The Quantum Theory of Radiation

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1 A. Einstein, Physikalishe Zeitshrift 18, The Quantum Theory of Radiation A. Einstein (Reeived Marh, 1917) The formal similarity of the spetral distribution urve of temperature radiation to Maxwell s veloity distribution urve is too striking to have remained hidden very long. Indeed, in the important theoretial paper in whih Wien derived his displaement law ( ) ν ρ = ν 3 f (1) T he was led by this similarity to a farther orrespondene with the radiation formula. He disovered, as is known, the formula [Wien s radiation formula] ρ = αν 3 e hν kt (2) whih is reognized today as the orret limiting formula for large values of ν T. Today we known that no onsideration whih is based on lassial mehanis and eletrodynamis an lead to a useful radiation formula; rather that the lassial theory leads to the Rayleigh formula. ρ = kα h ν2 T (3) After Plank, in his ground breaking investigation, established his radiation formula ρ = αν 3 1 ekt hν (4) 1 1

2 on the assumption that there are disrete elements of energy, from whih quantum theory developed very rapidly, Wien s onsiderations, from whih formula (2) evolved, quite naturally were forgotten. A little while ago I obtained a derivation, related to Wien s original idea, of the Plank radiation formula whih is based on the fundamental assumption of quantum theory and whih makes use of the relationship of Maxwell s urve to the spetral distribution urve. This derivation deserves onsideration not only beause of its simpliity, but espeially beause it appears to larify the proesses of emission and absorption of radiation in matter, whih is still in suh darkness for us. In setting down ertain fundamental hypotheses onerning the absorption and emission of radiation by moleules that are losely related to quantum theory, I showed that moleules with a distribution of states in the quantum theoretial sense for temperature equilibrium are in dynamial equilibrium with the Plank radiation; in this way, the Plank formula (4) was obtained in a surprisingly simple and general way. It was obtained from the ondition that the quantum theoreti partition of states of the internal energy of the moleules is established only by the emission and absorption of radiation. If the assumed hypotheses about the interation of matter and radiation are orret, they will give us more than just the orret statistial partition or distribution of the internal energy of the moleules. During absorption and emission of radiation there is also present a transfer of momentum to the moleules; this means that just the interation of radiation and moleules leads to a veloity distribution of the latter. This must early be the same as the veloity distribution whih moleules aquire as the result of their mutual interation by ollisions, that is, it must oinide with the Maxwell distribution. we must require that the mean kineti energy whih a moleule (per degree of freedom) aquires in a Plank radiation field of temperature T be kt 2 ; this must be valid regardless of the nature of the moleules and independent of frequenies whih the moleules absorb and emit. In this paper we wish to verify that this far reahing requirement is, indeed, satisfied quite generally; as a result of this our simple hypotheses about the emission and absorption of radiation aquire new supports. In order to obtain this result, however, we must enlarge, in a definite way, the previous fundamental hypothesis whih were related entirely to the exhange of energy. We are faed with this question; Does the moleule suffer a push, when it absorbs or emits the energy ɛ? As an example we onsider, 2

3 from the lassial point of view, the emission of radiation. If a body emits the energy ɛ, it aquires a bakward thrust [impulse] ɛ if all the radiation ɛ is radiated in the same diretion. If, however, the radiation ours through a spatially symmetri proess, for example, spherial waves. there is then no reoil at all. This alternative also plays a role in the quantum theory of radiation. If a moleule, in going from one possible quantum theoreti state to another, absorbs or emits the energy ɛ in the form of radiation, suh an elementary proess an be looked upon as partly or fully direted in spae, or also as a symmetri (non direted) one. It turns out that we obtain a theory that is free of ontraditions only if we onsider the above elementary proesses as being fully direted events; herein lies the prinipal result of the onsiderations that follow. Fundamental Hypotheses of the Quantum Theory Canonial Distribution of State Aording to the quantum theory, a moleule of a definite kind may, aside from its orientation and its translational motion, be in one only a disrete set of states Z 1, Z 2,... Z n... whose (internal) energies are ɛ 1, ɛ 2,... ɛ n.... If the moleules of this kind belong to a gas of temperature T, then the relative abundane W n of the state Z n is given by the statistial mehanial anonial partition funtion for states W n = p n e ɛ n kt (5) In this formula k = R N is the well-known Boltzmann onstant, p n a number that is independent of T and harateristi of the moleule and the state, whih we may all the statistial weight of the state. Formula (5) an be derived from the Boltzmann priniple or purely from thermodynamis. Equation (5) is the expression of the most far reahing generalization of the Maxwellian distribution of veloities. The latest important advanes in quantum theory deal with the theoretial determination of the quantum theoretial possible states Z n and their weights p n. For the prinipal part of the present investigation, it is not neessary to have a more detailed determination of the quantum states. 3

