On the Quantum Theory of Radiation.

Transcription

1 Physikalishe Zeitshrift, Band 18, Seite ) On the Quantum Theory of Radiation. Albert Einstein The formal similarity between the hromati distribution urve for thermal radiation and the Maxwell veloity-distribution law is too striking to have remained hidden for long. In fat, it was this similarity whih led W. Wien, some time ago, to an extension of the radiation formula in his important theoretial paper, in whih he derived his displaement law ρ = ν 3 f ν T ). 1) As is well known, he disovered the formula ρ = αν 3 e hν 2) whih is still aepted as orret in the limit of large values of ν T Wien s radiation formula). Today we know that no approah whih is founded on lassial mehanis and eletrodynamis an yield a useful radiation formula. Rather, lassial theory must of neessity lead to Rayleigh s formula ρ = kα h ν2 T. 3) Next, Plank in his fundamental investigation based his radiation formula ρ = αν 3 1 e hν 1 on the assumption of disrete portions of energy, from whih quantum theory developed rapidly. It was then only natural that Wien s argument, whih had led to eq. 2), should have beome forgotten. Not long ago I disovered a derivation of Plank s formula whih was losely related to Wien s original argument 1 and whih was based on the fundamental assumption of quantum theory. This derivation displays the relationship between Maxwell s urve and the hromati distribution urve and deserves attention not only beause of its simpliity, but espeially beause it seems to throw some light on the mehanism of emission and absorption of radiation by matter, a proess whih is still obsure to us. By postulating some hypotheses on the emission and absorption of radiation by moleules, whih suggested themselves from quantum theory, I was able to show that moleules with a quantum-theoretial distribution of states in thermal equilibrium, were in dynamial equilibrium with the Plank radiation; in this way, Plank s formula 4) ould be derived in an astonishingly simple and general way. It was obtained from the ondition that the internal energy distribution of the moleules demanded by quantum theory, should follow purely from an emission and absorption of radiation. But if these hypotheses on the interation between radiation and matter turn out to be justified, they must produe rather more than just the orret statistial distribution of the internal energy of the moleules: for Bern, Marh Verh. d. Deutshen physikal. Gesellshaft 18 Nr. 13/ ) 318. The arguments used in that paper are reprodued in the present disussion. 4)

2 2 A. Einstein: On the Quantum Theory of Radiation. there is also a momentum transfer assoiated with the emission and absorption of radiation; this produes, purely through the interation between the radiation and the moleules, a ertain veloity distribution for the latter. This must evidently be idential with the veloity distribution of the moleules whih is entirely due to their ollisions among themselves, i.e. it must agree with the Maxwell distribution. It has to be required that the mean kineti energy of a moleule per degree of freedom) should be equal to 1 2 in a Plank radiation field of temperature T. This requirement should hold independently of the nature of the moleules under onsideration and independently of the frequenies emitted or absorbed by them. We want to demonstrate in the present paper that this far-reahing requirement is in fat satisfied quite generally, thus lending new support to our simple hypotheses onerning the elementary proesses of emission and absorption. To obtain suh a result however requires a ertain extension of the hypotheses, whih had been up to now solely onerned with an exhange of energy. The question arises: does the moleule suffer an impulse when it emits or absorbs energy ɛ? As an example, let us onsider the emission of radiation from the point of view of lassial eletrodynamis. When a body emits an energy ɛ, it has a reoil momentum) ɛ/, provided the whole of the radiation is emitted in the same diretion. If, however, the emission proess has spatial symmetry, suh as in the ase of spherial waves, no reoil is produed at all. This seond possibility is also of importane in the quantum theory of radiation. If a moleule absorbs or emits energy ɛ in the form of radiation during its transition from one quantum-theoretially possible state to another, suh an elementary proess an be thought of as being partially or ompletely diretional, or else symmetrial non-diretional). It will beome apparent that we shall only then arrive at a theory whih is free from ontraditions, if we onsider suh elementary proesses to be perfetly diretional; this embodies the main result of the subsequent disussion. 1 Fundamental Hypotheses of the Quantum Theory Canonial Distribution of State In quantum theory a moleule of a given kind an only exist in a disrete set of states Z 1, Z 2,... Z n...m with internal) energies ɛ 1, ɛ 2,... ɛ n..., apart from its orientation and translatory motion. If suh moleules belong to a gas at temperature T, the relative frequeny W n of suh states Z n is given by the formula W n = p n e ɛ n 5) whih orresponds to the anonial distribution of states in statistial mehanis. In this formula, k = R N is the well-known Boltzmann onstant, and p n is a number, independent of T and harateristi for the moleule and its nth quantum state, whih an be alled the statistial weight of this state. Formula 5) an be derived from Boltzmann s priniple, or from purely thermodynamial onsiderations. It expresses the most extreme generalisation of Maxwell s veloity distribution law. The latest fundamental developments in quantum theory are onerned with a theoretial derivation of the quantum-theoretially possible states Z n and their weights P n. For the present basi investigation, a detailed determination of the quantum states is not required. 2 Hypotheses about the Energy Exhange Through Radiation Let Z n and Z m be two quantum-theoretially possible states of the gas moleule, whose energies are ɛ n and ɛ m, respetively, and satisfy the inequality ɛ m > ɛ n. Let us assume that the moleule is apable of a transition from state Z n into state Z m with an absorption of radiation energy ɛ m ɛ n ; that, similarly, the transition from state Z m to state Z n is possible, with emission of the same radiative energy. Let the radiation absorbed or emitted by the moleule have frequeny ν whih is harateristi for the index ombination m,n) that we are onsidering.

