.-.-. AMERICAN AAPT JOURNAL n -v-51 r-i f :i. r.-.* i.: v o/PHYSICS
Einstein's Proposal of the Photon Concept—a Translation of the Annalen der Physik Paper of 1905 A. B. Arons and M. B. Peppard
Citation: Am. J. Phys. 33, 367 (1965); doi: 10.1119/1.1971542 View online: http://dx.doi.Org/10.1119/1.1971542 View Table of Contents: http://ajp.aapt.Org/resource/1/AJPIAS/v33/i5 Published by the American Association of Physics Teachers
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Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission AMERICAN JOURNAL of PHYSICS A Journal Devoted to the Instructional and Cultural Aspects of Physical Science
VOLUME 33, NUMBER 5 MAY 1965
Einstein's Proposal of the Photon Concept—a Translation of the Annalen der Physik Paper of 1905*
A. B. ARONSf AND M. B. PEPPARD. Amherst College, Amherst, Massachusetts (Received 2 December 1964)
Of the trio of famous papers that Albert Einstein sent to the Annalen der Physik in 1905 only the paper proposing the photon concept has been unavailable in English translation. The American Journal of Physics is publishing the following translation in recognition of the sixtieth anniversary of the appearance of the original work. Physics teachers may take particu lar interest in the following aspects: (1) Einstein's keen awareness of the heuristic character of his new conception. (2) His demonstration from thermodynamic and statistical considerations that electromagnetic radiation might be conceived as consisting of finite numbers of discrete corpuscles of energy hv. (3) His prediction of the linear relation between the stopping potential of photoelectrons and the frequency of the incident light. This latter aspect of the photo electric effect was not included among Lenard's early investigations. It remained for Millikan and others to develop the elegant experimental techniques that confirmed Einstein's bold pre diction. Readers interested in pursuing the background in greater depth will find it rewarding to refer to the critical analyses by Martin J. Klein in "Einstein's First Paper on Quanta," in The Natural Philosopher (Blaisdell Publishing Company, New York, 1963), Vol. II, and "Einstein and the Wave-Particle Duality," in The Natural Philosopher, Vol. Ill, 1964. We are grateful to Professor Klein for his criticism and advice regarding this translation and for his generosity in making available to us an unpublished translation of his own.
CONCERNING AN HEURISTIC POINT OF spatial functions to describe the electromagnetic VIEW TOWARD THE EMISSION AND state of a given volume, and a finite number of TRANSFORMATION OF LIGHT parameters cannot be regarded as sufficient for BY A. EINSTEIN the complete determination of such a state. Ac PROFOUND formal distinction exists be cording to the Maxwellian theory, energy is to A tween the theoretical concepts which physi be considered a continuous spatial function in cists have formed regarding gases and other the case of all purely electromagnetic phenomena ponderable bodies and the Maxwellian theory of including light, while the energy of a ponderable electromagnetic processes in so-called empty object should, according to the present concep space. While we consider the state of a body to tions of physicists, be represented as a sum be completely determined by the positions and carried over the atoms and electrons. The energy velocities of a very large, yet finite, number of of a ponderable body cannot be subdivided into atoms and electrons, we make use of continuous arbitrarily many or arbitrarily small parts, while * Ann.. Physik 17, 132 (1905); Translation published the energy of a beam of light from a point source with the permission of Annalen der Physik. (according to the Maxwellian theory of light or, f Department of Physics. % Department of German. more generally, according to any wave theory) is 367
Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissic 368 A. B. ARONS AND M. B. PEPPARD continuously spread over an ever increasing be a number of electrons which are bound to volume. widely separated points by forces proportional The wave theory of light, which operates with to their distances from these points. The bound continuous spatial functions, has worked well in electrons are also to participate in conservative the representation of purely optical phenomena interactions with the free molecules and electrons and will probably never be replaced by another when the latter come very close. We call the theory. It should be kept in mind, however, that bound electrons "oscillators"; they emit and the optical observations refer to time averages absorb electromagnetic waves of definite periods. rather than instantaneous values. In spite of the According to the present veiw regarding the complete experimental confirmation of