4 Hypotheses about the Energy Exhange Through Radiation Let Z n and Z m be two possible quantum theoretial states of a gas moleule whose energies ɛ n and ɛ m respetively, satisfy the inequality ɛ m > ɛ n Let the moleule be able to pass from the state Z n to the state Z m by absorbing the radiation energy ɛ m ɛ n, similarly let the transition from state Z n to the state Z m be possible through the emission of this amount of energy. Let the radiation emitted or absorbed by the moleule for the given index and ombination (m, n) have the harateristi frequeny ν. We now introdue ertain hypotheses about the laws whih are deisive for these transitions. These hypotheses are obtained by arrying over the known lassial relations for a Plank resonator to the unknown quantum theoretial relations. Emission A Plank resonator that is vibrating radiates energy aording to Hertz, in a known way independently of whether it is stimulated by an external field or not. In aordane with this, let a moleule be able to pass from the state Z m to the state Z n with the emission of radiant energy ɛ m ɛ n of frequeny ν without being exited by any external ause. Let the probability dw for this to happen in the time dt be dw = A n m dt (A) where A n m is a harateristi onstant for the given index ombination. The assumed statistial law orresponds to that of a radioative reation: that elementary proess of suh a reation in whih only γ-rays are emitted. We need not assume that this proess requires no time; this time need only be negligible ompared to the times whih the moleule spends in the states Z 1, and so on. Inident Radiation If a Plank resonator is in a radiation field, the energy of the resonator hanges beause the eletromagneti field of the radiation does work on the resonator; this work an be positive or negative depending on the phases of the resonator and the osillating field. In aordane with this, we introdue the following quantum theoretial hypothesis. Under the ation of the 4

5 radiation density ρ of the frequeny ν a moleule in state Z n an go over to state Z m absorbing the radiation energy ɛ m ɛ n in aordane with the probability law dw = B m n ρ dt (B) In the same way, let the transition Z m Z n under the ation of the radiation also be possible, whereby the radiation energy ɛ m ɛ n is emitted aording to the probability law dw = B n m ρ dt (B ) Bn m and Bm n are onstants. We all both proesses hanges of states through inident radiation. The question presents itself now as to the momentum that is transferred to the moleule in these hanges of state. We begin with the events assoiated with inident radiation. If a direted bundle of rays does work on a Plank resonator, then an equivalent amount of energy is removed from the bundle. this transfer of energy results, aording to the law of momentum, to a momentum transfer from the beam to the resonator. The latter therefore experienes a fore in the diretion of the ray of the radiation beam. If the energy transferred is negative, the fore ating on the resonator is opposite in diretion. In the ase of the quantum hypothesis, this learly means the following. If, as the result of inident radiation, the proess Z n Z m ours, then an amount of momentum ɛ m ɛ n is transferred to the moleule in the diretion of propagation of the bundle of radiation. If we have the proess Z m Z n for the ase of inident radiation, the magnitude of the transferred momentum is the same, but it is in the opposite diretion. If a moleule is simultaneously exposed to many bundles of radiation, we assume that the total energy Z m Z n is taken from or added to just one of these bundles, so that even in this ase the momentum ɛ m ɛ n is transferred to the moleule. In the ase of emission of energy by radiation by a Plank resonator, there is no net transfer of momentum to the resonator beause, aording to lassial theory, of emission ours as a spherial wave. However, we have already noted that we an arrive at a ontradition free quantum theory 5

6 only if we assume that the proess of emission is a direted one. Every elementary proess of emission (Z m Z n ) will then result in a transfer to the moleule of an amount of momentum ɛ m ɛ n. If the moleule is isotropi, we must take every diretion of emission as equally probable. If the moleule is not isotropi, we arrive at the same result if the orientation hanges in a random way in the ourse of time. We must, in any ase, make suh an assumption also for the statistial laws (B) and (B ) for inident radiation sine otherwise the onstants Bn m and Bm n would have to depend on diretion, whih we an avoid by assuming isotropy or pseudo isotropy (through setting up temporal mean values). Derivation of the Plank Radiation Law We now enquire about those effetive radiation densities ρ whih must prevail in order that the energy exhange between moleules and radiation as a result of the statistial laws (A), (B) and (B ) shall not disturb the distribution of moleular states present as a onsequene of equation (5). For this, it is neessary and suffiient that on the average, per unit time, as many elementary proesses of type (B) take plae as proesses (A) and (B ) together. This ondition gives as a result of (5), (A), (B), (B ), for the elementary proesses orresponding to the index ombination (m, n) the equation p n e ɛ n kt B m n ρ = p m e ɛ m kt (B n m ρ + A n m) If, further, ρ is to beome infinite as T does, the onstants Bn m and Bm n must satisfy the relation p n Bn m = p m Bm n (6) We then obtain as the ondition for dynamial equilibrium the equation ρ = e An m/b n m ɛ m ɛ n kt 1 This is the dependene of the radiation density on the temperature that is given by the Plank law. From the Wien displaement law (1) it then following immediately that A n m Bm n = αν 3 (8) 6 (7)

