefundamental frequency w. For simplicity weshall here and hereafter idealize in one indexthe multiplicity of quantum numbers which willbe required in general to specify the typicalquantum state, just as in the classical treatmentwe shall typify in a simple Fourier series, witha single action and angle variable, the severalsets of such conjugate quantities which will beneeded in actuality to describe the motion. InEqs. (10) it is not stated whether the funda-mental frequency and the Fourier componentsXL. are to be evaluated for that classical motionwhich possesses bh units of action, or for someintermediate classical orbit. This ambiguity is,of course, typical of the correspondence prin-ciple, and the associated uncertainty in thetranscription can evidently only then be as-sumed to be negligible, when the quantum num-bers b and b - k are very large in comparison tothe quantum change, k.
Proceeding according to the correspondenceof Eq. (fO), just as in the case of single-quantumradiation, we transcribe the matrix element ofEq. (3) for double-photon emission into classicalterms, obtaining
Mba. (O) -*h v Xb-vXva/ [ (v-a)w- w]+h ±rZ Xb-vKXv-a/[(va)W 2 ]
= hs, k Xb-a-k\KkK/(kw -wi)
- ' Ysk XkKXb-a-kX/(ko -w 1 ) = 0. (11)
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JOHN A. WHEELER
This unexpected and zero result shows that wehave to go to a higher order of approximation inthe application of the correspondence principleto obtain the desired classical formula, and sug-gests that this formula when obtained will have aless direct connection with the quantum treat-ment than was the case for single-photonemission.
We would have obtained in a much morefamiliar part of quantum theory a result likewiseequal to zero and likewise misleading if we hadapplied the correspondences of Eq. (10) withoutfurther consideration. We should have found inthis way for the commutator of Heisenberg,
it-1 57 - Xfl&V Vf1) (12)
a value identically equal to zero, instead of thecorrect figure of unity. Both in this example andin the present problem the means of correctionis simple. We need only insert for a quantitysuch as p the m-v Fourier element of theclassical quantity p(t) in the mth quantum state,and for xvm the v-m Fourier element of x(t) inthe vth quantum state. We need not considerfurther the example of the Heisenberg commuta-tion relations, where the appropriate analysishas already been given elsewhere,4 and shallonly recall the following general points. First,the Fourier series for the classical variables givetheir expression in terms of conjugate actionand angle variables, J= (2t)-1fcyiep(t)dx(t) andw=cot, in the form
x(E, t) = s£,h X(J) exp(ikw). (13)
Second, in the equation connecting the anglevariable, w, with , the circular frequency,=co(J), is given by the derivative of the energy
with respect to the action variable, J. Third, theenergy of the kth quantum state is to. be con-sidered as given by setting J=kk in the equationE=E(J). Fourth, owing to the invariance of theHamiltonian equations with respect to change ofconjugate variables, we may write the Poissonbracket of any two dynamical quantities, B andC, in any one of the following forms:
4 See, for example, Ruark and Urey, Atoms, Mfoleculesand Quanta (McGraw-Hill Book Company, Inc., NewYork, 1930), p. 585.
{B C}
= (aB/ap)(aC/ax) - (aC/p)(aB/Ox)
= (aB/OE) (a C/at) - (a C/aE) (B/at)
= (aB/OJ)(aC/w)-(aC/OJ)(aB/aw). (14)
Proceeding in accordance with these prin-ciples, we now replace the terms in the matrixelement of Eq. (3) as follows, using as our stand-ard of reference the quantities of motion in thebth quantum state:
Xbv->Xb-v,
X +a+Xv-a± (v - b)h(dXv..a/dJ)
EV-*Eb+ (v-b)hco+2(v-b) 2 h2(dw/dJ). (15)
Then we find for the desired matrix element theexpression
Mba 24[k- wi]-2[ (b-a-k)Xb- a_lkX,
X (dw/dJ) ]+ [kc- w1]-1E (dXb a,_/dJ)
XkX," - (dXkK/dJ) (b-a-)Xbaixj. (16)
To obtain an expression for M which hasmore evident meaning than (16), though nolonger its manifest symmetry between the twodirections of polarization, we define for eachcomponent of displacement, x(t), of the radiat-ing particle a new associated classical dynamicalvariable, s(t), through the equation
+1C +iwl S=X, (17)
and the supplementary condition that s beperiodic in time with the same period, T whichappertains to the displacement, x. Furthermore,we use the Poisson bracket notation of (14),and employ the subscript b-a in connectionwith any dynamical quantity, Q(t), to denotethe (b-a)th Fourier coefficient of that dy-namical variable:
Qb-a= T-lf dt exp(-i(b-a)wt)Q(t). (18)
Then we have for the desired correspondenceprinciple transcription of the matrix element ofEq. (3) the very simple formula
MbaIX{X)X, SKIba.- (19)
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DOUBLE-PHOTON EMISSION
With this result we have achieved our aim ofexpressing in terms of the quantities charac-teristic of a single classical orbit the probabilityfor double-photon emission. It is remarkable thatthe final formula [the result of Eq. (19) insertedinto Eq. (2)] contains no mention of the quantumof action, in contrast to the expression (8) for theprobability of single-photon emission. As alreadynoted, no easily interpreted visualization of (19)is evident.
