24 OPTICS NEWS/Winter 1979

Einstein's Researches on the Nature of Light*

EMIL WOLF The fundamental contributions made by Albert Einstein toward our present-day understanding of the nature of light are reviewed. In order to place this work into proper perspective, the theories of Hooke, Huygens, Newton, Fresnel, and Maxwell are first summarized, and a brief account is given of the researches of Kirchhoff, Wien, Rayleigh, Jeans, and Planck on equilibrium radiation laws. The main publications of Einstein that led to the formulation of the quantum theory of radiation are then discussed. These include his investigations on the particle aspects of radiation, on the wave-particle duality, on the elementary processes of energy exchange between gas molecules and a radiation field, and on photon statistics.

"For the rest of my life I will reflect on what light is!"

A. Einstein, ca 1917

INTRODUCTION

In our time of ever-increasing special­ization, there is a tendency to concern ourselves with relatively narrow sci­entific problems. The broad founda­tions of our present-day scientific knowledge and its historical develop­ment tend to be forgotten too often. This is an unfortunate trend, not only because our horizon becomes rather limited and our perspective somewhat distorted, but also because there are many valuable lessons to be learned in looking back over the years during which the basic concepts and the fun­damental laws of a particular scientific discipline were first formulated.

To scientists and nonscientists alike, the name Albert Einstein is associated with a theory that has profoundly revolutionized man's ideas of space and time. His theory of relativity implied as basic a change in our con­ception of the universe as that which was brought about by Newton's theory of universal gravitation or Kep¬pler's theory of the planetary system. For this work alone, Einstein will cer­tainly always be remembered as one of the greatest geniuses of all times. What is not so well appreciated—even by many physicists—is that, quite apart from the theory of relativity, Einstein also made most basic contributions to our understanding of the nature of light and radiation in general. The present article is concerned with this aspect of Einstein's work. To place this work into a proper perspective, it seems appropriate first to recall certain

Copyright © 1979, Optical Society of America

milestones from the history of optics and radiation theory.

EARLY THEORIES CONCERNING THE NATURE OF RADIATION1

In the seventeenth century, two theories were put forward about the nature of light: the wave theory, whose chief proponents were Robert Hooke and Christian Huygens, and the corpuscular (or emission) theory, put forward by Isaac Newton.

According to the wave theory, light consists of rapid vibrations that are propagated in an elastic ether in a somewhat similar manner as a distur­bance is propagated on the surface of water. According to the corpuscular theory, on the other hand, light is propagated from a luminous body by minute particles.

Winter 1979/OPTICS NEWS 25

Christian Huygens (1629-1695)

The wave theory, as then formu­lated, appeared to be incapable of explaining the phenomenon of polar­ization, discovered by Huygens himself in studying the refraction of light by crystals. Newton, on the other hand, was able to account for polarization on the basis of his corpuscular theory. It was largely for this reason that the wave theory was rejected for over a century in favor of the corpuscular theory.

In 1801 Thomas Young discovered the principle of interference of light. Seventeen years later Augustin Fresnel showed in a celebrated memoir that, by combining Young's principle of interference with a basic postulate of Huygens's theory, one is led to a wave theory of light that explains dif­fraction of light, a phenomenon that was not comprehensible on the basis of Newton's corpuscular theory. With­in a few years after the publication of Fresnel's memoir and after experi­mental demonstrations of certain un­suspected predictions of his theory, Fresnel's wave theory became gen­erally accepted and Newton's corpus­cular theory fell into oblivion.

The formulation of the wave theory of light culminated in the work of

The author is with the Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627

Isaac Newton (1642-1727)

James Clerk Maxwell, who succeeded in 1865 in embodying all the laws of electricity and magnetism then known into a celebrated set of differential equations—now called Maxwell's equa­tions. One of the consequences of these equations was a prediction that time-dependent electric and magnetic effects are transmitted from one re­gion of space to another by means of waves—known now as electromagnetic waves. The speed of these waves in free space could be calculated from the results of purely electrical meas­urements, and it turned out to be of the order of magnitude of the speed of light, as then known from other exper­iments. This led Maxwell to conjecture that light waves are electromagnetic waves. In 1888 Heinrich Hertz demon-

Augustin Fresnel (1788-1827)

Thomas Young (1773-1829)

strated the existence of electro­magnetic waves experimentally.

We may thus summarize this part of our brief historical introduction by saying that, toward the end of the nineteenth century, it appeared firmly established that light is an electro­magnetic wave phenomenon.

Rather independently of the devel­opments just mentioned, investigations were carried out concerning thermal or heat radiation, which eventually also turned out to be of fundamental im­portance for elucidating the nature of light.

In the period 1814-1817 Joseph Fraunhofer discovered dark lines in the solar spectrum, which have since been named after him. On the basis of experiments by Robert Bunsen and

James Clerk Maxwell (1831-1879)

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26 OPTICS NEWS/Winter 1979

Heinrich Hertz (1857-1894)

Gustav Kirchhoff, they were inter­preted around 1860 as absorption lines of certain gases in the solar atmo­sphere. In the course of his investiga­tions of the solar spectrum, Kirchhoff derived from thermodynamics a num­ber of fundamental results relating to radiation in thermal equilibrium with bodies at a fixed temperature, say T. Even under equilibrium conditions, when the system is thermally insulated from its surroundings, the bodies will emit and absorb radiation, or, as we say these days, there will be inter­action between matter and the radi­ation field. The capacity of a body to emit and absorb radiation at some fixed frequency v may be charac­terized by certain quantities known as the emission coefficient, єv, and the absorption coefficient, αv. One of the laws, which Kirchhoff derived in 1859,

Table 1: Some of the Main Early Contri­butions Toward Elucidating the Nature of Light

Robert Hooke 1665 Wave theory Christian Huygens 1678

Wave theory

Isaac Newton 1671 Corpuscular theory

Thomas Young 1801 Interference of light

Augustin Fresnel 1818 Modernized scalar wave theory

Clerk Maxwell 1865 Electromag­netic wave theory

Heinrich Hertz 1888 Detection of electromag­netic waves

Joseph Fraunhofer (1787-1826)

asserts that, under equilibrium condi­tions, the ratio of the emission and the absorption coefficients is independent of the nature of the bodies that inter­act with the radiation field, i.e., their ratio is some function KV(T) that depends only on the frequency and the temperature:

Gustav Kirchhoff (1824-1887)

bodies in equilibrium with the radi­ation field, a great deal of effort has gone into determining its form. The four decades that followed Kirchhoff's discovery of the law that we just dis­cussed witnessed many unsuccessful attempts to derive the correct form of the function KV(T) or, what according to Eq. (3) amounts to the same thing, the spectrum of blackbody radiation as a function of frequency and temper­ature. Table 2 lists three of the most important formulas proposed for the function KV(T) in this period.