7 and ɛ m ɛ n = hν (9) where α and h are universal onstants. To obtain the numerial values of α and h we must have an exat theory of eletrodynami and mehanial proesses; we ontent ourselves for the moment with the Rayleigh law in the limit of high temperatures, where the lassial theory is valid in the limit. Equation (9) is, as we know, the seond prinipal rule in Bohr s theory of spetra, about whih we may assert, following upon Sommerfeld s and Epstein s ompletion of the theory, that it belong to the most fully verified domain of our siene. It also ontain impliitly the photohemial equivalent law, as I have already shown. Method for Calulating the Motion of Moleules in Radiation Fields We now turn our attention to the investigation of the motion imparted to our moleules by the radiation field. we make use in this of a method that is known to us from the theory of Brownian motion and whih I have often used in investigating motions in a region ontaining radiation. To simplify the alulation, we shall arry it through for the ase in whih the motion ours only along the X-diretion of the oordinate system. We further ontent ourselves with alulating the mean value of the kineti energy of the translation motion, and thus dispense with proof that these veloities v are distributed aording to the Maxwell law. Let the mass M of the moleule be large enough so that higher powers of v an be negleted relative to lower ones; we an then apply the usual mehanis to the moleule. Moreover, without any loss in generality, we may arry out the alulation as through the states with indies m and n were the ones the moleule an be in. The momentum Mv of a moleule undergoes two kinds of hanges in the short time τ. Even though the radiation is the same in all diretions, the moleule, beause of its motion, will experiene a resistane to its motion that stems from the radiation. Let this opposing fore be Rv, where R is a onstant to be determined later. This fore would ultimately bring the moleule to rest if the randomness of the ation of the radiation field were not suh as to transfer to the moleule a momentum of alternating sign and varying magnitude; this random effet will, in opposition to the previous one, sustain a ertain amount of motion of the moleule. At the end of the 7

8 given short time τ the momentum of the moleule will equal Mv Rvτ + Sine the veloity distribution is to remain onstant in time, the mean of the absolute value of the above quantity must equal that of the quantity Mv; thus, the mean values of the squares of both quantities averaged over a long time or a large number of moleules must be equal: (Mv Rvτ + ) 2 = (Mv) 2 Sine we have taken into aount the influene of v on the momentum of the moleule separately, we must disard the mean value v. On developing the left hand side of the equation we thus obtain 2 = 2RMv 2 τ (10) The mean value v 2 whih the radiation of temperature T by its interation imparts to the moleule must just equal the mean value v 2 whih the gas moleule aquires at temperature T aording to the gas law and the kineti theory of gases. For otherwise the presene of our moleules would disturb the thermal equilibrium between thermal radiation and an arbitrary gas of the same temperature. We must therefore have Equation (10) thus goes over into Mv 2 2 = kt 2 (11) 2 τ = 2RkT (12) The investigation is now to be arried through as follows. For a given radiation density (ρ(ν)) we shall be able to ompute 2 and R by means of our hypotheses about the interation between radiation and moleules. If we put this result into (12), this equation will have to be identially satisfied when ρ is expressed as a funtion of ν and T by means of Plank s equation (4). Computing R Let a moleule of given kind be in uniform motion with speed v along the X-axis of the oordinate system K. We inquire about the momentum transferred on the average from the radiation to the moleule per unit time. To 8