Application of our general result to the caseof the one-dimensional harmonic oscillator isstraightforward. The displacement, x(t), varieswith time in a pure sinusoidal manner, with noovertones. Accordingly, the associated dynamicalquantity s(t) of Eq. (17) will likewise contain inits Fourier analysis only the two terms sinwt andcoswt. Consequently s will be representable as alinear combination of the displacement, x(t), andthe momentum, p(t). The x part of s will dropout when we evaluate the Poisson bracket {x, s},and the p part will give a constant in virtue ofthe relation { x, p = -1. Thus { x, s } is equal toa constant. But all the coefficients in the Fourieranalysis of a constant are zero, with the excep-tion of the constant term, in which we are notinterested. Consequently double-photon emis-sion is forbidden for all transitions in the caseof a harmonic oscillator.
In the case of any one-dimensional potentialsymmetric with respect to the interchange of xand -x, the Fourier series for the displacementx(t) will lack all even harmonics. As a conse-quence, the dynamical quantity, s(t), whateverbe the value of xi, will also contain odd har-monics exclusively.'The Poisson bracket, {x, s},being biquadratic in the two quantities, willtherefore be built up entirely of even harmonics.Therefore double-photon emission will be for-bidden between two states unless their quantumnumbers, b and a, differ by an even integer.
In the special case where the potential con-sists of an infinite wall at x = AhL, and vanishesbetween these limits, the calculations may becarried through explicitly. In terms of the period,
T, of the motion, we take
x = (4L/T) (T/4 -I) for O <t < T/2,x = (4L/T) (t-3T/4) for T/2 <t < T. (20)
Writing 0 as an abbreviation for w1T we find
S-ix/cll = (4L/co2T) { - 1 +i[sin(0/2)]-1XE[-1 +exp(-i0/2)2 exp(ico1t)J }, (21)
in the interval 0<t< T/2, and in the next half-period a similar expression, obtained by re-placing the first -1 by + 1, and the second -1by -exp(-iO).
Whilst x(t) and s(t) repeat themselves onlyafter the interval T, and are continuous func-tions, the Poisson bracket {x, s} is discontinuousat 0, T/2, T, etc., and repeats itself after theinterval T/2. In terms of the time r=t-T/4,t-3T/4, etc., measured from the center of anysuch interval, we have
(mwi/16T) {x, s} = -0-+exp(iwir)X { [2 sin(0/4)/O sin(0/2)]
- sin3(0/4)/sin2(0/2)]-(ir/T)E2 sin(0/4)/sin(0/2)]}. (22)
The coefficients in the Fourier expression of (22)are readily found, and yield for the probabilityper second of double-photon emission in thejump from a given level to a state two quantumunits lower the result
dA - (1638 4 /92r2 ) (e4 /m 2 c6 )co2dw
X Esin4(0/4) /sin 4(0/2)]{1+2 sin(0/2)X [(47r- O)-1 0-i]} .2 (23)
Comparison of this result with an order-of-magnitude lower limit estimate of the proba-bility per second for single-photon emission,
A I. (e2 /m 3 ) C02,
shows the very low probability of double-quantum emission. It will be noted, however,that the probability given by (23) diverges when,keeping c01 +CO2=
2W, we let oi approach the
fundamental frequency co. Obviously the con-cept of double-quantum emission breaks downwhen one approaches the values of frequencyappropriate to the normal emission of two suc-cessive quanta.
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