ALBERT EINSTEIN AND LIGHT QUANTA*

Max Born

I regard the invitation to speak here on the occasion of the fiftieth anniversary of the appearance of Einstein’s first paper on light quanta (“On a Heuristic Point of View Concerning the Production and Transformation of Light”) as a great honor, and I considered it a matter of duty to accept this invitation. For, first, my friendship with Einstein has already lasted almost fifty years, and, second, Berlin University was the first university where I received a professorship and the only place where, for several years, I worked side by side with Einstein and lived in constant contact with him. However, alongside the desire to accept this invitation, objections also arose. When, a year and a half ago, I left my chair in Edinburgh on reaching the age limit, I decided no longer to take an active part in physical science (with the exception of completing two books that had been begun long ago and one of which had managed to appear). I settled in a quiet place where there is no university and no library, and sold a large part of my own books and journals. Thus I lacked the literary resources that could have refreshed my memory. And yet I decided not to decline the invitation. In fact, only a few theoretical physicists remain alive among those who, from the very beginning, saw and experienced the majestic picture of the rise of modern physics. If, therefore, you take into account that my account is essentially based on recollections and in some points may not be entirely accurate, I shall nevertheless try to describe the first years of this century in physics and Einstein’s part in its development, especially in the development of the idea of light quanta.

In the same year 1905, and in the same volume of the journal Annalen der Physik, two other papers by Einstein also appeared: on the theo—

*) Report delivered at the Berlin Physical Society on March 18, 1955, Berlin—Charlottenburg. Published in Naturwissenschaften 42, 425 (1955).

120

MAX BORN

...the theory of relativity and on Brownian motion. However different these topics may seem, there is nevertheless a connection between these works. In particular, Einstein’s theory of fluctuations and Brownian motion rests on considerations completely analogous to his justification of the idea of light quanta. This I shall try to explain as an example of the striving for a uniform, all-embracing conception of nature, which guided Einstein right up to the present day*).

Allow me, however, first to say a few words about Einstein’s early biography and about his sudden appearance as a new star in the firmament of physics. Einstein was born on March 14, 1879, in Ulm. His parents soon moved to Munich, and later to Pavia in Italy. He himself remained at the Gymnasium in Munich; however, he found the spirit of the school so deadening that in 1894 he left it of his own accord without taking the final examination and went to his parents, who at that time were living in Milan. The same spirit of independence prompted Einstein to leave the Jewish religious community and German citizenship. In 1895 he moved to Switzerland, entered the cantonal school in Aarau, where free winds were blowing and where he found friends as free-thinking as himself.

Having passed the school-leaving examination, Einstein entered the Higher Technical School in Zurich in 1896 in order to study mathematics and physics. Among his teachers was Hermann Minkowski, who subsequently played an outstanding role in the development of the theory of relativity. Minkowski was also my teacher (in Göttingen); after the appearance of Einstein’s work on the theory of relativity, he once said to me: “To tell the truth, I did not expect this from Einstein...” In 1900 Einstein passed the diploma examination for the title of teacher, but had difficulty obtaining a position. For some time he was an assistant teacher in Winterthur and Schaffhausen until, in 1902, he entered the Swiss Patent Office in Bern as an “expert, third class.” After the hardship in which he had spent his school and student years, this small but secure position seemed to him a paradise, and the eight hours a day that he was obliged to spend in the Office at his desk, going through dry patent applications, did not interfere with his reflections on physical and philosophical problems. In this environment, far from the stimuli of higher education, his great works arose. When, many years later, I asked for his help in finding a place for some capable student, I usually received an answer of approximately the following content: “Suggest that he become a shoemaker or a mechanic; if he has science in him

) Born’s lecture was delivered while Einstein was still alive, exactly one month before his unexpected death. (Ed. note*)

ALBERT EINSTEIN AND LIGHT QUANTA

in his blood and that he is capable of something—he will find his own way.” Nevertheless, he always helped when he could. However, the idea that the need for creativity must have nothing in common with earning one’s bread never left him, and even now it is alive, but for another reason, to which I shall return.