3 Physikalishe Zeitshrift, Band 18, Seite ) 3 For the laws governing this transition, we introdue a few hypotheses whih are obtained by arrying over the known situation for a Plank resonator in lassial theory to the as yet unknown one in quantum theory. a) Emission of radiation Aording to Hertz, an osillating Plank resonator radiates energy in the well-known way, regardless of whether or not it is exited by an external field. Correspondingly, let us assume that a moleule may go from state Z m to a state Z n and emit radiation energy ɛ m ɛ n with frequeny µ, without exitation from external auses. Let the probability dw for this to happen during the time interval dt, be dw = A n m dt where A n m is a onstant haraterising the index ombination under onsideration. The statistial law whih we assumed, orresponds to that of a radioative reation, and the above elementary proess orresponds to a reation in whih only γ-rays are emitted. It need not be assumed here that the time taken for this proess is zero, but only that this time should be negligible ompared with the times whih the moleule spends in states Z 1, et. b) Absorption of radiation If a Plank resonator is loated in a radiation field, the energy of the resonator is hanged through the work done on the resonator by the eletromagneti field of the radiation; this work an be positive or negative, depending on the phases of the resonator and the osillating field. We orrespondingly introdue the following quantum-theoretial hypothesis. Under the influene of a radiation density ρ of frequeny ν, a moleule an make a transition from state Z n to state Z m by absorbing radiation energy ɛ m ɛ n, aording to the probability law A) dw = B m n ρ dt B) We similarly assume that a transition Z m Z n assoiated with a liberation of radiation energy ɛ m ɛ n, is possible under the influene of the radiation field, and that it satisfies the probability law dw = B n m ρ dt B ) Bn m and Bm n are onstants. We shall give both proesses the name hanges of state due to irradiation. We now have to ask our selves what is the momentum transfer to the moleule for suh hanges of state. Let us first disuss the ase of absorption of radiation. If a radiation bundle in a given diretion does work on a Plank resonator, the orresponding energy is removed from the radiation bundle. To this transfer of energy there also orresponds a momentum transfer from radiation bundle to resonator, by momentum onservation. The resonator is thus ated upon by a fore in the beam diretion of the radiation bundle. If the energy transfer is negative, then the fore ats on the resonator in the opposite diretion. If the quantum hypothesis holds, we an obviously interpret the proess in the following way. If the inident radiation bundle produes the transition Z n Z m by absorption of radiation, a momentum ɛm ɛn is transferred to the moleule in the diretion of propagation of the beam. For the absorption proess Z m Z n the momentum transfer has the same magnitude, but is in the opposite diretion. For the ase where the moleule is ated upon simultaneously by several radiation bundles, we assume that total energy ɛ m ɛ n assoiated with an elementary proess is removed from, or added to, a single suh radiation bundle. Thus here, too, the momentum transferred to the moleule is ɛm ɛn. For an energy transfer by emission of radiation in the ase of a Plank resonator, no momentum transfer to the resonator takes plae, sine emission ours in the form of a spherial wave, aording to lassial theory. As was remarked previously, a quantum theory free from ontraditions an only be obtained if the emission proess, just as absorption, is assumed to be diretional. In that ase, for eah elementary