the theory origin of light, the radiation in the space we are as applied to diffraction, reflection, refraction, considering (radiation which is found for the case dispersion, etc., it is still conceivable that the of dynamic equilibrium in accordance with the theory of light which operates with continuous Maxwellian theory) must be identical with the spatial functions may lead to contradictions blackbody radiation—at least if oscillators of all with experience when it is applied to the phe the relevant frequencies are considered to be nomena of emission and transformation of light. present. It seems to me that the observations associated For the time being, we disregard the radiation with blackbody radiation, fluorescence, the emitted and absorbed by the oscillators and production of cathode rays by ultraviolet light, inquire into the condition of dynamical equilib and other related phenomena connected with the rium associated with the interaction (or collision) emission or transformation of light are more of molecules and electrons. The kinetic theory of readily understood if one assumes that the energy gases asserts that the average kinetic energy of of light is discontinuously distributed in space. an oscillator electron must be equal to the aver In accordance with the assumption to be con age kinetic energy of a translating gas molecule. sidered here, the energy of a light ray spreading If we separate the motion of an oscillator electron out from a point source is not continuously into three components at right angles to each distributed over an increasing space but consists other, we find for the average energy E of one of a finite number of energy quanta which are of these linear components the expression localized at points in space, which move without dividing, and which can only be produced and E=(R/N)T, absorbed as complete units. where R denotes the universal gas constant, N In the following I wish to present the line of denotes the number of "real molecules" in a thought and the facts which have led me to this gram equivalent, and T the absolute tempera point of view, hoping that this approach may be ture. The energy E is equal to two-thirds the useful to some investigators in their research. kinetic energy of a free monatomic gas particle because of the equality between the time average 1. CONCERNING A DIFFICULTY WITH REGARD TO values of the kinetic and potential energies of the THE THEORY OF BLACKBODY RADIATION oscillator. If through any cause—in our case We start first with the point of view taken in through radiation processes—it should occur that the Maxwellian and the electron theories and the energy of an oscillator takes on a time- consider the following case. In a space enclosed average value greater or less than E, then the by completely reflecting walls, let there be a collisions with the free electrons and molecules number of gas molecules and electrons which are would lead to a gain or loss of energy by the gas, free to move and which exert conservative forces different on the average from zero. Therefore, in on each other on close approach; i.e. they can the case we are considering, dynamic equilibrium collide with each other like molecules in the is possible only when each oscillator has the kinetic theory of gases.1 Furthermore, let there average energy E. 1 This assumption is equivalent to the supposition that known that, with the help of this assumption, Herr Drude the average kinetic energies of gas molecules and electrons derived a theoretical expression for the ratio of thermal are equal to each other at thermal equilibrium. It is well and electrical conductivities of metals.
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We shall now proceed to present a similar argu and in the limit we obtain ment regarding the interaction between the oscillators and the radiation present in the R8TT 2 pvdv— -T ' I vHv = cavity. Herr Planck has derived the condition N U Jo for the dynamical equilibrium in this case under the supposition that the radiation can be con 2. CONCERNING PLANCK'S DETERMINATION sidered a completely random process.3 He found OF THE FUNDAMENTAL CONSTANTS We wish to show in the following that Herr Planck's determination of the fundamental
where (Er) is the average energy (per degree constants is, to a certain extent, independent of of freedom) of an oscillator with eigenfrequency his theory of blackbody radiation. 4 v, L the velocity of light, . the frequency, and Planck's formula, which has proved adequate pAv the energy per unit volume of that portion up to this point, gives for p„ of the radiation with frequency between v and v-\-dv. If the radiation energy of frequency v is not continually increasing or decreasing, the follow a = 6.iOXlO-56, ing relations must obtain: fl =4.866X10-".