In that same year, 1900, when Einstein was taking his diploma examination, Max Planck reported to the German Physical Society his radiation formula (October 19)1 and its interpretation by means of the quantum hypothesis2. I have no personal recollections of the impression made by this communication, since at that time I was still at school and was only preparing for my final examination. In the autumn of 1901 I began to study at the university, but did not immediately concentrate on mathematics and physics; I studied such varied subjects as zoology, political economy, astronomy, and philosophy. Yet in the following years too, when in Breslau, Heidelberg, Zurich, and Göttingen I attended lectures by outstanding mathematicians and did a little physics, I learned—mainly from Minkowski—about the difficulties in the electrodynamics of moving systems, which led to the creation of the theory of relativity, but knew absolutely nothing about quanta. The latter were the property of specialists in the field of thermal radiation and, despite the experimental confirmation of Planck’s formula, were regarded as a peculiar ad hoc hypothesis, not to be taken seriously.

Einstein, however, did take it seriously. In years he was four years older than I, but in intellectual age (as psychologists say), probably eight years older or more. Already between the ages of 12 and 16 he had mastered differential and integral calculus from books and enthusiastically studied other branches of mathematics, whereas I learned these things as a student in the normal way—in lectures—though also with enthusiasm. And although Einstein in the Zurich Polytechnic surprised his teachers by irregular attendance at lectures and by a certain casualness, the reason for this was that he already knew almost everything presented in those lectures and was occupied with deeper problems.

In the interval between 1900 and 1905 the quantum theory apparently made no progress. Likewise, in the fundamental and broadly comprehensive book by E. Whittaker, A History of the Aether and Electricity (Sir Edmund Whittaker, A History of the Theories of Aether and Electricity. Edinburgh, Th. Nelson & Sons Ltd., 1953), the recently published second volume of which covers the period from 1900 to 1926, nothing is reported about this period.

A different picture emerged when, 50 years ago, Einstein’s first paper appeared3. The first six paragraphs of this paper contained theoretical considerations which, with the exception of a few specialists, attracted little attention. However, the last three paragraphs were devoted to an entirely new application:

quanta, namely—to the explanation of Stokes’ rule in luminescence, of the photoelectric effect, and of the ionization of gases. The common point of view was that in all these cases what is involved is the transformation of the kinetic energy \(E\) of an electron into a light quantum \(h\nu\), or conversely, so that a linear relation of the form

\[ E = h\nu + \mathrm{const.} \]

must hold. On the \((E,\nu)\) diagram this relation is represented by a straight line with slope \(h\), where \(h\) is Planck’s universal constant, the value of which had been determined by Planck from measurements of radiation. Thus these assertions were accessible to experimental verification, and there is nothing surprising in the fact that experimental physicists seized upon this problem.

It is not my task here to set forth the history of experimental investigations in this direction. It is enough to say that Einstein’s work gave rise to numerous experiments and for the first time brought order into the domain of the most diverse phenomena connected with the emission and destruction of radiation. I should, however, like to dwell on the six first paragraphs mentioned above, since it is precisely in them that the depth and convincing force of Einstein’s ideas were contained.

Einstein himself never tires of emphasizing that there is no unambiguous logical path from the facts of experience to the theoretical systems of physics; the latter, in his opinion, are essentially the children of free imagination. And yet the value of a theory is the greater, and our confidence in it the greater, the less freedom of choice there is in it, the greater its logical compulsion. It seems to me that the combined efforts of Max Planck and Albert Einstein satisfy this ideal. I shall try to explain this assertion, using the phraseology and notation now customary.

In many years of preliminary work Planck understood the statistical character of black radiation and developed methods for its theoretical investigation. In doing so he became convinced that entropy, because of its connection, discovered by Boltzmann, with the probability of a state, is the decisive quantity for describing the behavior of thermostatic systems. As for the density of radiation \(\rho\), it was known that it must have the form prescribed by Wien’s displacement law,

\[ \rho = \nu^{3} f(\nu/T) \]

(from which follows Stefan–Boltzmann’s law for total radiation), and that there are two limiting cases: for high temperatures Rayleigh’s law holds (proportionality of \(\rho\) to the temperature \(T\)); for low temperatures, Wien’s law (proportionality to

\[ \exp\left(-\frac{a}{T}\right) \]
).

Planck’s first step consisted in replacing the investigation of radiation by the investigation of a system of identical linear harmonic oscilla-

tors of frequency \(\nu\), in statistical equilibrium with the radiation. A purely electrodynamic calculation gives, for the ratio of the radiation density \(\rho\) to the mean energy of the oscillator \(U\), the quantity

\[ \frac{8\pi\nu^2}{c^3}. \]

Planck then calculates the second derivative of the entropy \(S\) with respect to the energy \(U\); I shall introduce for this quantity an abbreviated notation, putting

\[ \frac{d^2 S}{dU^2}=-k\gamma, \tag{1} \]

where \(k\) is the gas constant per particle of an ideal gas; in what follows, instead of the temperature I shall use the quantity

\[ \beta=\frac{1}{kT}. \]

In this case a simple thermodynamic consideration (see Appendix A, p. 133) gives

\[ \gamma=-\left(\frac{dU}{d\beta}\right)^{-1}. \tag{2} \]

The advantage of this notation is that in both of the limiting cases mentioned the energy \(U\), referred to one oscillator, is a particularly simple function of \(\beta\), so that one can immediately write down an explicit expression for (2). Indeed, we have:

\[ \begin{array}{ll} T\ \text{large (Rayleigh)} & U=\left\{\begin{array}{l} \beta^{-1},\\ U_0 e^{-\beta\varepsilon_0}; \end{array}\right. \qquad \frac{dU}{d\beta}=\left\{\begin{array}{l} -U^2,\\ -\varepsilon_0 U. \end{array}\right. \\[1ex] T\ \text{small (Wien)} \end{array} \tag{3} \]

Planck took a bold step: to interpolate between the two limiting cases by forming the sum

\[ -U^2-\varepsilon_0 U, \]

which, in the limiting cases of large or small radiation densities, passes into one of the two corresponding expressions. With this interpolation expression, by integration, taking into account Wien’s law for the radiation density, one obtains the following expression for the mean energy of the oscillator (Appendix B):

\[ U=\frac{\varepsilon_0}{e^{\beta\varepsilon_0}-1}, \qquad \varepsilon_0=h\nu, \tag{4} \]

where \(h\) is a constant. This expression, with the aid of the indicated conversion factor

\[ \frac{8\pi\nu^2}{c^3}, \]

leads to Planck’s radiation formula.

Of course, the considerations set forth can hardly be regarded as a derivation, owing to the arbitrariness of the choice of the quantity \(\gamma\) for interpolation.

In the second paper² Planck interprets the oscillator energy (4) as the mean energy for a Boltzmann distribution with finite

energy quanta, \(\varepsilon_n = n\varepsilon_0 = h\nu n \quad (n=0,1,2,\ldots)\):

\[ U= \frac{\displaystyle\sum_{n=0}^{\infty}\varepsilon_n e^{-\beta\varepsilon_n}} {\displaystyle\sum_{n=0}^{\infty} e^{-\beta\varepsilon_n}} = -\frac{d}{d\beta}\ln \sum_{n=0}^{\infty} e^{-\beta n\varepsilon_0} = \frac{\varepsilon_0}{e^{\beta\varepsilon_0}-1}. \]

Only someone who, like me, grew up in the classical tradition can fully appreciate the boldness of this idea. Planck himself was inclined to regard the distribution of energy in finite quanta not as a property of the radiation itself, but as the result of its interaction with oscillators. Here Einstein entered the scene.^3 He discovered the simple physical meaning of the quantity \(\gamma^{-1}\) used by Planck for interpolation, which leads directly to the conception of independent light quanta and, moreover, justifies the interpolation method itself. Einstein bases his reasoning on Boltzmann’s relation between entropy \(S\) and probability \(P\):

\[ S = k \ln P. \tag{5} \]

Since at that time this relation was by no means generally accepted, Einstein first gives a simple derivation of it (which even now seems to me the best one). The following considerations are very closely connected with Einstein’s theory of fluctuations and Brownian motion.

The main idea of the derivation is to invert formula (5), regarding probability as a function of entropy,

\[ P = e^{S/k}, \tag{6} \]

and to use the thermodynamic properties of entropy.

Let us imagine that a system in thermodynamic equilibrium is divided into small equal parts; then, owing to the statistical nature of heat, the energy \(E\) in any part will not be equal to the mean energy \(\overline{E}\) falling to each of the parts, but fluctuations \(\Delta E = E - \overline{E}\) will occur. The entropy of the part under consideration (at constant volume) may be regarded as a function of \(E\) and expanded in powers of \(\Delta E\). In this case the total entropy will not contain a linear term, since for an adiabatically isolated system the sum over all parts \(\sum(\Delta E)=0\). In this case, taking into account the abbreviated notation (1), we have:

\[ \sum S = \mathrm{const} - \frac{1}{2}k\gamma \sum(\Delta E)^2 + \ldots, \]

From (6) it now follows approximately

\[ P=e^{\frac{S}{k}}=P_0 e^{-\frac{1}{2}\gamma(\Delta E)^2}, \tag{7} \]

whence, by (2), we obtain directly (see Appendix B):

\[ \overline{(\Delta E)^2}=\frac{1}{\gamma}=-\frac{dU}{d\beta}. \tag{8} \]

This is the basic formula for energy fluctuations used in Einstein’s theory of Brownian motion. If, for example, one considers a material system at constant volume, then one obtains the well-known Einstein law

\[ \overline{(\Delta E)^2}=kT^2 c_v, \tag{9} \]

where

\[ c_v=\frac{dU}{dT} \]

is the specific heat.

In the case of radiation, \(E\) denotes the total energy of radiation of frequency \(\nu\), contained in a small volume chosen as unity, i.e. a quantity on average equal to \(\rho\). It is convenient, instead of the latter, to consider the quantity

\[ U=\rho\,\frac{c^3}{8\pi\nu^2}, \]

without bearing in mind here that \(U\) can be defined as the mean energy of an oscillator in equilibrium with the radiation. However, it later turned out that \(U\) has a meaning for the radiation itself. Namely, the radiation field can be decomposed by Fourier; then each Fourier component corresponds to an oscillator, and it turns out that

\[ \frac{8\pi\nu^2}{c^3} \]

is equal to the number of radiation oscillators per unit volume for a frequency interval equal to 1, while \(U\) is the energy of one of these oscillators.

Einstein considers the (Wien) case of low temperatures, where the radiation energy \(U\), in the indicated measure, is expressed as

\[ U=U_0 e^{-\varepsilon_0\beta}; \]

in that case, from (8) it follows that

\[ \overline{(\Delta E)^2}=\varepsilon_0 U \qquad (T\text{ small}). \]

If one regards \(E\) as a multiple \(n\) of small “energy quanta” \(\varepsilon_0\),

\[ E=n\varepsilon_0, \tag{10} \]

then one obtains \(U=\varepsilon_0 \overline{n}\), where \(\overline{n}\) is the mean number of quanta, and for the mean square fluctuation of \(n\) we shall have:

\[ \overline{(\Delta n)^2}=\overline{n} \qquad (T\text{ small}). \tag{11} \]

But this formula means, on the basis of the known statistical laws, that the radiation energy behaves as if it consisted of independent particles of magnitude \(\varepsilon_0=h\nu\).

This, too, is what the theoretical part of Einstein’s work consists in, in its essential features. Allow me, however, to take one more step.

The same reasoning as that just carried out leads, in the (Rayleigh) limiting case of high temperatures, where \(U=\beta^{-1}\), to the formula

\[ \overline{(\Delta E)^2}=U^2 \qquad (T \text{ large}) \tag{12} \]

or

\[ \overline{(\Delta n)^2}=\bar n^2 \qquad (T \text{ large}). \tag{13} \]

Planck’s interpolation, by adding both values that occur in the limiting cases, is justified by the simple observation that the mean square of a fluctuation is additive if the fluctuations are caused by independent influences.

What, then, are these processes? One of them was interpreted by Einstein: at low temperatures, i.e. at small energy densities, radiation behaves like an ideal gas. But what does the second process consist in, the one that appears in pure form at high temperatures? H. A. Lorentz showed that a statistical ensemble of plane waves with arbitrary amplitudes and phases leads precisely to expression (13) for energy fluctuations.

Thus we see that the wave-particle dualism is already fully contained, implicitly, in Einstein’s work. In the case of small energy densities, particles manifest themselves; their number undergoes fluctuations according to law (11); for large densities, (13) holds; and, finally, for intermediate densities,

\[ \overline{(\Delta n)^2}=\bar n+\bar n^2=\bar n(\bar n+1). \tag{14} \]

Here, therefore, there appears that peculiarity of quantum effects whereby the correct quantum formulas are obtained from the classical formulas if \(n^2\) is replaced by \(n(n+1)\).

How were these ideas received? I shall allow myself to speak of my own experience. In Göttingen, as far as I recall, I heard nothing about quanta; likewise in Cambridge, where in the spring and summer of 1906 I attended the lectures of J. J. Thomson and Larmor for several months and went through the experimental course at the Cavendish Laboratory. Only then, when in the autumn of 1906 I came to Breslau to Lummer and Pringsheim, did I enter a real quantum atmosphere. For both of them had made an essential contribution to the experimental study of black radiation. But although Planck’s formula stood at the center of the discussion, those discussing it were inclined to regard Planck’s hypothesis of the quantization of the oscillator’s energy as a preliminary working hypothesis, and Einstein’s light quanta were not taken seriously. After all, Lummer was in fact a great specialist in the field of wave optics—recall Lummer’s plate—and it would have been too much to demand of someone who every ...

ALBERT EINSTEIN AND LIGHT QUANTA

day observed interference in order for him to believe in the revival of the corpuscular theory. Attempts were made to understand the law \(E=h\nu\) classically. I, too, am guilty of this sin; I shall give you my reasoning, for amusement’s sake. Imagine an apple tree in which the length \(l\) of the apple stems decreases in proportion to the square of the height \(H\) above the ground; then \(\nu \sim \dfrac{1}{\sqrt{l}} \sim H\). If one now shakes the apple tree with a definite frequency, then the apples hanging at a definite height will enter into resonance, fall down, and reach the ground with a kinetic energy proportional to the height from which they fell, and therefore also proportional to the frequency \(\nu\): Voilà! Now this reasoning seems to us naive, not to say childish. But in my defense I can report that Max Planck himself presented this model in one of his lectures. Where he got it from, I do not know.

At the beginning of 1907 Einstein took a second essential step in the theory of quanta. I have in mind his theory of specific heat.^4 The latter clearly shows that the matter is by no means reducible to an exchange of energy between radiation and particles, as Planck had thought, but that what is involved is a very general principle embracing electrodynamics and mechanics. I cannot recall exactly how this view took shape in my mind. By that time I had made Einstein’s acquaintance, and in conversations with him I learned more than from his articles. Outwardly, my adherence to Einstein’s point of view was manifested in the theory of the specific heat of a crystalline lattice that I published jointly with my friend Theodor von Kármán—a theory in which Einstein’s considerations were refined by taking account of the vibrational spectrum, in which, however, Debye anticipated us by several weeks. But this was already in 1912.

In the following year a plan matured in the Berlin Academy of Sciences, consisting in inviting Einstein to Berlin; at that time he was a professor at the Higher Technical School in Zurich. To support this plan, the four most outstanding German physicists—Planck, Nernst, Rubens, and Warburg—made a submission to the Prussian Ministry of Education, in which they listed and assessed Einstein’s merits. In it there occurred an astonishing passage, about which I may tell you without risking immodesty, since it is printed in Carl Seelig’s excellent book Albert Einstein and Switzerland.* The submission in question says: “To sum up, it may be said that among the major problems with which modern physics is so rich, there is not a single one with respect to which Einstein has not taken a notable position. The fact that in his reasoning he sometimes goes beyond the mark, as, for example, in his hypothesis of light quanta, should not be held too strongly against him.

* Carl Seelig, Albert Einstein und die Schweiz. Zürich: Europa—Verlag, 1952.

“…or, without daring to take risks, one cannot accomplish anything truly new even in the most exact natural science.”

This quotation illustrates the skepticism toward the notion of light quanta that was universal at that time. It is not my task to describe the gradual change in attitudes. I wish to note only a few of the most important events. Among these belongs, first of all, J. J. Thomson’s observation, made in 1907, that the number of atoms ionized by X-rays decreases with distance from the source of the rays, whereas for each act of ionization one and the same energy must be supplied. From this he concluded that the radiation consists not of spherical waves, but rather behaves like a shower of particles. Later (1916) Einstein substantiated this idea of “needle radiation” (Nadelstrahlung) statistically,* by calculating the momentum that a suspended particle in a light field receives and gives up in absorption and emission. It turned out that the requirement that normal Brownian motion not be disturbed in the process is fulfilled only if the radiation behaves like a gas of particles with energy \(h\nu\) and momentum \(h\nu/c\).

Meanwhile, beginning in 1913, Bohr’s works on the quantum theory of electronic orbits in atoms and on the origin of spectra began to appear. From then on it became clear that in the physics of the future the idea of quanta must play a dominant role. Amid the discussion of quantum conditions and selection rules necessary for describing the motion of electrons, the quantum structure of light was almost forgotten until, in 1922, it again became topical thanks to the discovery of the Compton effect and its explanation as a consequence of collisions between photons and electrons. This interpretation is based on the application of the laws of conservation of energy and momentum, and here both of Einstein’s discoveries are used—those which we especially commemorate in these days,* namely the relativistic formulas

\[ E = mc^2,\qquad p = mc,\qquad m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \]

for the electron, and the quantum formulas

\[ E=h\nu,\qquad p=\frac{h\nu}{c} \]

for the photon.

To the same circle of ideas belongs the work of Smekal, who in the following year, 1923, drew attention to the fact that, according to the law of conservation of energy, in the case of a system consisting of an atom (or molecule) and a photon, one should expect the existence of a new kind of

Born’s lecture was given on the occasion of the fiftieth anniversary of the special theory of relativity. (Editor’s note.*)

scattering of light, namely scattering with a change in wavelength. This gave Landén, Kramers, and Heisenberg occasion to revise the theory of dispersion from the point of view of quantum ideas—an important step in the slow development which ultimately led to quantum mechanics. The phenomenon predicted by Smekal, which is now called the Raman effect*), was discovered only in 1928 by Raman in molecules and by Landsberg and Mandelstam in crystals.

As for the radiation formula, Einstein derived it in 1916 on the basis of Bohr’s postulates on stationary states\({}^{6}\), considering the emission and absorption of light quanta by analogy with radioactive decay. The symmetry of the transition probabilities \(A_{mn}\) between the states \(m\) and \(n\), \(A_{mn}=A_{nm}\), which occurs here, was also an important precursor of quantum mechanics. This derivation of the radiation formula from transition probabilities corresponds, in the theory of gases, to the use of Boltzmann’s collision formula. But the laws of equilibrium of a gas can also be derived without considering collision processes, by finding the most probable state. Is it not possible likewise to obtain the radiation formula as the energetic equation of state of a photon gas? Einstein’s initial arguments connected with fluctuations went in this direction; however, for the case of small radiation densities they led only to formula (11), characteristic of a gas, whereas in the general case the more complicated formula (14) holds. This question occupied and troubled Einstein for a long time, and when in 1924 the Indian physicist Bose found the solution, taking into account in the statistics the fundamental indistinguishability of photons, Einstein immediately adopted this idea and extended it to material particles\({}^{7}\). At that time he already knew of de Broglie’s bold ideas, which lay at the foundation of wave mechanics (de Broglie’s papers were published as short communications beginning in 1923, and in 1924 were collected in his famous dissertation). Bose–Einstein statistics was Einstein’s last positive contribution to the development of quantum mechanics. Here I may break off my historical survey. From this moment on, Einstein’s attitude toward the very foundations of quantum theory that he himself had laid became increasingly critical and skeptical.

He himself revealed the paradoxical wave–particle dualism. The main task of theoretical physics consisted in overcoming this apparent dualism. This was accomplished in two ways: through a generalization of Bohr’s ideas about electronic orbits there arose matrix mechanics, which was founded by Heisenberg and developed by him jointly with P. Jordan and with me, and also independently of us by Dirac; from de Broglie’s hypothesis there arose wave

*) In Soviet literature the term “combination scattering” has been adopted. (Ed. note)

mechanics of Schrödinger. Soon, however, it turned out that both methods are merely different representations of one and the same theory.

The formalism of the theory had already been well developed and substantiated before a reasonable interpretation could be found. This interpretation differs from classical theory in that it renounces the exact prediction of physical situations; it permits only the calculation of probabilities. The overwhelming majority of physicists accepted this interpretation—especially the experimenters, since this interpretation corresponds exactly to the empirical state of affairs in the study of atoms.

But Einstein considered the statistical interpretation unsatisfactory and tried again and again to refute it. In this connection, however, the interpretation of the square of the wave function as a probability belongs to Einstein himself. It was he who expressed the idea that the mean density of photons in a light beam must coincide with the energy density of the electromagnetic waves describing this beam. I advanced this same idea in 1927 for the interpretation of Schrödinger’s wave function; with the corresponding generalizations it is now generally accepted. The apparent contradiction in the simultaneous use of wave and corpuscular representations was removed by Heisenberg’s uncertainty relations. The concept of complementarity put forward by Niels Bohr gave the whole edifice of quantum mechanics an epistemological foundation.

The photon itself is, of course, a special particle: it has no rest mass and it always moves with one and the same velocity. It belongs, strictly speaking, not to quantum mechanics but to quantum electrodynamics. Already in the first papers of Heisenberg, Jordan, and myself, the electromagnetic field was quantized by establishing commutation relations between the components of the field, and, as the most important application, the fluctuation formula (14) for the electromagnetic field was derived by interference of quantized waves, with the sum \(n + n^2\) appearing automatically. This was the beginning from which the modern refined quantum electrodynamics of Schwinger, Feynman, and others arose.

But neither Bohr’s philosophy, nor the enormous successes of ordinary quantum mechanics, nor the astonishing accuracy of the results obtained with the aid of quantum electrodynamics could induce Einstein to acknowledge these theories. He did not deny their applicability, but regarded them as incomplete, preliminary auxiliary means which in the future would have to be replaced by something better.

Einstein’s position rested on his philosophical convictions. I shall quote two passages from his letters to me which I, incidentally, with his permission, published in one of my books (Natural Philosophy of Cause and Chance, Clarendon Press, Oxford, 1949).

In a letter of November 7 he wrote:

“In our scientific views we have developed into antipodes. You believe in a God who plays dice, while I believe in complete lawfulness in the world

objectively existing, which I am trying to grasp in a purely speculative way. I hope that someone will find a more realistic path and, accordingly, a more tangible foundation for such a view than I have managed to do. The great initial successes of quantum theory could not make me believe in the game of dice lying at its basis.”

From December 3, 1947, i.e.:

“My physical position I cannot justify to you in such a way that you would recognize it as at all reasonable. Of course, I understand that the fundamentally statistical point of view, the necessity of which was for the first time clearly realized by you, contains a considerable share of truth. However, I cannot seriously believe in it because this theory is incompatible with the basic proposition that physics must represent reality in space and in time without mystical action at a distance... What I am firmly convinced of is this: in the end one will stop at a theory in which the things lawfully connected will be not probabilities but facts, as until recently it was taken to be self-evident. In support of this conviction I can cite not logical grounds, but my little finger as witness—that is, an authority that inspires no confidence beyond the limits of my skin.”

A little over a year ago I again had a correspondence with Einstein concerning a short article in which he develops, by means of a concrete example, the same thought: the rejection of probabilities as the sole object of physical theory. In analyzing this model I began to doubt whether classical mechanics can in fact make deterministic assertions. As a result, I became convinced that mechanical determinism rests on an assumption that contradicts the method of thought of modern physics, founded by Einstein himself—namely, it contradicts the postulate according to which statements that are in principle inaccessible to experimental verification have no meaning. I recently published a short communication on this subject in Physikalische Blätter. In it I justify the idea that classical mechanics, too, can express only probabilistic propositions.

Therefore Einstein’s objection to the statistical interpretation of quantum mechanics seems to me groundless. However, from the quoted excerpts from the letters, and also from the later correspondence, it follows that Einstein’s rejection of contemporary quantum physics is conditioned not so much by the question of determinism as by his faith in the objective reality of physical being independently of the observer. Elsewhere I have shown that Einstein’s objections can be answered if one analyzes the concept of the reality of physical objects and, in doing so, correspondingly uses the mathematical concept of invariance with respect to transformations. Einstein did not confine himself to criticizing the statistical interpretation

of quantum mechanics, but he strove continuously to create another foundation for physics. His starting point in this was the general theory of relativity, which he tried to generalize in the hope of arriving, in the end, at an explanation of quantum phenomena and elementary particles. He obtained no positive results, and physicists knew little of his great and difficult labors.

Thus Einstein found himself in an isolation that would have been tragic, were it not for his joyful, optimistic temperament, which guarded him against bitterness. After all, he had always been a solitary man. He sought knowledge neither for material gain nor for fame. The tragedy of his life is the tragedy of science in general—the tragedy of the abuse of science in the political competition of states. What he himself thought on this matter is shown by his brief letter to the editor of the American newspaper The Reporter:

“You ask me what I think of your articles concerning the situation of natural scientists in America. Instead of trying to analyze the problem, I want to express my feeling in the form of a brief remark: if I were once again a young man and had to choose a profession, I would not try to become a scientific worker—a scholar or a teacher. I would prefer to be a tinsmith or a peddler, in the hope of obtaining that modest independence which, under present circumstances, is still possible.” He expressed this very same thought to me in a comparatively calm time, more than 40 years ago, as advice to a young physicist. A more modest and at the same time more trenchant form of condemnation of the contemporary application of science—which serves in this world not spirit, but force—could hardly be possible. However, this does not concern our subject.

The characterization of Einstein as a scholar and thinker would, however, be incomplete if at least a few words were not said about Einstein as a human being. Allow me, therefore, to conclude with a few lines from one of his letters to me:

“The sense of how one ought to act and how one ought not to act grows and dies like a tree, but no fertilizer can play an essential role in this. What every human being must do is to set an example of purity and to have the courage to preserve ethical convictions seriously in a society of cynics. For a long time I have been striving to act in this way—with varying success.”

These are the words of disappointment of one who believed in predestination also in human affairs, but who took into account the influence of the ethical personality. Probably this peculiar method of saving moral freedom is, philosophically, very debatable. But Einstein’s philosophy of physics also seems to me debatable. This should not disturb our friendship, for what is at issue is not doctrines, but purity and honesty in thoughts and feelings. And in both respects we honor Einstein as an exemplar and as a teacher.

ALBERT EINSTEIN AND LIGHT QUANTA

APPENDICES

A. For every thermodynamic system at constant volume one has

\[ dS=\frac{dU}{T}=k\beta\,dU; \tag{1} \]

therefore

\[ \left. \begin{aligned} -k\gamma&=\frac{d^2S}{dU^2}=-\frac{d}{dU}(k\beta)=k\frac{d\beta}{dU},\\ \gamma&=-\left(\frac{dU}{d\beta}\right)^{-1}. \end{aligned} \right\} \tag{2} \]

B. If, together with Planck, we put

\[ \frac{dU}{d\beta}=-U^2-\varepsilon_0 U, \tag{3} \]

then, carrying out the elementary integration and taking into account the boundary condition consisting in the fact that for \(\beta=0\) Rayleigh’s law \(U=\beta^{-1}\) holds, we obtain:

\[ U=\frac{\varepsilon_0}{e^{\beta\varepsilon_0}-1}. \tag{4} \]

From Wien’s displacement law

\[ \rho=\nu^3 F\left(\frac{\nu}{T}\right) \]

it follows for

\[ U=\rho\,\frac{c^3}{8\pi\nu^2}, \]

\[ U=\nu F(\nu\beta), \tag{5} \]

i.e., it must be

\[ \varepsilon_0=h\nu. \tag{6} \]

C. (7) is the Gaussian distribution in the variables \(\Delta E=x\). If the total probability is normalized to 1,

\[ \int_{-\infty}^{+\infty} P\,dx = P_0\int_{-\infty}^{+\infty} e^{-\frac12\gamma x^2}\,dx = P_0\sqrt{\frac{2\pi}{\gamma}} =1, \tag{7} \]

whence

\[ P_0=\sqrt{\frac{\gamma}{2\pi}}, \]

so that

\[ P(x)=\sqrt{\frac{\gamma}{2\pi}}\,e^{-\frac12\gamma x^2}. \tag{8} \]

Hence the following mean values are obtained:

\[ \overline{\Delta E}=\bar{x}=\int_{-\infty}^{+\infty} xP(x)\,dx=0, \]

\[ \overline{(\Delta E)^2}=\overline{x^2}=\sqrt{\frac{\gamma}{2\pi}}\int_{-\infty}^{+\infty} x^2 e^{-\frac{1}{2}\gamma x^2}\,dx=\frac{1}{\gamma}. \]

CITED LITERATURE

1. Planck M., Verh. deutsch. phys. Ges. 2, 202 (1900).
2. Planck M., Verh. deutsch. phys. Ges. 2, 237 (1900).
3. Einstein A., Ann. Phys. 17, 132 (1905); 20, 199 (1906); Phys. Zeits. 10, 185, 817 (1909).
4. Einstein A., Ann. Phys. 22, 180 (1907).
5. Einstein A., Mitt. phys. Ges., Zürich, No. 18 (1916); Phys. Zeits. 18, 121 (1917).
6. Einstein A., Ber. deutsch. phys. Ges. 18, 318 (1916).
7. Einstein A., Sitzungsber. preuß. Akad., Berlin, 261 (1924); 3, 18 (1925).