4 4 A. Einstein: On the Quantum Theory of Radiation. emission proess Z m Z n a momentum of magnitude ɛm ɛn is transferred to the moleule. If the latter is isotropi, we shall have to assume that all diretions of emission are equally probable. If the moleule is not isotropi, we arrive at the same statement if the orientation hanges with time in aordane with the laws of hane. Moreover, suh an assumption will also have to be made about the statistial laws for absorption, B) and B ). Otherwise the onstants Bn m and Bm n would have to depend on the diretion, and this an be avoided by making the assumption of isotropy or pseudo-isotropy using time-averages). 3 Derivation of the Plank Radiation Law We now look for that partiular radiation density ρ, for whih the exhange of energy between radiation and moleules in keeping with the probability laws A), B), and B ) does not disturb the moleular distribution of states given by eq. 5). For this it is neessary and suffiient that the number of elementary proesses of type B) taking plae per unit time should, on average, be equal to those of type A) and B ) taken together. From this ondition one obtains from 5), A), B), B ) the equation p n e ɛ n B m n ρ = p m e ɛ m B n m ρ + A n m) for the elementary proesses assoiated with the index ombination m,n). If, in addition, ρ tends to infinity with T, as will be assumed, the relation p n B m n = p m B n m 6) has to hold between the onstants B m n and B n m. We then obtain from our equation, ρ = e An m/b n m ɛ m ɛ n 1 7) as the ondition for dynamial equilibrium. This expresses the temperature dependene of the radiation density aording to Plank s law. From Wien s displaement law 1) it follows immediately that A n m Bm n = αν 3 8) and ɛ m ɛ n = hν 9) where α and h are universal onstants. To ompute the numerial value of the onstant α, one would have to have an exat theory of eletrodynami and mehanial proesses; for the present, one has to onfine oneself to a treatment of the limiting ase of Rayleigh s law for high temperatures, for whih the lassial theory is valid in the limit. Eq. 9) is of ourse the seond prinipal rule in Bohr s theory of spetra. Sine its extension by Sommerfeld and Epstein, this may well be laimed to have beome a safely established part of our siene. It also ontains impliitly the photohemial priniple of equivalene, as has been shown by me.