(R/N)T=E=(E,) = (LV8«-^)P„ For large values of T/v; i.e. for large wavelengths 2 PV=(R/N){$TTV /U)T. and radiation densities, this equation takes the These relations, found to be the conditions of form 2 dynamic equilibrium, not only fail to coincide Pr=(a//3> r. with experiment, but also state that in our model It is evident that this equation is identical with there can be no talk of a definite energy distribu the one obtained in Sec. 1 from the Maxwellian tion between ether and matter. The wider the and electron theories. By equating the coefficients range of wavenumbers of the oscillators, the of both formulas one obtains greater will be the radiation energy of the space, (R/N)(STr/D) = (a/$) 8 M. Planck, Ann. Physik 1, 99 (1900). 3 This problem can be formulated in the following man or ner. We expand the Z component of the electrical force (2) at an arbitrary point during the time interval between t =0 N= (0/a) (frrR/L*) = 6.17 X1023 and t = T in a Fourier series in which A,^. 0 and OSja,.^ 2ir\ the time T is taken to be very large relative to all the periods of oscillation that are present: i.e., an atom of hydrogen weighs 1/N grams = 1.62 X 10~24 g. This is exactly the value found Z = 2 A, sin ( 2irv-j,-\-a,\. by Herr Planck, which in turn agrees with values »-i found by other methods. If one imagines making this expansion arbitrarily often at a given point in space at randomly chosen instants of time, We therefore arrive at the conclusion: the one will obtain various sets of values of A, and ay. There then exist for the frequency of occurrence of different sets greater the energy density and the wavelength of values of A, and a, (statistical) probabilities dW of the of a radiation, the more useful do the theoretical form: principles we have employed turn out to be; for dW = f(Ai,Af • -ai,a2- • -)dAidAi' • -daidas- • •. small wavelengths and small radiation densities, Thejradiation is then as disordered as conceivable if however, these principles fail us completely. f(A A*- • -a^ar • •) = F { A )F {Ai)---f { )f { ) ••-, u l i l i 1 ai i a% In the following we shall consider the experi i.e., if the probability of a particular value of A or a is independent of other values of A or a. The more closely mental facts concerning blackbody radiation this condition is fulfilled (namely, that the individual pairs without invoking a model for the emission and of values of A, and a„ are dependent upon the emission and absorption processes of specific groups of oscillators) propagation of the radiation itself. the more closely will radiation in the case being considered approximate a perfectly random state. M. Planck, Ann. Physik 4, 561 (1901).
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3. CONCERNING THE ENTROPY OF RADIATION increases by dT The following treatment is to be found in a famous work by Herr W. Wien and is introduced dS: dpdv, here only for the sake of completeness. T(T) Suppose we have radiation occupying a volume or, since d
/dp)dE. the radiation density p{v) is given for all fre- Since dE is equal to the heat added and since quencies.5 Since radiations of different fre the process is reversible, the following statement quencies are to be considered independent of each also applies other when there is no transfer of heat or work, the entropy of the radiation can be repre dS= (l/T)dE. sented by By comparison one obtains S—vi Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission EINSTEIN'S PROPOSAL OF THE PHOTON CONCEPT 371 If we confine ourselves to investigating the de and pendence of the entropy on the volume occupied W=WvWi. by the radiation, and if we denote by So the The last equation says that the states of the two entropy of the radiation at volume v , we obtain 0 systems are independent of each other. S-So=CE//3y)ln(»/w0). From these equations it follows that This equation shows that the entropy of a 4>(Wi-Wi) = Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission 372 A. B. ARONS AND M . B. PEPPARD For this probability, which is a "statistical We still wish to compare the average mag probability," one obviously obtains: nitude of the energy quanta of the blackbody radiation with the average translational kinetic W=(v/v )n. 0 energy of a molecule at the same temperature. By applying the Boltzmann Principle, one then The latter is %(R/N)T, while, according to the obtains Wien formula, one obtains for the average mag nitude of an energy quantum S-S0 = R(n/N) ln(»/»0). / /•- N It is noteworthy that in the derivation of this equation, from which one can easily obtain the I Jo R8v law of Boyle and Gay-Lussac as well as the If the entropy of monochromatic radiation analogous law of osmotic pressure thermody 6 depends on volume as though the radiation were namically, no assumption had to be made as to a discontinuous medium consisting of energy a law of motion of the molecules. quanta of magnitude Rffv/N, the next obvious step is to investigate whether the laws of emission 6. INTERPRETATION OF THE EXPRESSION FOR and transformation of light are also of such a THE VOLUME DEPENDENCE OF THE EN TROPY OF MONOCHROMATIC RADIA nature