The first formula, derived by W. Wien in 1893 on the basis of thermo­dynamics, is known as Wien's displace­ment law. According to this law, the function Kv(T) is equal to the product of the third power of the fre­quency and some function F(v/T) of the ratio of the frequency v to the temperature T. However, thermo­dynamics did not specify the form of this function.

Using a somewhat heuristic argu­ment, Wien proposed three years later an explicit form for the function

Table 2 The Formulas of Wien and of Rayleigh and Jeans for the Specific Intensity KV(T) = (c/8π)uv(T) of Black-body Radiation

Wien's displacement law (1893)

Wien's radiation law (1896)

Rayleigh-Jeans law (1900, 1905)

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This is clearly an important law, since it brings into evidence a universal func­tion associated with radiation under equilibrium conditions.

In the special case in which the body is such that it absorbs all the radiation that falls on it, the absorption co­efficient αv

has the value unity. Such a body is said to be a blackbody, and the radiation emitted by it is said to be blackbody radiation.

It follows at once from Kirchhoff's law (1) that, for a blackbody,

i.e., its emission coefficient, con­sidered as a function of frequency and temperature, is precisely the universal function KV(T). One can show that KV(T) is related to the energy density uv(T) of blackbody radiation by the formula

c being the speed of light in vacuum. Because of the universality of the

function KV(T), that is, because of its independence of the nature of the

Winter 1979/OPTICS NEWS 27

Wilhelm Wien (1864-1928)

F(v/T), leading to the so-called Wien's radiation law. It is listed in the second row in Table 2. The values of the two constants α and β that appear in this radiation law did not follow from Wien's argument and had to be deter­mined empirically.

Subsequently, in 1900, Lord Rayleigh showed that the application of the equipartition theorem of statis­tical mechanics to electromagnetic vibrations in a cavity leads to quite a different formula. When later cor­rected by J. H. Jeans for a simple error, it has the form shown in the third row of Table 2 (where k is the Boltzmann constant and c is the speed of light in vacuo), known as the Ray­leigh-Jeans law.

We note that both Wien's radiation law and the Rayleigh-Jeans law are of the form required by Wien's displace­ment law. However, careful measure­ments carried out around the turn of the century, notably by O. Lummer and E. Pringsheim and by H. Rubens and F. Kurlbaum, revealed that neither of them completely agrees with the experimentally determined energy dis­tribution in the spectrum of black-body radiation. It turned out that Wien's radiation law was a good approximation for sufficiently high frequencies and low-enough tempera­tures, whereas the Rayleigh-Jeans law was a good approximation in the other extreme case of low frequencies and high temperatures. The other law, Wien's displacement law, is, of course,

Lord Rayleigh (1842-1919)

incomplete because it contains the un­known function F(v/T).

This, then, was the situation in the closing years of the last century. Both thermodynamics and the theory of electricity and magnetism failed to predict the energy distribution in the spectrum of blackbody radiation.

The correct blackbody radiation law—one that is in complete agreement with experiment—was discovered by Max Planck in 1900, just a few months after the publication of Rayleigh's paper; and an assumption that went into its derivation marked the birth of one of the greatest revolutions that took place in science—the birth of the quantum theory.

Planck's radiation law may be writ­ten in the form

James H. Jeans (1877-1946)

the Rayleigh-Jeans law. Having quick­ly realized that his new law was in an extremely good agreement with exper­iment, Planck set about to find a more satisfactory theoretical basis for it. Within a few weeks he succeeded, and he outlined the essential features of his new derivation at a meeting of the

Max Planck (1858-1947)

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where h and k are constants, the first of which is now known as Planck's constant and the other is Boltzmann's constant. An essential ingredient of Planck's original derivation of his radi­ation law was a fortunate guess relat­ing to the entropy of a linear oscil­lator, which interacts with radiation. This guess was, in fact, equivalent to an interpolation between certain thermodynamic expressions appro­priate to Wien's radiation law and to

28 OPTICS NEWS/Winter 1979

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Albert Einstein in the Swiss Patent Office in Bern, ca 1905.

German Physical Society in Berlin on December 14, 1900.

According to Kirchhoff's law and formula (3), the equilibrium en­ergy density uv(T) is independent of the exact nature of the bodies that interact with the radiation field. Planck chose one of the simplest pos­sible models for these bodies, namely, harmonic oscillators. He then showed that the radiation law that he discov­ered a few weeks earlier may be de­rived in a systematic way from the laws of electromagnetic theory and thermodynamics if one makes the fol­lowing assumption: An oscillator, vibrating with frequency v, can take

on only one of the energy values E0, 2E0, 3E0,..., where E0 = hv. Thus, in deriving the correct blackbody radi­ation law, Planck found it necessary to introduce the notion of a quantum of energy E0 = hv, which represents the smallest amount of energy that an oscillator can emit or absorb. The need for introducing such a quantum of energy was in flat contradiction to Maxwell's electromagnetic theory, which places no restriction on the amount of energy that an oscillator can emit or absorb.

It may readily be shown that, when v/T>> 1 (i.e., in the limit of high frequencies and low temperatures),

Planck's radiation law (4) reduces to

where

The formula (5) is just Wien's radia­tion law, but we now also obtain ex­plicit expressions for the constants a and β. When v/T >> 1 (i.e., in the limit of low frequencies and high temper­atures), Eq. (4) reduces to

which is nothing but the Rayleigh-Jeans law.

PARTICLE ASPECTS OF RADIATION

Even though Planck's introduction of the concept of an energy quantum led eventually to one of the greatest sci­entific revolutions of all times, his theory did not at first attract much attention. One of the first scientists who clearly recognized that Planck's discovery initiated a new era in physics was a young man, Albert Einstein, who around that time—in 1902—at the age of 23 was appointed to a post at the Swiss Patent Office. His appoint­ment carried the title "Technical Ex­pert, Third Class." A photograph of this young man from that particular period appears on this page.