5 Physikalishe Zeitshrift, Band 18, Seite ) 5 4 Method for Calulating the Motion of Moleules in Radiation Fields We now turn to a disussion of the motion of our moleules under the influene of radiation. For this we shall make use of a method whih is well known from the theory of Brownian motion, and whih I employed on several oasions for numerial omputations of motions in a radiation field. To simplify the alulation we shall only onsider the ase where the motions take plae in just one diretion, the X- diretion of the oordinate system. Furthermore, we shall onfine ourselves to a alulation of the average value of the kineti energy of the progressive motion, and we shall thus not attempt to prove that suh veloities v obey the Maxwell distribution law. The mass M of the moleule is assumed suffiiently large, so that higher powers of v/ an be negleted in omparison with lower ones; we an then apply the laws of ordinary mehanis to the moleule. Finally, no real loss of generality is introdued if we perform the alulations as if the states with index m and n were the only possible states for the moleule. The momentum Mv of a moleule undergoes two different types of hange during the short time interval τ. Although the radiation is equally onstituted in all diretions, the moleule will nevertheless be subjeted to a fore originating from the radiation, whih opposes the motion. Let this be equal to Rv, where R is a onstant to be determined later. This fore would bring the moleule to rest, if it were not for the irregularity of the radiative interations whih transmit a momentum of hanging sign and magnitude to the moleule during time τ; suh an unsystemati effet, as opposed to that previously mentioned, will sustain some movement of the moleule. At the end of the short time interval τ, the momentum of the moleule will have the value Mv Rvτ + Sine the veloity distribution is supposed to remain onstant with time, the average of the absolute value of the above quantity must be equal to Mv; the mean values of the squares of these quantities, taken over a long time interval or over a large number of moleules, must therefore be equal: Mv Rvτ + ) 2 = Mv) 2 Sine we were speifially onerned with the systemati effet of v on the momentum of the moleule, we shall have to neglet the average value v. Expanding the left-hand side of the equation, one therefore obtains 2 = 2RMv 2 τ. 10) The mean square value v 2, whih the radiation of temperature T produes in our moleules by interating with them, must be of the same size as the mean square value v 2 obtained from the gas laws for a gas moleule at temperature T in the kineti theory of gases. For the presene of our moleules would otherwise disturb the thermal equilibrium between the thermal radiation and an arbitrary gas held at the same temperature. We must therefore have Mv 2 = 2 2 Equation 10) thus beomes 11) 2 τ = 2R 12)

6 6 A. Einstein: On the Quantum Theory of Radiation. The investigation is now ontinued as follows. For a given radiation ρν)), 2 and R an be alulated, using our hypotheses on the interation between radiation and moleules. If the results are inserted in eq. 12), this equation must beome an identity, if ρ is expressed as a funtion of ν and T, using Plank s equation 4). 5 Calulation of R Consider a moleule of the kind disussed above, moving uniformly with veloity v along the X-axis of the oordinate system K. We wish to find the average momentum whih is transferred from the radiation field to the moleule per unit time. In order to alulate it, we have to desribe the radiation in a oordinate system K whih is at rest relative to the moleule in question. For we had only formulated our hypotheses on emission and absorption for the ase of stationary moleules. The transformation to the [oordinate] system K has been arried out in a number of plaes in the literature, partiularly aurately in Mosengeil s Berlin dissertation. For ompleteness, however, I shall reprodue these simple arguments at this point. Referred to K, the radiation is isotropi, i.e. the radiation of frequeny range dν per unit volume, assoiated with a given infinitesimal solid angle dκ relative to its diretion of propagation, is given by ρdν dκ 4π where ρ depends only on the frequeny ν but not on the diretion. To this partiular radiation there orresponds a partiular radiation in K whih is similarly haraterised by a frequeny range dν and a ertain solid angle dκ. The volume density of this partiular radiation is given by ρ ν, φ )dν dκ 13 ) 4π This defines ρ. It depends on the diretion, whih we shall define in the usual way by means of the angle ϕ with the X -axis and the angle ψ whih the projetion in the Y Z -plane makes with the Y -axis. To these angles orrespond the angles ϕ and ψ, whih determine the diretion of dκ in K in an analogous manner. First of all it is lear that the transformation law between 13) and 13 ) must be the same as that for the squares of the amplitudes, A 2 and A 2, of a plane wave with orresponding diretion. We therefore find, to the desired approximation, that 13) or ρ ν, ϕ )dν dκ ρν)dνdκ = 1 2 v os ϕ 14) ρ ν, ϕ ) = ρν) dν dκ dν dκ 1 2 v ) os ϕ 14 ) The theory of relativity further gives the following 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)