that they can be interpreted or explained TION IN ACCORDANCE WITH by considering light to consist of such energy BOLTZMANN'S PRINCIPLE quanta. We shall examine this question in the following. In Sec. 4, we found the following expression for the dependence of the entropy of monochro 7. CONCERNING STOKES'S RULE matic radiation on the volume According to the result just obtained, let us S-Sa=(E/pv)ln(v/v0). assume that, when monochromatic light is trans formed through photoluminescence into light of If one writes this in the form a different frequency, both the incident and N E) (E S-S0=(R/N) \nl(v/v0y ' - '^, emitted light consist of energy quanta of magni tude Rj3v/N, where v denotes the relevant fre and if one compares this with the general formula quency. The transformation process is to be for the Boltzmann principle interpreted in the following manner. Each inci dent energy quantum of frequency v\ is absorbed S~S0=(R/N)logW, and generates by itself—at least at sufficiently one arrives at the following conclusion: low densities of incident energy quanta—a light If monochromatic radiation of frequency v and quantum of frequency v^; it is possible that the energy E is enclosed by reflecting walls in a absorption of the incident light quantum can give volume Vo, the probability that the total radiation rise to the simultaneous emission of light quanta energy will be found in a volume v (part of the of frequencies v%, vit etc., as well as to energy of volume VQ) at any randomly chosen instant is other kinds, e.g., heat. It does not matter what intermediate processes give rise to this final re W=(v/vo)(NIB)-(E'M. sult. If the fluorescent substance is not a per From this we further conclude that: Mono petual source of energy, the principle of conserva chromatic radiation of low density (within the tion of energy requires that the energy of an range of validity of Wien's radiation formula) emitted energy quantum cannot be greater than behaves thermodynamically as though it con that of the incident light quantum; it follows sisted of a number of independent energy quanta that of magnitude R0v/N. Rpv2/N^RpVl/N 6 or If £ is the energy of the system, one obtains: -d(E-TS) =pdv = TdS=RT(n/N)(dv/v); therefore pv = R(n/N)T. This is the well-known Stokes's Rule. Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissic EINSTEIN'S PROPOSAL OF THE PHOTON CONCEPT 373 It should be strongly emphasized that accord quanta penetrate into the surface layer of the ing to our conception the quantity of light body, and their energy is transformed, at least in emitted under conditions of low illumination part, into kinetic energy of electrons. The sim (other conditions remaining constant) must be plest way to imagine this is that a light quantum proportional to the strength of the incident light, delivers its entire energy to a single electron; we since each incident energy quantum will cause an shall assume that this is what happens. The elementary process of the postulated kind, inde possibility should not be excluded, however, that pendently of the action of other incident energy electrons might receive their energy only in part quanta. In particular, there will be no lower limit from the light quantum. for the intensity of incident light necessary to An electron to which kinetic energy has been excite the fluorescent effect. imparted in the interior of the body will have According to the conception set forth above, lost some of this energy by the time it reaches deviations from Stokes's Rule are conceivable in the surface. Furthermore, we shall assume that the following cases: in leaving the body each electron must perform an amount of work P characteristic of the sub 1. when the number of simultaneously inter stance. The ejected electrons leaving the body acting energy quanta per unit volume is so large with the largest normal velocity will be those that that an energy quantum of emitted light can were directly at the surface. The kinetic energy receive its energy from several incident energy of such electrons is given by quanta; 2. when the incident (or emitted) light is not RPv/N-P. of such a composition that it corresponds to blackbody radiation within the range of validity If the body is charged to a positive potential of Wein's Law, that is to say, for example, II and is surrounded by conductors at zero when the incident light is produced by a body of potential, and if II is just large enough to prevent such high temperature that for the wavelengths loss of electricity by the body, it follows that: under consideration Wien's Law is no longer Ue = R0v/N~P valid. The last-mentioned possibility commands es where e denotes the electronic charge, or pecial interest. According to the conception we IlE=Rpp-P' have outlined, the possibility is not excluded that a "non-Wien radiation" of very low density can where E is the charge of a gram equivalent of a exhibit an energy behavior different from that of monovalent ion and P' is the potential of this a blackbody radiation within the range of quantity of negative electricity relative to the validity of Wien's Law. body.8 If one