9 alulate this we must onsider the radiation from a oordinate system K that is at rest with respet to the given moleule. For we have formulated our hypotheses about emission and absorption only for moleules at rest. The transformation to the system K has often been performed in the literature. Nevertheless, I shall repeat the simple onsiderations here for the sake of larity. Relative to K the radiation is isotropi, that is, the quantity of radiation in a solid angle dκ in the diretion of the radiation in a frequeny range dν is ρdν dκ (13) 4π where ρ depends only on the frequeny ν but not on the diretion of the radiation. This speial beam orresponds to a speial beam in the system K whih is also haraterized by a frequeny range dν and a solid angle dκ. The volume density of this speial beam is ρ (ν, φ )dν dκ 4π (13 ) This defines ρ. It depends on the diretion of the radiation whih, in the familiar manner, is defined by the angle φ it makes with the X axis and whih its projetion on the Y, Z plane makes with the Y axis. These angles orrespond to the angles φ and ψ whih in an analogous manner determine the diretion of dκ in K. To begin with, it is lear that the same transformation law between (13) and (13 ) must hold as between the amplitudes A 2 and A 2 of a plane wave moving in the orresponding diretion. Hene, to our desired approximation we have ρ (ν, φ )dν dκ = 1 2 v os φ (14) ρ(ν)dνdκ or ρ (ν, φ ) = ρ(ν) dν dν dκ (1 dκ 2 v ) os φ (14 ) The relativity theory further gives the formulae, valid to the desired approximation, ν = ν (1 v ) os φ (15) os φ = os φ v + v os2 φ (16) ψ = ψ (17) 9

10 from (15) it follows, to the same approximation that ν = ν ( 1 + v os φ ) or Hene, again to the desired approximation ρ(ν) = ρ (ν + v ) ν os φ ρ(ν) = ρ(ν ) + ρ(ν ) ν ( v ν os φ ) Further, aording to (15), (16), and (17) dν dν = (1 + v ) os φ dκ dκ = sin φ dφ dψ sin φ dφ dψ = d(os φ) d(os φ ) = 1 2 v os φ (18) As a result of these two equations and equation (18), equation (14 ) goes over into [ ρ (ν, φ ) = (ρ) ν + v ( ) ] ρ ν os φ (1 3 v ) ν os φ (19) With the aid of (19) and our hypotheses about the radiation from and radiation onto moleules, we an easily alulate the average momentum transferred to the moleule per unit time. Before we an do this, however, we must say something to justify our proedure. It may be objeted that equations (14), (15), (16) are based on maxwell s theory of the eletromagneti field that is not onsistent with the quantum theory. This objetion deals, however, more with the form than with the substane of the problem. For, no matter how the theory of eletromagneti proesses may be formulated, in any ase the Doppler priniple and the law of aberration still remain, and hene also the equations (15) and (16). Moreover, the validity of the energy relationship (14) ertainly extends beyond that of the wave theory; this transformation law is also valid, for example, aording to relativity theory, for the energy density of a mass of infinitesimally small rest density that is moving with the [quasi-] speed of light. We may therefore assert the validity of equation (19) for any theory of radiation. The radiation belonging to the solid angle dκ would, aording to (B), give rise to B m n ρ (ν, φ ) dκ 4π 10 ν

11 elementary proesses per seond of radiation events of the type Z n Z m if the moleule after eah suh proess immediately returned to state Z n. Atually, however, the time of lingering in state Z n, aording to (5), is where we have used the abbreviation 1 S p ne ɛ n kt S = p n e ɛ n kt + p m e ɛ m kt (20) The number of these proesses per seond is therefore atually 1 S p ne ɛ n kt B m n ρ (ν, φ ) = dκ 4π. In eah of these elementary proesses the momentum ɛ m ɛ n os φ is transferred to the moleule in the diretion of the X -axis. In an analogous manner we find, based on (B ) that the orresponding number of elementary proesses of radiation events of type Z m Z n per seond is 1 S p me ɛ m kt Bmρ n (ν, φ ) dκ 4π and in eah suh elementary proess the momentum ɛ m ɛ n os φ is transferred to the moleule. The total momentum transferred to the moleule per unit time by inident radiation is, keeping in mind (6) and (9), hν S p nb m n ( e ɛ n kt e ɛ m kt ) ρ (ν, φ ) os φ dκ where the integration is to be taken over all solid angles. Carry this out, and we obtain with the aid of (19) value hν ( 2 S ρ (1/3) ν ρ ν ) p n B m n ( 4π e ɛ n kt e ɛ m kt ) v. 11