In the spring of 1905, Einstein, then quite unknown to the scientific world, wrote to a former fellow student and friend, Conrad Habicht, asking for a copy of a thesis that Habicht had recently completed. Einstein's letter contains a passage that, in English translation, reads2:

"But why have you not sent me your thesis? Don't you know, you wretch, that I would be one of the few fellows who would read it with interest and pleasure? I can promise you in return

Winter 1979/OPTICS NEWS 29

four works, the first of which . . . is very revolutionary...."

Three of these four papers that Ein­stein mentioned in this letter were published in one and the same volume, Volume 17 (4th series) of the journal Annalen der Physik, in 1905, having all been submitted within a period of only three and a half months. As Max Born noted,3 this is one of the most remarkable volumes in the whole scientific literature; for each of the three papers of Einstein that appear in it is acknowledged as a masterpiece and the starting point of a new branch of physics. In our present-day termi­nology the subjects of these three papers, in the order in which they appear, are: (1) the particle nature of radiation, (2) the theory of Brownian motion, and (3) the special theory of relativity.

We will review only the first paper, since the other two deal with topics that do not come within the scope of this article. However, it seems appro­priate to mention that the second paper probably did more than any other work to convince physicists of the reality of atoms and molecules and of the fundamental role that proba­bility theory plays in the formulation and the interpretation of the basic laws of physics; and that in the third paper Einstein took the monumental steps by which the intuitive notions of absolute space and absolute time were abolished and replaced by the concept of a four-dimensional space-time con­tinuum. This step has had the most profound consequences for our pres­ent-day understanding of the funda­mental physical laws of the universe.

The first of these papers4 has, in translation, the title: "On a heuristic point of view concerning the creation and conversion of light." It is this paper that Einstein himself referred to as "very revolutionary" in the letter to his friend Conrad Habicht. In modern textbooks it is usually referred to as "Einstein's paper on the photoelectric effect." Actually, this paper contains appreciably more. In fact, Einstein's whole discussion of the photoelectric effect covers less than four pages; but, as in most of his writing, Einstein was able to get to the root of a problem in

Title page of Vol. 17 (4th series) of Annalen der Physik, 1905, containing Einstein's three epoch-making papers-on the particle nature of radiation, the theory of Brownian motion, and the special theory of relativity.

a few lines, with simple language that was remarkably free of complicated mathematics.

Essentially what Einstein did in this paper was to put forward a great deal of evidence that not only do the pro­cesses of emission and absorption of radiation take place in discrete amounts of energy (as appears to have been established by Planck) but that radiation itself behaves under certain circumstances as if it consisted of a collection of particles, which in mod­ern language are called photons. Thus in this paper Einstein reintroduced a corpuscular theory of light—first ad­

vocated by Newton in the 17th cen­tury. As we already noted, the corpus­cular theory was completely discredit­ed by Fresnel's wave theory, formu­lated almost 90 years before the publication of Einstein's paper; and the wave theory of light itself ap­peared to have been put on firm foun­dation by Maxwell about 40 years before the appearance of Einstein's paper.

In the introduction to his paper, Ein­stein discusses the success of the wave theory of light, which deals with continuous functions in space. Then Einstein goes on to say that neverthe-

30 OPTICS NEWS/Winter 1979

Phillip E. A. Lenard (1862-1947)

less it is possible that this theory will lead to a contradiction with experi­ence if it is applied to the phenomena of generation and conversion of light. He then continues as follows:

"In fact it seems to be that the observations on blackbody radiation, photoluminescence, the production of cathode rays by ultraviolet light, and other phenomena involving the emis­sion or conversion of light, can be better understood on the assumption that the energy of light is distributed discontinuously in space. According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not con­tinuously distributed over an ever-in­creasing volume, but it consists of a finite number of energy quanta, local­ized in space, which move without being divided and which can be ab­sorbed or emitted only as a whole."

We will now present the essence of some of the examples given by Ein­stein to support these views.

Suppose that we throw particles into a box of volume V. Let us select in the box some subregion of volume A V (not necessarily small). If we throw one particle into the box, the probability that it will land in the chosen sub¬region is clearly

If we repeat this procedure n times, i.e., if we throw n particles into the

Robert A. Millikan (1868-1953)

box, one at a time, then, by an ele­mentary rule of probability theory, the probability that all the n particles will end up in the subregion is

Now instead of a system of particles, let us suppose that the box contains radiation under equilibrium conditions at temperature T. The energy density of the radiation field is given by Planck's law, but we know that for sufficiently high frequencies or suf­ficiently low temperatures it may be approximated by Wien's radiation law. Hence under these conditions the total energy of the radiation in the box is given by the formula

where α and β are the constants given by Eq. (6).

We have already noted that, even under equilibrium conditions, there are fluctuations in the radiation field and hence in the course of time the energy is being redistributed through­out the box. Consequently, in the chosen subregion the energy will have sometimes a larger and sometimes a smaller value than its mean value. In fact there is some probability that at a given instant all the energy E will be concentrated in this subregion. By using Wien's radiation law [Eq. (10)] and only some general principles of

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thermodynamics, Einstein showed that this probability is given by

Comparison of this result with Eq. (9) shows that this probability is the same as if the radiation field consisted of n particles, where

i.e., of n particles, each carrying energy hv. In Einstein's (translated) words:

"Monochromatic radiation of low density (within the range of validity of Wien's radiation law) behaves, in a thermodynamic sense, as if it consisted of mutually independent quanta of energy of magnitude5 hv."

Another example that Einstein gave in this paper in support of his view regarding the corpuscular nature of radiation was, as already mentioned, the photoelectric effect. This is the phenomenon of ejection of electrons from a metal when electromagnetic radiation of short enough wavelength impinges on the metal surface. The effect was discovered in 1887 by Hein¬rich Hertz in the course of experi­ments referred to earlier, which played a decisive role in confirming the cor­rectness of Maxwell's electromagnetic theory of light. In retrospect, there is some irony in this situation, since later, when the photoelectric effect was studied quantitatively, it was not possible to reconcile it with Maxwell's electromagnetic theory.

Some puzzling facts about the photoelectric effect were revealed largely by systematic experiments of P. Lenard, carried out in the period 1899-1902. Lenard found that the energy of an ejected electron is in­dependent of the intensity of the light that illuminates the metal surface, but that it depends on the frequency of the light; and also that, when the light intensity is increased, there is an in­crease in the number of the ejected electrons, but not in their energies. The difficulties of trying to explain these observations by the wave theory of light become apparent when we re­call that, according to the wave

Winter 1979/OPTICS NEWS 31

theory, the energy that is carried by a light wave is measured by the light intensity. Hence, as the intensity is increased, more energy becomes avail­able to be imparted to the electrons, and hence their energy should also in­crease, contrary to Lenard's observa­tions.