takes E = 9.6X103, then IM0~8 is the 8. CONCERNING THE EMISSION OF CATHODE potential in volts which the body assumes when RAYS THROUGH THE ILLUMINATION irradiated in a vacuum. OF SOLID BODIES In order to see whether the derived relation The usual conception, that the energy of light yields an order of magnitude consistent with is continuously distributed over the space experience, we take P' = 0, J> = 1.03X1015 (corre through which it propagates, encounters very sponding to the limit of the solar spectrum serious difficulties when one attempts to explain toward the ultraviolet) and (3 = 4.866X10-". We the photoelectric phenomena, as has been pointed obtain II-107 = 4.3 volts, a result agreeing in out in Herr Lenard's pioneering paper.7 order magnitude with those of Herr Lenard.9 According to the concept that the incident 8 If one assumes that the individual electron is detached light consists of energy quanta of magnitude from a neutral molecule by light with the performance of a Rpv/N, however, one can conceive of the ejection certain amount of work, nothing in the relation derived above need be changed; one can simply consider P' as the of electrons by light in the following way. Energy sum of two terms. 9 P. Lenard, Ann. Physik 8, pp. 165, 184, and Table I, 7 P. Lenard, Ann. Physik 8, 169, 170 (1902). Fig. 2 (1902). Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission 374 A. B. ARONS AND M . B. PEPPARD If the derived formula is correct, then II, when 9. CONCERNING THE IONIZATION OF GASES represented in Cartesian coordinates as a func BY ULTRAVIOLET LIGHT tion of the frequency of the incident light, must We shall have to assume that, in the ionization be a straight line whose slope is independent of of a gas by ultraviolet light, an individual light the nature of the emitting substance. energy quantum is used for the ionization of an As far as I can see, there is no contradiction individual gas molecule. From this it follows between these conceptions and the properties of immediately that the work of ionization (i.e., the the photoelectric effect observed by Herr Lenard. work theoretically needed for ionization) of a If each energy quantum of the incident light, molecule cannot be greater than the energy of independently of everything else, delivers its an absorbed light quantum capable of producing energy to electrons, then the velocity distribution this effect. If one denotes by J the (theoretical) of the ejected electrons will be independent of the work of ionization per gram equivalent, then it intensity of the incident light; on the other hand follows that: the number of electrons leaving the body will, Rfrv^J. if other conditions are kept constant, be pro portional to the intensity of the incident light.10 According to Lenard's measurements, however, Remarks similar to those made concerning the largest effective wavelength for air is approxi 5 hypothetical deviations from Stokes's Rule can mately 1.9X10~ cm; therefore: can be made with regard to hypothetical bound i?/3. =6.4X1012erg^J. aries of validity of the law set forth above. In the foregoing it has been assumed that the An upper limit for the work of ionization can energy of at least some of the quanta of the also be obtained from the ionization potentials of 12 incident light is delivered completely to in rarefied gases. According to J. Stark the smallest dividual electrons. If one does not make this observed ionization potentials for air (at plati 18 obvious assumption, one obtains, in place of the num anodes) is about 10 V. One therefore ob 12 last equation: tains 9.6 X10 as an upper limit for J, which is nearly equal to the value found above. UE+P%R/3v. There is another consequence the experimental For fluorescence induced by cathode rays, testing of which seems to me to be of great which is the inverse process to the one discussed importance. If every absorbed light energy above, one obtains by analagous considerations: quantum ionizes a molecule, the following rela tion must obtain between the quantity of ab nE+P'^Rpv. sorbed light L and the number of gram molecules of ionized gas j: In the case of the substances investigated by Herr Lenard, PE10* is always significantly greater j = L/Rpv. than Rfiv, since the potential difference, which the cathode rays must traverse in order to If our conception is correct, this relationship produce visible light, amounts in some cases to must be valid for all gases which (at the relevant hundreds and in others to thousands of volts.11 It frequency) show no appreciable absorption with is therefore to be assumed that the kinetic energy out ionization. of an electron goes into the production of many Bern, 17 March 1905 light energy quanta. Received 18 March 1905. 10 P. Lenard, Ref. 9, p. 150 and p. 166-168. 12 J. Stark, Die Elektrizitdt in Gasen (Leipzig, 1902), p. 57. 10a Should be HE (translator's note). 13 In the interior of gases the ionization potential for 11 P. Lenard, Ann. Physik 12, 469 (1903). negative ions is, however, five times greater. Downloaded 18 Jan 2013 to 128.103.149.52. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission