THE LIGHT QUANTUM HYPOTHESIS

ITS DEVELOPMENT AND CURRENT STATE¹

On the twenty-fifth anniversary of the light quantum hypothesis (1905–1930)

P. Jordan, Hamburg

Introduction. I. Initial development of the light quantum hypothesis. 1. Einstein’s works of the period 1905–1909. 2. Debye’s derivation of Planck’s law. 3. Einstein’s investigations of the period 1916–1917. 4. Construction of the theory of light quanta. II. Light quanta and wave theory. 5. The crisis of the theory of light. 6. The Bohr–Kramers–Slater theory. III. Matter waves. 7. Bose–Einstein statistics. 8. de Broglie waves. 9. de Broglie waves and Einstein’s theory of gases. IV. Influence of the light quantum hypothesis on the development of quantum mechanics. 10. Schrödinger waves. 11. Wave mechanics and matrix theory. 12. Further development of the theory. V. Quantum-mechanical theory of radiation. 13. Quantum mechanics of radiation fluctuations. 14. Interaction of atoms with radiation. 15. The many-body problem and quantum mechanics. 16. Waves and particles. 17. Relativistic construction of the theory.

Introduction

Planck first justified the quantum hypothesis by the following reasoning. If one imagines that an electron is elastically bound to some equilibrium position, about which it is capable of oscillating with frequency $\nu$, and if one then assumes that it is subject to the influence of electro-

¹ Ergebnisse der exakten Naturwissenschaften. Vol. VII. Berlin: Springer, 1929.

of magnetic radiation, the mean energy of the oscillating electron \(W_\nu\) remains unchanged for a long time in the case where its value is related to the radiation density \(\rho_\nu\) by the relation

\[ W_\nu=\frac{c^3}{8\pi \nu^2}\rho_\nu \tag{1} \]

(\(c\) is the speed of light). This can be shown by applying classical mechanics to the oscillating electron, Maxwell’s theory to the radiation field, and Lorentz’s theory to the interaction between them. But according to experimental data

\[ \rho_\nu=\frac{8\pi h\nu^3}{c^3}\cdot \frac{1}{e^{\frac{h\nu}{kT}}-1} \tag{2} \]

It was precisely this formula that Planck obtained by interpolating between the radiation formulas of Rayleigh–Jeans and Wien, and which was then confirmed experimentally before Planck found its theoretical explanation. From the equalities (1) and (2) it follows that the well-known statistical theorem on the uniform distribution of energy over degrees of freedom, which follows from classical theory and gives, for the energy of oscillators in thermal equilibrium, the value \(W_\nu=kT\), cannot be valid, since together with equality (1) it leads to the Rayleigh–Jeans radiation law. On the contrary, according to equalities (1) and (2), the following relation must hold:

\[ W_\nu=\frac{h\nu}{e^{\frac{h\nu}{kT}}-1} \tag{3} \]

From this formula Planck drew his astounding conclusion about the existence of discrete energy levels in oscillators of this kind. This conclusion, obviously, destroyed the very deepest foundations of all theories used for deriving formula (1)—namely, the assumption of the continuous course of natural phenomena. Criticism was often directed against this proof, which seemed logically so easily vulnerable, but I must say that I never could

look upon this train of thought otherwise than with deep reverence, if one may so express it. A remarkable instinct enabled Planck to find precisely the point at which the classical theories, despite their complete unsuitability for solving the problem considered by Planck, yielded, by way of exception, a correct and decisive formula. The special relation (1) is correct, although the general theories from which it was derived are incorrect: an entirely special combination of the content of these theories with the special properties of the harmonic oscillator led, essentially by chance, to the correct result.

We shall encounter the same situation at other decisive points in the historical development of quantum theory. The foundations of classical theory, previously understood as something self-evident, proved false when one penetrated into the domain of quantum phenomena. Thus at first there was in general no possibility of making logically irreproachable conclusions about the laws of this completely unknown domain, and progress could be made only by such a path as in Planck’s reasoning cited above; namely, by taking as a basis certain laws and formulas of classical theory, with respect to which one could suppose that they had a broader range of applicability than the conceptions from which they had originally been derived.

The efforts gradually to adapt, in this sense, theoretical conceptions to the peculiar facts of this new domain of phenomena, by generalizing and developing classical concepts, are essentially grouped around two principal cycles of ideas: Einstein’s hypothesis of light quanta and Bohr’s correspondence principle. The directions of quantum theory characterized by them developed for some time separately, and had comparatively little influence on one another. Only in the new quantum mechanics did they both merge into something unified. Here we shall trace only the development of the hypothesis of light quanta.

I. THE INITIAL DEVELOPMENT OF THE LIGHT-QUANTA HYPOTHESIS

1. Einstein’s works of the period 1905–1909

The development of the light-quanta hypothesis begins with the work published in 1905 by Einstein: “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (“On a heuristic point of view concerning the production and transformation of light”). In this work Einstein for the first time advanced the hypothesis that the light flux—at least in the case of small radiation densities—consists of discrete, corpuscular quanta of energy \(h\nu\); he justified this hypothesis by two different lines of reasoning.

On the one hand, he discerned a basis for his hypothesis in experiments relating to the production and transformation of light: Stokes’s rule (according to which the frequency of fluorescence light—if exceptions are left aside—is always less than the frequency of the primary, exciting light) received a direct explanation on the basis of the law of conservation of energy.

The fact discovered by Lenard several years earlier, that the velocity of electrons emitted under the influence of light (in the so-called photoelectric effect) depends only on the frequency of the exciting light and not on its intensity, could likewise be explained by the new hypothesis, whereas for the classical wave theory it presented insurmountable difficulties. On the basis of the equality \(\frac{1}{2}mv^2 = h\nu - P\), Einstein also obtained a quantitatively satisfactory agreement between the theoretical and experimental order of magnitude of the maximum velocity \(v\) of the emitted electrons. Finally, Einstein’s hypothesis led to a quantitative relation between the then known, only very roughly minimal values of the electric ionization potential and the ionizing frequency of light for a gas; this relation was also satisfied with sufficient accuracy. Here we see the first beginnings of that direction of research work which was

LIGHT-QUANTUM HYPOTHESIS

then so fruitfully developed by Stark, Franck, Hertz, and others; the numerous and varied results of these investigations all find their explanation in the simple Einstein–Bohr frequency condition.

Einstein, however, soon after this (in 1906) pointed out that the hypothesis he had advanced could also be regarded as a natural development of Planck’s above-mentioned proposition concerning the possible discrete states of an electric oscillator: if only discrete states exist, then there must also exist discrete, jump-like changes of them, and from this we pass directly to the emission of light quanta \(h\nu\). Of course, only subsequent development could show whether these conclusions, at once primitive and bold, were really legitimate. It is well known how long Bohr was cautious about passing from the frequency condition and the assumption of quantum jumps in the atom to radical conceptions of light quanta.

However, Einstein gave the most important foundation for his hypothesis by his thermodynamic-statistical arguments. Einstein carried out the following thought experiment. Suppose that in a certain cavity with reflecting walls of volume \(V\) there is enclosed energy \(E\) in the form of electromagnetic radiation, all of which belongs to a narrow frequency interval \(\nu, \nu+\Delta\nu\). In some (comparatively very small) part \(V_0\) of the volume \(V\), the average amount of energy will then be

\[ E_0 = E \frac{V_0}{V}. \]

But this amount of energy will not constantly have a strictly definite value; on the contrary, it is subject to certain temporal oscillations (fluctuations). According to the classical wave theory, one can calculate in advance all the laws governing these fluctuations. Einstein showed that they can, on the other hand, also be calculated by means of thermodynamic arguments, without resorting to wave theory, if only one assumes that the relation established by Boltzmann between entropy and statistical probab-

ness¹ is preserved not only in classical, but also in quantum theory.