Einstein considered Lenard's obser­vations as a clear demonstration of the corpuscular nature of light: If light consists of energy quanta, each of amount hv, the quanta penetrate into the surface layer of the metal and their energy is at least partly transformed into the kinetic energy of the elec­trons. The simplest situation is that in which a light quantum transfers all its energy to a single electron and the energy is sufficient to set the electron free. On leaving the metal, the electron loses part of this energy because some work (W, say) is required to remove it from the metal. Einstein was thus led to the following formula for the maxi­mum kinetic energy of an ejected elec­tron:

This formula, known now as Einstein's photoelectric equation, at once ex­plains Lenard's observations.

We note that Einstein's photoelectric equation predicts that the maximum energy of the ejected photoelectrons is a linear function of frequency, whose slope is precisely Planck's constant. Hence the measurement of the de­pendence of the maximum electron energy on the frequency could be used to determine Planck's constant. In 1905, when Einstein put forward his photoelectric equation, quantitative studies of the photoelectric effect were in their infancy. It took nearly a decade of difficult experimentation before Einstein's equation could be fully tested. It was largely confirmed by the work of R. A, Millikan, who be­gan by completely disbelieving Ein­stein's theory. In a paper6 published in 1949 on the occasion of Einstein's seventieth birthday, this is what Mil­likan said: "I spent ten years of my life testing that 1905 equation of Ein­stein's and, contrary to all my expecta­tions, I was compelled in 1915 to assert its unambiguous experimental

Fig. 1. The dependence of the photocurrent potential on the frequency of the incident light in experiments on the photoelectric effect [From R. A. Millikan, Phys. Rev. 7,373 (1916)].

verification in spite of its unreason­ableness since it seemed to violate everything that we knew about the interference of light."

In Fig. 1 some of Millikan's experi­mental data are reproduced. They exhibit the linear relationship pre­dicted by Einstein's photoelectric equation, the slope of the line being indeed the value of Planck's constant.

THE WAVE-PARTICLE DUALITY

The 1905 paper of Einstein, which we just discussed, clearly revealed the fail­ure of classical physics to account for certain observed phenomena involving radiation. This paper showed a need for even more drastic changes than those brought about by Planck's assumption of quantized energy of the emitting and absorbing oscillators. Ein­stein's analysis indicated that not only do emission and absorption of energy take place in discrete energy quanta but that the radiation field itself be­haves, in certain situations, as if it con­sisted of such corpuscles of energy. The situation was puzzling, for on one hand there were phenomena such as inter­ference and diffraction of light, which seemed clearly to demonstrate the wave nature of radiation; but there were other phenomena such as the photoelectric effect, which reveal, as Einstein showed, that radiation has corpuscular nature.

In spite of the clarity and the sim-

plicity of Einstein's arguments, his views about the particle structure of radiation were strongly opposed at that time--and for a long time after­ward—by many eminent physicists, in­cluding Max Planck. Undeterred by the opposition, Einstein continued to explore the consequences of his cor­puscular theory and was probing more deeply into the nature of radiation.

In 1909, four years after his "photo­electric paper," Einstein published a paper7 with the title "On the present status of the problem of radiation," which became another milestone in the development of physics. In this publication Einstein showed, again by characteristically simple arguments typical of so much of his work, that Planck's radiation law itself implies that the radiation field exhibits not only wave features but also corpus­cular features. This result was the first clear indication of the so-called wave-particle duality that many years later became an accepted feature of modern quantum physics.

The essence of Einstein's argument may be described as follows: Consider again blackbody radiation at tempera­ture T in a thermally insulated cavity of volume V. Let us fix our attention on a subregion of volume V inside the cavity. As we noted earlier, there will be energy fluctuations inside this sub-region, caused by radiation moving in and out of it. Einstein inquired about the magnitude of these fluctuations.

Let E be the amount of energy in

32 OPTICS NEWS/Winter 1979

Back row: Einstein, Ehrenfest, de Sitter; front row: Eddington, Lorentz.

the subregion, at some fixed instant of time, in a frequency range (v, v + dv), say. The simplest measure of the energy fluctuations is the variance

where the bar denotes the statistical average.

Einstein first showed on the basis of thermodynamics that the variance may be expressed by a formula that can be written in the form

where, as before, k is the Boltzmann constant. Now according to Planck's radiation law [Eq. (4) ] , the mean energy in the subregion of volume V is given by the formula

where

On substituting from E q . (16) into E q . (15), Einstein obtained the fol low­ing expression for the variance:

Equation (18) is known today as Ein­stein's fluctuation formula for black-body radiation.

Einstein drew some far-reaching con­sequences from this formula. He argued as fol lows: If the radiation field inside the cavity consisted of purely electromagnetic waves, as most sci­entists of that time believed, one could explain the energy fluctuations in the fol lowing way: The radiation field could be represented as a mixture of plane waves (normal modes) wi th dif­ferent amplitudes, different phases, and different states o f polarization, propagated in all possible directions. The fluctuations of the energy in the subregion V could then be considered to be a manifestation o f short-term interference effects between the dif­ferent plane waves. Einstein indicated, by means o f a simple dimensional argument that covers only a few lines, that the energy fluctuations produced in this manner would be given by just the second term on the right-hand side of his f luctuation formula, a result that may be expressed in the form

Some years later H. A . Lorentz 8 veri­fied this result by long explicit calcula¬tions (which cover more than six page: in print).

Einstein argued now as fol lows: Since his f luctuation formula, (18), derived f rom Planck's law, con­tains more than just the term propor­tional to E2 (which alone is compre­hensible on the basis of the wave theory), something else must be going on apart f rom ordinary wave propaga­t ion. Suppose, Einstein argued, that the radiation consists of light quanta, each of energy hv, which behave like independent classical particles. If, at some given instant, there were n such quanta in the subregion V at some fixed time, then elementary statis­tics (involving the Poisson distri­bution) gives the fol lowing expression for the variance ( n)2 ≡(n-n)2 of their number n:

The total energy carried by the n quanta is

and the variance of this energy [ which we denote by ( E ) 2

p a r t i c l e s ] is therefore

Now, according to Eqs. (20) and (21), we have ( n ) 2 = E/hv and, on using this expression in Eq . (22), we obtain the formula

which is precisely the first term on the right-hand side of E q . (18).

It fol lows that Einstein's f luctuation formula for blackbody radiation may be expressed in the form

Thus Einstein showed that, in its sta­tistical behavior, blackbody radiation exhibits both wave features and parti­cle features. 9 This result is the first example of the famous wave-particle duality that much later (around 1926)

followed in a consistent manner from modern quantum mechanics.

Radiation carries not only energy, but also momentum, which is mani­fested as pressure. In the same paper Einstein provided additional evidence for the wave-particle duality from an analysis of momentum fluctuations in blackbody radiation.

In a lecture that Einstein delivered in the same year (1909) at a scientific meeting in Salzburg, he clearly indi­cated, in the following words, the con­clusions to which his investigations on the structure of radiation had led him so fa r 1 0 : "It cannot be denied that there is a broad group of facts con­cerning radiation, which show that light has certain fundamental prop­erties that can be much more readily understood from the standpoint of the Newtonian emission theory than from the standpoint of the wave theory. It is, therefore, my opinion that the next stage of the development of theoret­ical physics will bring us a theory of light which can be regarded as a kind of fusion of the wave theory and the emission theory . . . a profound change in our views of the nature and consti­tution of light is indispensible."

ELEMENTARY PROCESSES OF INTERACTION BETWEEN RADIATION AND MATTER

During the next few years Einstein concentrated his efforts in different directions, and he developed his gen­eral theory of relativity. But in 1917 he returned to the radiation problem once again, and he published another fundamental paper in this field. By that time much progress had been made toward the understanding of the spectrum of atomic elements, chiefly as a result of some work by Niels Bohr. Before discussing the 1917 paper of Einstein, it may be useful to review briefly some of the back­ground.

In 1911, Ernest Rutherford put for­ward a model of the atom according to which the atom consists of a small, heavy, charged central nucleus sur­rounded by a charge distribution of the opposite sign. However, the distri-

bution of the charge was not under­stood. Bohr, in a well-known paper published in 1913, assumed that the atom can exist permanently only in one of a series of states—known as stationary states--characterized by dis­crete values of the energy. When the atom emits or absorbs radiation it undergoes a transition from one such stationary state to another. Bohr assumed that, if the emitted or ab­sorbed radiation has frequency v, the change in the energy of the atom is

where E' and E" are the energies of the two stationary states and h is Planck's constant. Equation (25) is known as the Bohr frequency condi­tion. This basic assumption of Bohr's theory is in direct contradiction with classical electrodynamics, which does not impose any such restriction on the emitted or absorbed energy. However, the Bohr theory gave the correct values of the wavelengths of the spec­tral lines of the hydrogen atom and of the Rydberg constant and also eluci­dated some features of the periodic table of atomic elements. But the theory also contained some rather mysterious features and gave no in­sight whatsoever into the laws that govern the transitions from one allowed state of the atom to another. It was Einstein's 1917 paper, 1 1 en­titled "On the quantum theory of radi­ation," that provided the first real in­sight into this question.

In this paper Einstein considered a gas in equilibrium with radiation at temperature T. Bohr's theory suggests that the gas molecules can only exist in certain discrete states, with energies El,E2,E3,.. . Now even under equi­librium conditions there are fluctua­tions, caused by the exchange of energy between the molecules and the radiation field—but always in such a way that the equilibrium is main­tained. The detailed mechanism of this process was at that time completely unknown. Einstein put forward a cer­tain model for this energy exchange and showed that Planck's radiation law follows from it in a straightforward manner. This paper contains also a

Winter 1979/OPTICS NEWS 33

number of other remarkable results, as we will soon see.

Following Einstein, let us consider transitions between two possible states of the gas molecules, with energies En

and Em, where Em > E„. If a mole­cule makes a transition from the upper to the lower state, it will emit energy of the amount Em - En .If it under­goes a transition from the lower to the upper state, it will absorb energy of this amount. The frequency of the emitted or absorbed radiation will again be denoted by v. Einstein now made some assumptions about the mechanism responsible for such transi-

Niels Bohr (1885-1962), in early 1920's

tions. More specifically, he assumed that a transition between the allowed energy states occurs in one of three possible elementary processes of inter­action:

(a) Spontaneous Emission. It had been known from the study of radio­activity since about the early years of this century that atoms may disinte­grate into atoms of other kinds and that this process is completely un­affected by external circumstances—

34 OPTICS NEWS/Winter 1979

Fig. 2. The three elementary processes of interaction be­tween gas molecules and radiation, which Einstein11 as­sumed in his derivation of Planck's radiation law.

for example, the rate of disintegration is unaffected by temperature and pres­sure. From the analysis of experi­mental data on radioactive decay the following law was deduced: The prob­ability, d W ( 1 ) , that an atom disinte­grates in a small time interval dt is simply proportional to dt, i.e.,

where Anm is a constant, depending on the type of atom being considered. Einstein assumed that one of the ele­mentary processes of energy exchange between the gas molecules and the radiation field is governed by such a law. In other words, a molecule may emit radiation into the field, and the emission is not affected by external circumstances; in particular, it is not affected by the nature of the radiation field that surrounds the molecule. This process is called spontaneous emission.

(b) Induced Radiation Processes. If an oscillator of frequency v is placed in an electromagnetic field of the same frequency, then, according to classical electromagnetic theory, the energy of the oscillator may change as a result of transfer of energy between the oscil­lator and the field. Whether this change of energy is due to the emis­sion or absorption of radiation by the oscillator depends on the relative phase difference between the oscillator and the field. These two processes are called induced processes, because the change in the energy of the oscillator can now be attributed to the presence of the field that surrounds it. Ein­stein assumed that such an energy ex­change also takes place between the gas molecules and the equilibrium radi­ation field and, moreover, that such induced (or stimulated) emissions and absorptions are also governed by statis­tical laws However, the elementary

probabilities for the two induced pro­cesses, unlike those for spontaneous emission, can be expected to depend on the radiation field that surrounds the molecule. Specifically, Einstein postulated that the elementary proba­bility for a molecule to undergo an induced transition in a small time interval of duration dt is

for stimulated emission and

for absorption, where Bnm and Bmn

are constants (depending on the type of molecule being considered) and uv

is the energy density of the radiation field at the frequency v. These assump­tions of Einstein are indicated sche­matically in Fig. 2.

Einstein now asked the following question: Assuming that the energy exchange takes place in this manner, what can one deduce about the energy density uv of the radiation field? He argued essentially as follows: Suppose that, in equilibrium, Nm of the mole­cules will, on the average, have energy Em and Nnof them will have energy En. According to the principles of sta­tistical mechanics,

where k is again the Boltzmann con­stant and Cn and Cm are constants ("statistical weights") that are charac­teristic of the two quantum states of the molecules.