7 Physikalishe Zeitshrift, Band 18, Seite ) 7 With the same approximation, we have from 15) ν = ν 1 + v os ϕ ) Therefore, again to the same approximation, or ρν) = ρ ν + v ν os ϕ ) ρν) = ρν ) + ρν ) ν Moreover, from 15), 16) and 17), dν dν = 1 + v os ϕ ) v ν os ϕ ) 18) dκ dκ = sin ϕ dϕ dψ dos ϕ) sin ϕ dϕ = dψ dos ϕ ) = 1 2 v os ϕ By means of these two relations and 18), we an write 14 ) in the form ρ ν, ϕ ) = [ ρ) ν + v ) ] ρ ν os ϕ 1 3 v ν ν os ϕ ) 19) Using 19) and our hypothesis on the emission and absorption of radiation by the moleule, we an easily alulate the average momentum transferred into the moleule per unit time. Before doing so, however, we shall have to say a few words in justifiation of this approah. It ould be objeted that eqs. 14), 15), 16) are based on Maxwell s theory of the eletromagneti field whih annot be reoniled with quantum theory. But this objetion relates more to the form than to the real essene of the matter. For whatever the shape of a future theory of the eletromagneti proesses, the Doppler priniple and the aberration law will at all events remain preserved, and hene also eqs. 15) and 16). Furthermore, the validity of the energy relation 14) ertainly extends beyond wave theory; aording to the theory of relativity, this transformation law also holds, e.g., for the energy density of a mass moving with almost) the veloity of light and having infinitesimally small rest density. Eq. 19) an therefore lay laim to being valid for any theory of radiation. Aording to B), the radiation assoiated with the solid angle dκ would give rise to Bn m ρ ν, φ ) dκ 4π elementary absorption proesses of the type Z n Z m per seond, if the moleule were to be restored to the state Z n immediately after eah suh elementary proess. But in reality, the time for remaining in state Z n per seond is equal to 1 S p ne ɛn from 5), where the abbreviation S = p n e ɛ n + p m e ɛ m 20) has been used. The number of suh proesses per seond is thus really 1 S p ne ɛ n B m n ρ ν, ϕ ) = dκ 4π.

8 8 A. Einstein: On the Quantum Theory of Radiation. Far eah suh elementary proess a momentum ɛm ɛn os ϕ gets transmitted to the atom in the diretion of the positive X -axis. Analogously we find, starting from B ), that the orresponding number, per seond, of elementary proesses for an absorption of the type Z m Z n is 1 S p me ɛ m Bmρ n ν, ϕ ) dκ 4π and in suh a proess a momentum ɛm ɛn os φ is transferred to the moleule. The total momentum transfer to the moleule produed by the absorption of radiation is therefore, per unit time, hν S p nb m n e ɛ n e ɛ m ) ρ ν, φ ) os φ dκ 4π This follows from 6) and 9), and the integration extends over all elementary solid angles. On integrating, one obtains from 19) hν 2 ρ 1/3) ν ρ ) p n Bn m S ν e ɛ n e ɛ m ) v. Here the effetive frequeny is again denoted by ν instead of ν ). But this expression represents the whole of the average momentum transferred per unit time to a moleule moving with veloity v. For it is lear that the elementary radiative emission proesses, whih take plae in K without interation with the radiation field, have no preferred diretion, so that they annot transmit any momentum to the moleule, on average. The final result of our disussion is therefore R = hν 2 ρ 1/3 ν ρ ) p n Bn m e ɛ n S ν 1 e hν ) 21) 6 Calulating 2 It is muh simpler to alulate the effet of the irregularity of the elementary proesses on the mehanial behaviour of the moleule beause the alulation an be based on a moleule at rest, to the degree of approximation to whih we had restrited ourselves from the beginning. Consider an arbitrary event, ausing a momentum transfer λ to a moleule in the X-diretion. This momentum an be assumed of different sign and magnitude in different ases. Nevertheless λ is supposed to satisfy a ertain statistial law, suh that its average value vanishes. Now let λ 1, λ 2,... be the momenta transmitted to the moleule due to a number of mutually independent auses, so that the total momentum transfer is given by = Σλ ν Then, if the averages λ ν of the individual λ ν vanish, 2 = Σλ ν 2. 22) If the mean square values λ ν 2 of the individual momenta are equal = λ 2 ) and if l is the total number of events produing these momenta, the relation

9 Physikalishe Zeitshrift, Band 18, Seite ) 9 2 = lλ 2 22a) holds. Aording to our hypotheses, a momentum λ = hν os ϕ is transferred to the moleule for eah absorption and emission proess. Here, ϕ denotes the angle between the X-axis and a randomly hosen diretion. One therefore obtains λ 2 = 1 3 ) 2 hν. 23) Sine we assume that all the elementary proesses whih our an be regarded as mutually independent events, we are allowed to use 22a). Then l is the number of elementary events whih our in time τ. This is twie the number of absorption proesses Z n Z m taking plae in time τ. We therefore have l = 2 S p nb m n e ɛ n ρτ 24) and from 23), 24) and 22), 2 τ = 2 3S ) 2 hν p n Bn m e ɛ n ρ. 25) 7 Conlusion We now have to show that the momenta transferred from the radiation field to the moleule aording to our basi hypotheses, never disturb the thermodynami equilibrium. For this, we need only insert the values for 2 τ and R determined by 25) and 21), after replaing in 21) the expression ρ 1 / 3 ) ν ρ ) 1 e hν ) ν by ρhν 3RT, from 4). It is then seen immediately that our basi equation 12) is identially satisfied. We have now ompleted the arguments whih provide a strong support for the hypotheses stated in 2, onerning the interation between matter and radiation by means of absorption and emission proesses, or in- or outgoing radiation. I was led to these hypotheses by my endeavour to postulate for the moleules, in the simplest possible manner, a quantum-theoretial behaviour that would be the analogue of the behaviour of a Plank resonator in the lassial theory. From the general quantum assumption for matter, Bohr s seond postulate eq. 9) as well as Plank s radiation formula followed in a natural way. Most important, however, seems to me to be the result onerning the momentum transfer to the moleule due to the absorption and emission of radiation. If one of our assumptions about the momenta were to be hanged, a violation of eq. 12) would be produed; it seems hardly possible to maintain agreement with this relation, imposed by the theory of heat, other than on the basis of our assumptions. The following statements an therefore be regarded as fairly ertainly proved. If a radiation bundle has the effet that a moleule struk by it absorbs or emits a quantity of energy hv in the form of radiation ingoing radiation), then a momentum hv/ is always transferred to the moleule. For an absorption of energy, this takes plae in the diretion of propagation of the radiation bundle, for an emission in the opposite diretion. If the moleule is ated upon by several diretional radiation bundles,

10 10 A. Einstein: On the Quantum Theory of Radiation. then it is always only a single one of these whih partiipates in an elementary proess of irradiation; this bundle alone then determines the diretion of the momentum transferred to the moleule. If the moleule undergoes a loss in energy of magnitude hv without external exitation, by emitting this energy in the form of radiation outgoing radiation), then this proess, too, is diretional. Outgoing radiation in the form of spherial waves does not exist. During the elementary proess of radiative loss, the moleule suffers a reoil of magnitude hv/ in a diretion whih is only determined by hane, aording to the present state of the theory. These properties of the elementary proesses, imposed by eq. 12), make the formulation of a proper quantum theory of radiation appear almost unavoidable. The weakness of the theory Iies on the one hand in the fat that it does not get us any loser to making the onnetion with wave theory; on the other, that it leaves the duration and diretion of the elementary proesses to hane. Nevertheless I am fully onfident that the approah hosen here is a reliable one. There is room for one further general remark. Almost all theories of thermal radiation are based on the study of the interation between radiation and moleules. But in general one restrits oneself to a disussion of the energy exhange, without taking the momentum exhange into aount. One feels easily justified in this, beause the smallness of the impulses transmitted by the radiation field implies that these an almost always be negleted in pratie, when ompared with other effets ausing the motion. Far a theoretial disussion, however, suh small effets should be onsidered on a ompletely equal footing with more onspiuous effets of a radiative energy transfer, sine energy and momentum are linked in the losest possible way. For this reason a theory an only be regarded as justified when it is able to show that the impulses transmitted by the radiation field to matter lead to motions that are in aordane with the theory of heat.