12 Here we have represented the effetive frequeny again with ν and not with ν. This expression gives, however, the total momentum transferred on the average to a moleule moving with speed v. For it is lear that those elementary proesses of emission of radiation not indued by the ation of the radiation field have no preferred diretion as seen from system K and hene, on the average, annot transfer any momentum to the moleule. We thus obtain as the final of our onsiderations R = hν ( 2 ρ 1/3 ν ρ ) p n B mn e ɛ ( n kt 1 e hν ) kt (21) S ν Calulating 2 It is muh easier to alulate the random effet of the elementary proesses on the mehanial behavior of the moleule. For we alulate this for a moleule at rest for whih the approximation whih we have been using applies. Let some event ause the momentum λ to be transferred to a moleule in the X diretion. This momentum is to be of varying magnitude and diretion from moment to moment. However, let λ obey a statistial law suh that its average value vanishes. Then let λ 1, λ 2... be the momenta whih are transferred to the moleule in the X-diretion by various operating auses that are independent of eah other so that the total momentum that is transferred is = Σλ ν We then have (if for the individual λ ν vanish) 2 = Σλ ν 2 (22) If the mean values λ ν 2 of the individual momenta are all equal to eah other (= λ 2 ) and if l is the total number of proesses giving rise to momenta, we have the relation 2 = lλ 2 (22a) Aording to our hypothesis, in eah proess of inident radiation and outflowing radiation, the momentum λ = hν os φ 12

13 is transferred to the moleule. Here φ is the angle between the X-axis and some randomly hosen diretion. Hene, we obtain λ 2 = 1 3 ( ) hν 2. Sine we assume that all the elementary proesses that are present are to be onsidered sas events that are independent of eah other, we may apply (22a), l is then the number of all elementary proesses that our in the time τ. This is twie as large as the number of radiation-inident proesses Z n Z m in the time τ. We thus have From (23), (24) and (22) we thus obtain l = 2 S p nb m n e ɛ n kt ρτ (24) 2 τ = 2 3S ( ) hν 2 p n Bn m e ɛ n kt ρ (25) Results In order now to show that the momenta transferred from the radiation to the moleule aording to our basi hypotheses never disturb the thermodynami equilibrium, we need only introdue the values for 2 τ and R alulated in (25) and (21) respetively after the quantity ( ρ ( 1 / 3 ) ν ρ ) ( 1 e hν ) kt ν in (21) is replaed by ρhν 3RT from (4). We then see that our fundamental equation (12) is satisfied identially. The above onsideration lends very strong support to the hypotheses introdued earlier for the interation between matter and radiation by means of absorption and emission, and through inident and outgoing radiation. I was led these hypotheses in trying to postulate in the simplest possible way a quantum behavior of moleules that is analogous to the Plank resonators of lassial theory. We obtained, without effort, from the general 13

14 quantum assumption for matter, the seond Bohr rule (equation (9)) as well as Plank s radiation formula. Most important, however, appears to me the result about the momentum transferred to the moleule by inoming and outgoing radiation. If one of our hypotheses were altered, the result would be a violation of equation (12); it appears hardly possible, exept by way of our hypotheses, to be in agreement with this relationship whih is demanded by thermodynamis. We may therefore onsider the following as pretty well proven. If beam of radiation has the effet that a moleule on whih it is inident absorbs or emits an amount of energy h ν in the from of radiation by means of an elementary proess, then the momentum h ν / is always transferred to the moleule, and, to be sure, in the ase of absorption, in the diretion of the moving beam and in the ase of emission in the opposite diretion. If the moleule is subjet to the simultaneous ation of beams moving in various diretions, then only one of these taken part in any single elementary proess of inident radiation; this beam alone then determined the diretion of the momentum transferred to the moleule. If, through an emission proess, the moleule suffers a radiant loss of energy of magnitude h ν without the ation of an outside ageny, then this proess, too, is a direted one. emission in spherial waves does not our. Aording to the present state of the theory, the moleule suffers a reoil of magnitude h ν / in a partiular diretion only beause of the hane emission in that diretion. This property of elementary proesses as expressed by equation (12) makes a quantum theory of radiation almost unavoidable. The weakness of the theory lies, on the one hand, in its not bringing us loser to a union with the wave theory, and, on the other hand, that it leaves the time and diretion of the elementary proesses to hane; in spite of this, I have full onfidene in the trustworthiness of this approah. Only one more general remark. Almost all theories of thermal radiation rest on the onsiderations of the interation between radiation and moleules. But, in general, one is satisfied with dealing only with the energy exhange, without taking into aount the momentum exhange. One feels justified in this beause the momentum transferred by radiation is so small that it always drops out as ompared to that from other dynamial proesses. But for the theoretial onsiderations, this small effet is on an equal footing with the energy transferred by radiation beause energy and momentum are very intimately related to eah other; a theory may therefore be onsidered orret only if it an shown that the momentum transferred aordingly from the radiation to the matter leads to the kind of motion 14

15 that is demanded by thermodynamis. 15

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