Since equilibrium is assumed to be maintained under the energy ex­change, there must be on the average as many transitions per unit time from the upper to the lower state as from

the lower to the upper state. This requirement is expressed mathe­matically by the equation

On substituting into Eq. (30) for the three elementary probabilities from Eqs. (26)-(28) and for the average number of molecules from Eq. (29), one obtains the relation

Assuming next, as Einstein did, that the energy density uv of the field will increase to infinity with T, one obtains from Eq. (31) the relation

If Eq. (31) is solved for uv and use is made of the relation (32), one ob­tains the following expression for the energy density of the equilibrium radiation field:

Now in order that this expression agree with Wien's displacement law, which we discussed earlier (See Table 2), the following relations must ob­viously hold:

where and h are constants. On mak­ing use of these two relations in Eq. (33), one obtains for the energy density of the equilibrium radiation field the formula

Equation (36) has the form of Planck's radiation law [Eq. (4)], except that the value of the constant y has not been determined by this argument. It could be obtained, as Einstein also pointed out, by proceeding to the classical limit (hv/kT << 1); the ex-

pression (36) must then reduce to the Rayleigh-Jeans law (see Table 2). One then finds that a has indeed the expected value,

This derivation by Einstein of the Planck radiation law is remarkable for a number of reasons:

In the first place, Einstein showed that Planck's law follows directly from his model of interaction be­tween radiation and matter. The probabilities that Einstein assumed for each of the elementary processes tak­ing part in the interaction are ex­amples of what is known today as transition probabilities between states. In this connection we recall that Bohr's quantum theory of the hydro­gen atom did not give any indication of the laws governing such transitions. The concept of transition probability originated in this paper of Einstein.

Einstein's derivation gave also as a by-product the relations (32) and (34) between the A - and B coef­ficients that specify the probabilities. These relations played an important role in the later development of the quantum theory of radiation.

Einstein's derivation also gave as a by-product the relation (35), which will be recognized as the Bohr fre­quency condition [Eq. (25)] that up to then was simply assumed.

These results alone would mark this paper of Einstein as a fundamental contribution to physics. However, much more was contained in this paper. Einstein also considered the question of momentum transfer be­tween the gas molecules and the radi­ation field. He showed that, when a molecule absorbs or emits a quantum hv under the influence of external radiation from a definite direction, momentum of magnitude hv/c (=h/λ where X is the wavelength of the radia­tion associated with the frequency v) is transferred to the molecule; and that the change in the momentum of the molecule is in the direction of the incident radiation if the energy is absorbed and is in the opposite direc­tion if the energy is emitted. More surprisingly, Einstein showed that, if an energy quantum is emitted

Arthur H. Compton (1892-1962)

in the absence of any external in­fluence (spontaneous emission), the momentum transfer is also a directed process. In Einstein's words: "There is no radiation in spherical waves. In a spontaneous emission process the molecule suffers a recoil of magnitude hv/c in a direction that in the present state of the theory is deter­mined only by 'chance'." He goes on to say that "These properties of the ele­mentary processes . . . make it seem practically unavoidable that one must construct an essentially quantum mechanical theory of radiation."

Einstein's conclusions that a quan­tum of energy hv carries a momentum whose magnitude i s 1 2 hv/c and has a definite direction were verified experi­mentally some years later. The well-known experiments of A. H. Comp­ton 1 3 on scattering of x rays provided the first experimental confirmation of the correctness of some of these pre­dictions.

THE BOSE-EINSTEIN STATISTICS AND MATTER WAVES

Successful as Einstein's notion of quanta of a radiation field was in eluci­dating various phenomena involving the interaction of light and matter, many puzzles surrounded it. A l l the derivations of Planck's radiation law,

Winter 1979/OPTICS NEWS 35

Satyendranath Bose (1894-1974)

including Einstein's 1917 derivation, appealed at some point to classical electromagnetic theory. Yet the quan­tum features of the radiation field, which, as Einstein showed, are implicit in Planck's law, are in direct contradic­tion with the classical theory. Einstein himself was well aware of these dif­ficulties, and he stressed over and over again the need to formulate a basically new theory that would fuse the wave features and the particle features of radiation. Such a theory, namely, mod­ern quantum mechanics, was indeed formulated about eight years after the publication of the paper by Einstein that we have just discussed. Although today one mainly thinks of de Broglie, Schrödinger, Heisenberg, Born, Dirac, and Pauli as the founders of modern quantum mechanics, Ein­stein actually played a key role in pre­paring the ground for it, especially in connection with its wave-mechanical formulation. However, discussion of this development does not fall within the Scope of the present article. We will, therefore, only briefly indicate a few important steps that Einstein took in this direction and that themselves have a bearing on the question of the nature of light, particularly with re­gard to its statistical properties.

36 OPTICS NEWS/Winter 1979

Louis de Broglie (born 1892)

Erwin Schrödinger (1887-1961)

In the summer of 1924 Einstein received a letter from an Indian physicist, S. N. Bose, which began as fol lows 1 4 :

"Respected Sir, I have ventured to send you the accompanying article for your perusal and opinion. I am anx­ious to know what you think of it. You will see that I have tried to de­duce the coefficient 8πv2 /c 3 in Planck's law independent of classical electrodynamics, only assuming that the ultimate elementary region in phase space has the content h3 . I do not know sufficient German to trans­late the paper. If you think the paper

worth publication I shall be grateful if you arrange for its publication in Zeit-sehrift für Physik. Though a complete stranger to you, I do not feel any hesi­tation in making such a request. Be­cause we are all your pupils though profiting only by your teaching through your wri t ings.. . ."

In this manuscript Bose essentially treated the light quanta as particles of a gas, with the difference that those particles that belong to the same ele­mentary cell of phase space, of volume h3 (where h is Planck's constant), are intrinsically indistinguishable. This assumed property of the quanta led to a statistical procedure that differs from that of classical statistical me­chanics and provided indeed a com­plete derivation of Planck's radiation law, without any appeal to classical electromagnetic theory.

Einstein translated Bose's manu­script and sent it to the Editor of Zeit-schrift für Physik with a letter contain­ing the following remark (which was printed in German at the end of the translated paper1 5): "In my opinion Bose's derivation of Planck's formula signifies an important advance. The method used also yields the quantum theory of the ideal gas, as I will show elsewhere."

On studying Bose's manuscript, Einstein seems to have realized that Bose's approach had far-reaching consequences. In a paper 1 6 that Ein­stein presented to the Prussian Acad­emy only a few days after sending the translation of Bose's manuscript for publication, and in another paper 1 7

that Einstein published in the follow­ing year (1925), he applied Bose's method not to a gas of light quanta, but to a real gas, consisting of mon¬atomic molecules. This work of Ein­stein is another of his masterpieces, and Martin J . Klein, the distinguished Yale historian of physics, said about the second of these two papers that it contains "as many ideas in its dozen pages as many an annual volume of the journals of physics." 1 8

One of the many important results that Einstein derived in that paper is the following: He considered a gas in a vessel under equilibrium conditions. Let n be the number of molecules,

with energies in the range (E, E + dE), which are situated in some particular region of the vessel. This number will, of course, fluctuate, and Einstein showed that the variance of the fluctu­ations of n may be expressed in the same form as the variance of energy-fluctuations in blackbody radiation, which he derived in 1909 and which we discussed earlier. In other words, the variance is again expressible as the sum of two terms. The first term can be attributed to classical particles; this term may be understood on the basis of the Maxwell-Boltzmann sta­tistics of noninteracting molecules. The second term, which is analogous to the contribution from wave inter­ference in the radiation problem, can­not be understood from the classical (particle) theory. This is just the opposite situation to that encountered in the radiation case, where the inter­ference term followed from classical (wave) theory, whereas the particle term was new. Einstein suggested that the second term in the expression for the variance of the fluctuations in an ideal gas may be interpreted in a similar way (as due to wave interference) if one associates a radia­tion field with the gas, and he goes on to say: "I shall discuss this interpretation in greater detail, because I believe that more than a mere analogy is involved here."

The paper by Bose and Einstein's two papers on the quantum theory of an ideal gas are the foundations of what today is known as the Bose-Einstein statistics. It applies to photons and to many other elemen­tary particles, which were actually unknown in 1925. The formula for the variance, as the sum of contributions from classical particles and classical waves, is a fundamental consequence of the Bose-Einstein statistics.

Between the appearance of Ein­stein's first and second papers on the quantum theory of an ideal gas, a period spanning less than five months, another important development took place. In November 1924, Louis de Broglie submitted a thesis at the Sorbonne in which he put forward a theory according to which a wave-called a matter wave—was associated

with the mot ion of every par t i c le . 1 9

In this theory, the frequency v and the wavelength X of the matter wave were related to the energy E and to the magnitude p of the momentum of the particle by the formulas

These two relations wi l l be recognized as being identical wi th those that, according to Einstein, connect the particle features (E and p) of radiation with the frequency v and the wave­length λ of the radiation f ield. Thus the association of matter waves wi th particles (de Broglie's theory) and the association of particles wi th a radi­ation field (Einstein's theory) are gov­erned by the same basic relations, namely Eqs. (38).

Einstein received a copy of de Broglie's thesis from Paul Langevin 2 0

in December 1924 and at once realized that de Broglie's ideas were basically sound and that the appearance o f the wave-interference term in his (Ein­stein's) new fluctuation formula for an ideal gas may be attributed to de Broglie's matter waves. He immedi-

Nernst, Einstein, Planck, Millikan, and von Laue (Berlin, ca 1928)

Winter 1979/OPTICS N E W S 37

Table 3 Einstein's Researches on the Nature of Radiation and the Wave-Particle Duality.

1905 PARTICLE NATURE OF RADIATION Particle features of blackbody radiation The photoelectric effect

1909 WAVE-PARTICLE DUALITY Energy fluctuations of blackbody radiation

1917 ELEMENTARY PROCESSES OF INTERACTION Spontaneous and induced processes, transition probabilities Momentum of energy quanta, directionality of interactions and

their statistical aspects

1924, 1925 BOSE-EINSTEIN STATISTICS Quantum theory of an ideal gas; density fluctuations;

matter waves

ately wrote to Langevin, and his letter contains the prophetic remark that de Broglie "has l i f ted a corner of the great v e i l . " 2 1 The very same month (December 1924) Einstein submitted for publication his second paper on the quantum theory of an ideal gas. He included in it a few remarks indicating the importance that he attached to de Broglie's work and outlining the con­nection that he believed to exist be­tween de Broglie's hypothesis of matter waves and his own investiga­tions on the quantum theory of the ideal gas. These remarks of Einstein

are known to have stimulated Erwin Schrödinger only a few months later to develop one form of modern quan­tum mechanics—namely wave mechanics.22

Thus the structure that Einstein built wi th his 1905 "photoelectric paper," his 1909 paper containing the first clear evidence for the wave-particle duality, his 1917 paper on the elementary processes of interaction between molecules and radiation, and his 1924 and 1925 papers on the quan­tum theory of an ideal gas did not only elucidate the nature of light and

38 OPTICS NEWS/Winter 1979

Albert Einstein Ulm, March 14, 1879-Princeton, April 18, 1955

radiation in general, but also proved to be of fundamental importance to the formulation of wave mechanics.

It was a long journey and in some ways a rather lonely one, since Ein­stein encountered much opposition to his concept of quanta of radiation. We already noted the skepticism of Mil-

likan. Another example is provided by a passage from a letter, which Max Planck wrote in 1913, in which Ein­stein was being proposed for member­ship in the Prussian Academy of Sci­ences. 2 3 After writing about Einstein in very glowing terms, this is what Planck said: "That he may sometimes

have missed the target of his specu­lations, as for example in his hypoth­esis of light quanta, cannot really be held against him." How confident Einstein was to continue, in spite of such skepticism, to probe further and further along the path that he had entered upon as an unknown young

scientist in the Swiss Patent Office around 1905!

Practically all Einstein's results relat­ing to the nature of radiation have, in more recent times, been derived in a systematic manner from the modern theory of radiation, to the formulation of which Einstein contributed in such a fundamental way; but the hueristic models by means of which Einstein first obtained these results will forever remain wonderful examples of how truly great research in physics is done.

"All the fifty years of conscious brooding have brought me no closer to the answer to the question, 'What are light quanta?' Of course today every rascal thinks he knows the answer, but he is deluding himself. "

A. Einstein, 1951

*This article is based on lectures given during the past two years to the Local Sections of the Optical Society of America. In addition to Einstein's ori­ginal papers, a number of published accounts and discussions of Einstein's works influenced this presentation. I particularly wish to acknowledge my indebtedness to the following excel­lent sources:

F. Reiche, The Quantum Theory (Methuen, London, 1922).

M. Born, Natural Philosophy of Cause and Chance (Clarendon Press, Oxford, 1949; reprinted by Dover, New York, 1964).

M. J . Klein, "Einstein's first paper on quanta," in The Natural Philoso­pher (Blaisdell, New York), Vol . 2 (1963), pp. 57-86, and "Einstein and the wave-particle duality," ibid., Vol . 3 (1964), pp. 1-49.

M. Jammer, The Conceptual Devel­opment of Quantum Mechanics (McGraw Hill, New York, 1966).

REFERENCES 1. For references and fuller accounts of

the early theories, see E. T. Whittaker, A History of the Theories, of Aether and Electricity, Vol. I (The Classical Theories), revised and enlarged edition, 1951; Vol. II (The Modern Theories 1900-1926), 1953 (T. Nelson and Sons, London and Edinburgh). Excellent dis­cussions of the radiation laws are given in the first chapter of Max Jammer, The

Conceptual Development of Quantum Mechanics (McGraw Hill, New York, 1966) and in an article by Martin J. Klein, "Max Planck and the beginnings of the quantum theory," Archive for History of Exact Sciences 1,459 (1962).

2. Quoted in C. Seelig, Albert Einstein, A Documentary Biography, transl. M. Savill (Staples Press, London, 1956), p. 74.

3. Max Born, "Einstein's statistical theo­ries," in Albert Einstein: Philosopher-Scientist, P. A. Schlipp, ed. (the Library of Living Philosophers, Evan¬ston, I11. 1949), pp. 161-177; repub­lished by Harper & Brothers, New York, 1959.

4. A. Einstein, Ann. Physik, 4th Ser. 17, 132-148 (1905); English translations: A. B. Arons and M. B. Peppard, Am. J. Phys. 33, 367-374 (1965), and D. ter Haar, The Old Quantum Theory (Pergamon, Oxford and New York, 1967) , pp. 91-107.

5. Actually Einstein expressed the energy of the quanta as (R/N)ßv, where R is the universal gas constant, N is the Avo¬gadro number, and ß is the constant that appears in the exponential term of Wien's radiation law (second row of Table 2). The probable reason that Ein­stein did not make the formal identifica­tion of the proportionality factor (R/N)ß with Planck's constant h is dis­cussed in Sec. 6 of M. J. Klein, "Ein­stein's first paper on quanta," in The Natural Philosopher (Blaisdell, New York) Vol. 2 (1963), pp. 57-86.

6. R. A. Millikan, Rev. Mod. Phys. 21, 343-345 (1949).

7. A. Einstein, Phys. Z. 10, 185-193 (1909).

8. H. A. Lorentz, Les théories statistiques en thermodynamique (Tuebner, Leipzig, 1916), Appendix IX, pp. 114-120.

9. Similar calculations based on the Ray-leigh-Jeans law, (7), rather than on Planck's law, lead to the result

whereas calculations based on the Wien's radiation law (5) give

If we recall that the Rayleigh-Jeans law and Wien's radiation law are good approximations when hv/kT<< 1 and hv/kT>> 1 respectively, we see that in the low-frequency-high-temperature limit wave features are predominant, whereas in the high-frequency-low-temperature limit the particle features predominate.

10. A. Einstein, Ver. Dsch. Phys. Ges. 11, 482-500 (1909). The text of this lec­ture is also published in Phys. Z. 10, 817-825 (1909), where an account is also given of the discussion that fol­lowed Einstein's paper (ibid, pp. 825-826).

Winter 1979/OPTICS NEWS 39

11. A. Einstein, Phys. Z. 18, 121-128 (1917). English translations in D. ter Haar, The Old Quantum Theory (Perga-mon, Oxford and New York, 1967), pp. 167-183, and in B. L. van der Waerden, Sources of Quantum Mechanics (North-Holland, Amsterdam, 1967; reprinted by Dover, New York, 1968), pp. 63-77.

12. Suggestions that an energy quantum carries a momentum of magnitude hv/c were actually made earlier on the basis of the special theory of relativity and electromagnetic theory (cf. M. Jammer, Ref. 1, loc. cit., p. 37).

13. A. H. Compton, Bull. Nat. Res. Council (U.S.) 4 (20), Part 2, pp. 1-56 (1922). Actually Einstein's ideas regarding the momentum of the energy quanta are not mentioned in Compton's paper. Their relevance for the interpretation of such experiments was, however, noted shortly afterward by P. Debye, Phys. Z. 24, 161-166 (1923).

14. As quoted in M. Jammers, Ref. 1, loc. cit., p. 248.

15. (S. N.) Bose, Z. Phys. 26,178-181 (1924).

16. A. Einstein, Preuss. Akad. d. Wissen-schaften, Berlin, Phys. Math. KL, Sitz-ungsber., 1924, pp. 261-267.

17. A. Einstein, Preuss. Akad. d. Wissen-schaften, Berlin, Phys. Math. KL, Sitz-ungsber., 1925, pp. 3-14.

18. M. J. Klein, "Einstein and the wave particle duality," in The Natural Philos­opher (Blaisdell, New York),Vol. 3 (1964), p. 33.

19. A brief preliminary account of the theory appeared in L. de Broglie, C. R. Acad. Sci. 177, 507-510, 548-550, 630-632 (1923). An English summary of these three notes is given in Philos. Mag. 47, 446-458 (1924).

20. A fuller account of Langevin's role in bringing de Broglie's theory to Einstein's attention is given in M. Jammer, Ref. 1, loc. cit., pp. 248-249.

21. Louis de Broglie, New Perspectives in Physics, transl. A.J . Pomerans (Basic Books, New York 1962), p. 139.

22. " . . . The whole thing would certainly not have originated yet, and perhaps never would have, (I mean not from me), if I had not had the importance of de Broglie's ideas really brought home to me by your second paper on gas degen­eracy." [From a letter written by E. Schrödinger to A. Einstein on April 23, 1926, included in Letters on Wave Mechanics, K. Przibram, ed., translated and with an introduction by M. J. Klein (Philosophical Library, New York, 1967), p. 26.]

23. The letter was also signed by three other leading scientists from Berlin, W. Nernst, H. Rubens, and E. Warburg. [C. Kirsten and H.G. Korber, Physiker über Physiker (Akademie-Verlag, Berlin, 1975), pp. 201-208].