Planck’s radiation law (2) can thereby be introduced as an empirical law. In what follows the following turns out to be the case. At a large radiation density, which corresponds to the classical Rayleigh–Jeans law,² we naturally obtain the same results as are given (without the aid of thermodynamics) by the classical wave theory. But at a small radiation density thermodynamics requires effects that are incompatible with the classical wave theory. Einstein showed this in 1905 by the following example. Let the radiation density \(\rho_\nu\) in the frequency interval \(\Delta\nu\) under consideration be very small, which corresponds to Wien’s distribution law;³ the question is asked how great is the probability that, as an exception, at some instant the entire radiation energy \(E\) will turn out to be concentrated in the volume \(V_0\). Einstein’s result is that this probability is equal to

\[ W=\left(\frac{V_0}{V}\right)^{\frac{E}{h\nu}} \tag{4} \]

Here light quanta \(h\nu\) begin to play an explicit role; the quantity \(n=\dfrac{E}{h\nu}\), according to Einstein’s hypothesis, is the number of light particles, and \(\left(\dfrac{V_0}{V}\right)^n\) is the probability that they will all gather in the volume \(V_0\). One may assert that, in general, in the region of applicability of Wien’s law radiation behaves, with respect to density fluctuations, exactly like a classical ideal gas consisting of particles \(h\nu\). Formula (4), which we shall call the first Einsteinian

¹ According to Boltzmann \(S=k\lg W\), where \(S\) is entropy and \(W\) is the statistical probability of a certain definite state (\(k\) is Boltzmann’s constant).

² Rayleigh–Jeans law \(\rho_\nu=\dfrac{8\pi h\nu^3}{c^3}\cdot kT\). Valid for \(kT\gg h\nu\).

³ Wien’s law \(\rho_\nu=\dfrac{8\pi h\nu^3}{c^3}e^{-\frac{h\nu}{kT}}\). Valid for \(h\nu\gg kT\).

by the law of radiation, gives an extremely clear example of the applicability of this proposition.

In 1909 Einstein returned again to this question and showed the following. If, in some space of black radiation \(V_0\), the energy belonging to the frequency interval \(\nu,\ \nu+\Delta \nu\) is equal to \(E_0\), and if one sets \(E_0-E=\Delta E_0\), then one can again, purely thermodynamically, calculate the mean square of the fluctuation \((\Delta E_0)^2\).

\[ (\Delta E_0)^2=h\nu\cdot E_0+\frac{c^3}{8\pi\nu^2\cdot \Delta\nu}\cdot \frac{E_0^2}{V_0} \tag{5} \]

Hence, for the limiting case of the Rayleigh–Jeans law (\(\rho_\nu\) large) we obtain

\[ (\Delta E_0)^2=\frac{c^3}{8\pi\nu^2\cdot \Delta\nu}\cdot \frac{E_0^2}{V_0} \tag{5a} \]

and for the limiting case of Wien’s law (\(\rho_\nu\) small)

\[ (\Delta E_0)^2=h\nu E_0 \tag{5b} \]

Since the number of light quanta contained in the volume \(V_0\) is equal to \(n_0=\frac{E}{h\nu}\), it follows from formula (5b) that, for small \(\rho_\nu\),

\[ (\Delta n)^2=n_0, \tag{6} \]

and this again corresponds exactly to the laws of the classical ideal gas. Formula (5) we shall briefly call the second Einstein law of fluctuations.

Classical wave theory gives, for the square of the fluctuation, instead of expression (5), expression (5a); this was studied in detail by Lorentz.1 Thus, in this regularity too, only in the limiting case of large—

of the radiation density, classical wave theory gives correct results, i.e. results satisfying the requirements of thermodynamics.

The same was also found in another line of reasoning contained in Einstein’s 1909 paper. Einstein imagines that in a radiation field, enclosed in a certain cavity, there is a plate capable of moving freely; it must freely transmit all radiation lying outside the frequency range \(\nu, \nu+\Delta\nu\), and reflect the radiation of this frequency range. Moving relative to the hollow space, this plate must on average experience a constant resistance, which is the result of light pressure; if only this resistance acted, it would brake the motion of the plate and in the end bring it to rest. But by virtue of fluctuations in the intensity of the radiation, fluctuations also occur in this radiation pressure, which will not allow the plate to come to complete rest: it will execute Brownian motion. The statistical laws of this Brownian motion had already been studied earlier (Einstein–Smoluchowski); precisely because of the generality of statistical principles they must always be one and the same, independently of the special mechanism that produces them in a given particular case. In the region of applicability of the classical Rayleigh–Jeans law, of course, everything remains in order: the fluctuations of radiation pressure calculated according to wave theory turn out to be just as strong as is required in order to produce the correct Brownian motion. But where deviations from the Rayleigh–Jeans law become noticeable, the fluctuations following from classical wave theory already prove insufficiently strong to be able to produce Brownian motion. Einstein’s investigations in this case too make it possible to trace quantitatively the deviations from classical wave theory that are required by thermodynamic statistics. In this way the picture of light quanta becomes much more vivid.

According to Maxwell’s classical theory, a plane wave carries with it a store of energy \(E\) and a store of momentum \(\dfrac{E}{c}\). Since the formulae for the magnitude of Maxwell’s radiation pressure that follow from this are undoubtedly correct both empirically and thermodynamically, it is entirely natural to suppose that a light quantum with energy \(h\nu\) possesses momentum \(\dfrac{h\nu}{c}\). In agreement with this, the kinematics of the special theory of relativity also requires, for a particle moving with the speed of light and having energy \(E\), the momentum \(\dfrac{E}{c}\).

Einstein showed that this assumption concerning the momenta \(\dfrac{h\nu}{c}\) is in remarkable agreement with the fluctuations of radiation pressure studied on a movable plate.^1 Indeed, if one again considers the limiting case of small radiation densities (the region of Wien’s law), then the oscillations caused by these fluctuations turn out to be precisely of the same kind as if a rain of corpuscular particles with momentum \(\dfrac{h\nu}{c}\) were falling upon the plate for a long time.

In this connection let us note once more that Wien’s radiation law itself may also be derived from the conception of an ideal gas of light quanta; the formula of Wien’s law,

\[ \rho_\nu=\frac{8\pi h\nu^3}{c^3}e^{-\frac{h\nu}{kT}} \tag{7} \]

corresponds directly to the Maxwellian distribution of velocities:

\[ \rho(v)=\text{const.}\,v^2\cdot e^{-\frac{v^2}{2kT}}, \tag{8} \]

where \(\rho(v)\) measures the number of atoms with velocity \(v\).

^1 Einstein, however, at this point did not explicitly state the assumption that the momenta are equal to \(\dfrac{h\nu}{c}\); he noted only that we again encounter here quanta with energy \(h\nu\).

Thus, as a result of Einstein’s work, the picture of radiation in empty space appears to us as extraordinarily vivid, but at the same time deeply enigmatic. Purely phenomenologically, using thermodynamic arguments and Boltzmann’s principle, we can, on the basis of Planck’s experimentally verified formula, not only calculate in advance its stationary temperature equilibrium, but also predict the regularity of all effects that may be observed as a result of fluctuations. But only in two limiting cases is it possible, on the basis of classical conceptions, to give with the aid of a model a visual interpretation of all these laws; and moreover the models that we must use in these two cases prove to be entirely different: in the Rayleigh–Jeans case it is an oscillating wave field, while in Wien’s case it is a corpuscular ideal gas.

2. Debye’s derivation of Planck’s law

Following the historical sequence, we shall mention here first of all Debye’s work, published in 1910. In this work, undoubtedly for the first time, an attempt was made to apply to the field of electromagnetic radiation those conceptions which had been developed in the example noted above of the Planck oscillator, were then confirmed in Einstein’s ideas concerning the theory of the specific heat of solids, and two years later entered as one of the most essential foundations into Bohr’s atomic theory. The oscillating electromagnetic field in empty space is a superposition of purely harmonic normal oscillations, each of which, taken separately, represents in a certain sense a harmonic oscillator. Debye’s bold assumption consisted in the fact that these normal oscillations are subject to quantization in the same way as material oscillators, and that they possess discrete energy levels \(0, h\nu, \ldots\).

This assumption leads directly to Planck’s radiation law. Indeed, the mean energy of such a quantum oscillator is determined by formula (3).

But in a cavity of volume \(V\) there exist precisely

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

normal modes with frequency lying in the interval \(\nu,\nu+\Delta\nu\). Thus the total radiant energy of this frequency region in the whole volume is equal to

\[ V\cdot \frac{8\pi\nu^{2}}{c^{3}}\,\Delta\nu\, \frac{h\nu}{e^{\frac{h\nu}{kT}}-1}; \tag{9} \]

dividing this expression by \(V\,\Delta\nu\), we obtain for the radiation density \(\rho_\nu\) the expression corresponding to formula (2).

In addition to this result, from de Broglie’s ideas, as Rubinowicz and Flamm showed, there follows the possibility of interpreting the Bohr frequency condition without the aid of the radical hypothesis of light quanta. If an atom emits light by a quantum jump, this means that it transfers to the ether a supply of energy \(\Delta W\). This supply of energy is expended by the ether in order to carry the appropriate normal mode into a quantum state, one step higher than before. Consequently, the frequency \(\nu\) of this normal mode must satisfy the condition \(\Delta W=h\nu\).

Just as in Einstein’s ideas, in de Broglie’s the energy of the cavity is the sum of discrete elements of energy \(h\nu\). But whereas Einstein assumes a spatial localization of these elements, de Broglie assigns to each of the normal modes, extending over the entire cavity, a definite number of quanta \(h\nu\). Further development led to the merging of both these conceptions. Although de Broglie’s work already in 1910 contained a considerable share of truth, it was not destined to become the starting point of further progress. Given the state of affairs at that time, its results had to be exhausted by the points noted above. There was then no possibility of seeing in it the tool with whose help one could overcome the difficulties of the wave theory revealed by Einstein in fluctuation phenomena. Only 15 years later did quantum mechanics proceed to the end along the path on which de Broglie had set foot.

3. Einstein’s Investigations of the Period 1916–1917

Einstein’s best-known work on the quantum theory of radiation belongs to the period 1916–1917. In it he established his famous probability laws for the interactions between radiation and matter. This work shed bright light on the elementary processes of emission and absorption of radiation by individual atoms or molecules, and indicated the existence, in these processes as well, of light quanta, which had earlier been discovered in the fluctuations of pure radiation.

Let two stationary states of an atom or molecule have energies \(W_1, W_2\), \((W_2 > W_1)\). Then, for an atom free from external influences and situated at the moment \(t\) on the upper of these levels, there exists a definite probability \(w\) that before the moment \(t + dt\) it will spontaneously pass to the lower level, emitting a light quantum \(h\nu = W_2 - W_1\). The probability \(w\), as in the process of radioactive decay, does not depend on how long the atom has already been in this state. Einstein himself, however, noted quite clearly the fundamental difficulty of this assumption. If it is correct, then we are deprived of any hope of ever interpreting atomic emission as a strictly causal process. Here, however, we are dealing with a difficulty created by nature itself. The experimental validity of the law of radioactive decay is not subject to any doubt, and this law represents so obvious and unquestionable a case of a non-causal phenomenon that up to now no one has even attempted to reduce it to any causal mechanism.

The influence of the radiation field on the atom further creates, according to Einstein, further transition probabilities, namely the probabilities of positive or negative absorption for atoms situated respectively on the lower or upper level. Taken together, for transitions from top to bottom one obtains the probability

\[ w_- = C\left(8\pi h\frac{\nu^2}{c^3} + \rho_\nu\right), \tag{10a} \]

whereas for transitions from below upward the probability is

\[ w^{+}=C\rho_\nu, \tag{10b} \]

where \(\nu\) is the transition frequency, calculated from the formula \(h\nu=W_1-W_2\).

If \(N_1\) and \(N_2\) are the numbers of atoms situated at the lower and upper levels, then in thermodynamic equilibrium the relation must hold\(^1\)

\[ N_2=N_1 e^{-\frac{h\nu}{kT}}, \tag{11} \]

and equilibrium with one and the same number of upward and downward transitions is possible if and only if \(\rho_\nu\) is determined precisely by Planck’s formula (2) for black radiation.

Einstein further investigated the influence of emission and absorption processes on translational motion; it was found that, just as in the case of the moving reflecting plate, correct thermal equilibrium can occur only in the case when every act of emission or absorption is associated in the atom with a change of momentum.

Here, after a long interval, the momenta of light quanta, \(h\nu/c\), again appeared before the eyes of physicists, and from then on they have never ceased to be a subject of discussion. However, for several more years they were still regarded as something more or less unreliable and unclear, until, finally, A. H. Compton and Debye found, in the scattering of X-radiation, the possibility of experimentally detecting these recoil impulses. The laws of this Compton effect and its connection with the hypothesis of light quanta are well known, and we shall therefore confine ourselves only to pointing to them.

\(^1\) From \(N_1w^{+}=N_2w^{-}\) it follows that

\[ 1=e^{-\frac{h\nu}{kT}}\left(\frac{8\pi h}{c^3}\frac{\nu^3}{\rho_\nu}+1\right), \]

whence formula (3) follows.

4. Construction of the Theory of Light Quanta

Einstein’s treatment of the interaction of radiation of free atoms became the model for a number of works that studied, from the standpoint of quantum theory, various elementary acts taking place in collisions of material particles with light quanta or with one another (absorption, emission or scattering of light, excitation, ionization, chemical transformations, etc.). The general results of all the investigations (Pauli, Einstein–Ehrenfest, Klein–Rosseland, Milne, Fowler, Becker, and others) were summarized by Dirac in his work, which appeared in 1924. The most general elementary act of this kind is evidently such that in it there occurs a collision of a certain number of (identical or different) material particles and, at the same time, a certain number of light quanta is absorbed. After the process some other (or changed) material particles arise and, in addition, again some number of light quanta. The probability of such a process—up to a certain factor depending on the special character of the process—is determined by the product of the concentrations of all colliding particles, multiplied by the factor \(\rho_\nu\) for each absorbed quantum \(h\nu\), and by the factor \(\dfrac{8\pi h\nu^3}{c^3}+\rho_\nu\) for each newly emitted quantum.

Moreover, Einstein’s formulas (10a) and (10b) take on a more transparent form if, instead of \(\rho_\nu\), one introduces into them the quantity

\[ n_\nu=\frac{c^3}{8\pi h\nu^3}\rho_\nu . \tag{12} \]

In the spirit of Debye’s ideas, the quantity \(n_\nu\) may be regarded as the mean number of quanta of energy \(h\nu\) for each natural oscillation of frequency \(\nu\).

In this case, for emission the probability is proportional to \(n_\nu+1\), while for absorption it is proportional to \(n_\nu\). This is a result quite naturally following from symmetry considerations, as was (admittedly later) discovered by the investigations of Bothe and Dirac. If one again takes Debye’s ideas as the basis and investigates the probability

of the fact that from a separate, perfectly definite natural oscillation a quantum \(h\nu\) is absorbed by an atom, it will turn out that this probability is proportional to the number of quanta in this natural oscillation before the process, and after the process this number is decreased by one. We shall obtain, therefore, complete symmetry if, for the inverse process, we assume that the probability of the emission of one \(h\nu\) in a definite oscillator of our hollow space is proportional to the number of quanta of this natural oscillation after the process, i.e., is proportional to a number one greater than the number of quanta before the process. This explains the proportionality of the probabilities of emission and absorption to the different numbers \(n_\nu + 1\) and \(n_\nu\), if one takes into account that the mean number of quanta \(n_\nu\) of the various natural oscillations present must be considered without allowance for the changes caused by the process itself, i.e., before the process.

II. THE LIGHT-QUANTUM HYPOTHESIS AND THE WAVE THEORY

5. Crisis of the Theory of Light

All the successes of the light-quantum hypothesis—on the one hand, the naturalness in the development of theoretical ideas and the simplicity of the results thereby obtained, and, on the other hand, the experimental confirmation of its consequences in the Compton effect—at first seemed only to make the crisis of the theory of light ever more serious. To the infinite number of experiments which, apparently, satisfactorily confirmed the wave theory of light, there were opposed proofs of the corpuscular nature of radiation which could no longer be evaded. It is not the task of this article to outline in greater detail the anxiety and confusion that were generated by this result; nor can the numerous and peculiar ways by which attempts were made to resolve these difficulties be noted here, or even merely mentioned. Many of the questions and experiments discussed at that time are so closely connected with conceptions now almost forgotten that we can scarcely

we can even understand them. Thus, for example, much astonishment was caused by the fact that some spectral lines give light of an extraordinarily high degree of monochromaticity, for which, even with a path difference of the order of one meter, a noticeable interference is obtained; with this were connected speculations concerning the “length” of the light quantum. Others thought that with the aid of a Nicol one could “divide in half” a light quantum, and one author made a canal ray pass by a narrow slit, while he thought that not the whole light quantum emitted by a rapidly moving atom would have time to pass through the slit, and sought fractional parts of the quantum with a correspondingly altered frequency.

The content of the experiments available by that time may be characterized approximately as follows: on the one hand, in numerous effects the corpuscular nature of light manifests itself, while on the other—in all classical interference experiments—the wave theory makes it possible to make correct predictions. The attempts then being made to understand this enigmatic dual nature of radiation amounted to attempts to construct a model of light operating within the framework of classical conceptions of time and space and, if possible, with classical causality—for example, a model of energy quanta describing definite trajectories in the sense of classical kinematics and connected with some interference field. Even Einstein tried for some time to construct such a model by modifying Maxwell’s wave equations of light. However, because of the complete opposition between the classical conceptions of waves, on the one hand, and corpuscles, on the other, all attempts of this kind were bound in advance to seem even more hopeless than, for example, the earlier attempts to construct a mechanical model of the ether. And indeed, whereas the mechanical theories of the ether had at least partial successes, these attempts led to no results whatever.

Nevertheless, the question remained serious as to whether cases, unnoticed up to then, might not be found in which the clas-

classical wave theory and, from the standpoint of interference phenomena, would lead to false results. Here, above all, three special formulations of this question deserve attention. First, it might have seemed doubtful whether the usual laws of interference remain valid also at extremely small radiation densities, with individual light quanta manifestly far apart from one another in space. It was natural to think that perhaps interference arises only as the result of the interaction of a large number of quanta.¹

Experiment, however, has provided no grounds here for supposing a dependence of interference on the absolute value of the brightness; on the contrary, on this question it confirms the invariable correctness of the classical wave theory.

Secondly, the “one-sidedness” of the emission of light by an atom, manifested in the recoil impulse \(\frac{h\nu}{c}\), and the directedness of the emission of a light quantum in each elementary act make it natural to suppose that—unlike the consequences of classical ideas about a spherical wave—there can be no coherence between radiation coming from one and the same light source in different directions. But here, too, experiment (Zeleny, Schrödinger, Gerlach–Lande) has shown that the classical wave theory is right.

Thirdly, it might have seemed possible that in experiments with rapidly moving light sources or interference apparatuses the classical theory would fail. Indeed, we possess only a very small number of experiments of this kind, and the theory of light quanta, associated with classical corpuscular notions, gave grounds to expect the failure of wave theory precisely under such circumstances. Einstein for a long time thought that, in such arrangements, one might expect the discovery of new effects, and he made an at—

¹ From the modern point of view one would have to say: in cases where the light quanta are manifestly far apart from one another in space, if in such cases they have any definite position at all.

detailed proposal for setting up experiments of this kind. However, carried out by Rupp, these experiments again confirmed only the predictions of the classical wave theory. Meanwhile, Einstein, independently of this, came to the conviction that in this case as well no departures from the classical theory were to be expected. Indeed, classical optics is so strictly closed in itself as a system of principles and propositions that one cannot assume the presence in this theory of special irregularities in some particular case without thereby casting doubt on its most general and unshakable principles. Therefore, as a result, there had to become established a unanimous conviction that, with respect to interference experiments, the classical wave theory is impeccable and exhaustive.

6. The Bohr–Kramers–Slater Theory

The historian of quantum theory would give only a very imperfect picture of the work of Niels Bohr if he confined himself to a consecutive enumeration of all his individual discoveries and results. What we owe to Bohr consists not only in particulars, but above all in the general spirit of his work; we owe to him that mental attitude which alone makes it possible to see individual problems in the proper light. In quantum theory, moreover, the rule was confirmed which says that a problem proves to be, to a large extent, solved as soon as its correct formulation is given. Before attempting to clarify here the significance of the work of Bohr, Kramers, and Slater in the development of the theory of radiation, we must devote a few words to the general course of Bohr’s ideas.

The essential difference between Bohr’s conceptions and the then-current views of the other most eminent theorists is characterized by one expression of Sommerfeld’s, who called the correspondence principle a “magic wand,” with the aid of which extraordinarily numerous and important results can be obtained. In this definition there is clearly manifested the un-

THE LIGHT-QUANTUM HYPOTHESIS

which was underestimated, and there is no doubt that at that time the majority of theorists felt that the correspondence principle, despite all its fruitfulness, ought to be regarded as a purely phenomenological, heuristic principle, making it possible to know only, in a certain sense of the word, the external connection of things, and not the true roots of the facts predicted with its aid.

Although individual acts of emission occur discontinuously, by jumps, nevertheless the statistical mean values of the intensity of radiation can be calculated with the aid of classical concepts—for example, the concept of the emission of a continuous spherical wave. Naturally, the majority of theorists took up the question of why this should be so: what remarkable and mysterious mechanism determines these peculiar relations? However, such a formulation of the question was unfruitful. And it was precisely Bohr who not only established the peculiar interrelation between quantum and classical laws, formulated by the correspondence principle, but—what is no less important—expressed himself quite clearly in favor of the view that this interrelation must be regarded not as something in need of explanation, but as a primary given fact.

From this standpoint there also developed the work of Bohr, Kramers, and Slater. Without in any way attempting to shield it from deserved reproaches, one may say that its main feature consists in the fact that it does not resolve the difficulties of the theory of radiation, but denies them. For the purposes of research, the elimination of apparent problems is just as important and often requires the same expenditure of intellectual energy as the solution of real problems. The major achievement, one that has retained its significance, attained by Bohr, Kramers, and Slater consisted in establishing that the problem of a joint explanation of light quanta and interference phenomena is to a significant degree an apparent problem, which finds its resolution in a simple, unprejudiced interpretation of the facts. Characteristic of this standpoint is the first sentence of the paper by Bohr–Kramers–Slater: “In attempts at a theor—

theoretically to interpret the interaction of radiation and matter we encounter two different points of view, which seem to contradict each other.

Let us consider some classical interference experiment, in which light emitted by a monochromatic source, through an arrangement of mirrors, prisms, etc., falls on some receiving screen. The methods of classical optics enable us to predict a definite distribution of the intensity of light on this screen. If we now imagine that the source emits only one single quantum of light, then from the experiment (the validity of the laws of interference for small radiation densities), without resorting to too bold an extrapolation, we can conclude only one thing: there is a certain probability that this quantum will fall on a definite small area of the screen, and this probability is precisely determined by the illumination of the corresponding place, calculated according to the classical theory. What is new, as established by Bohr, Kramers, and Slater, consists simply in the fact that one can confine oneself to the establishment of this fact. If experiments are performed with an individual light quantum, using macroscopic optical instruments, then these experiments cannot give an answer to any other question except the question of the probability of perceiving this quantum, under the given conditions, at one place or another. Consequently—and this is precisely the decisive turn—nothing more can be demanded of the theory. One can be satisfied with this state of affairs. At the same time all hopeless questions, such as the question of the “trajectory” of a light quantum in a wave field, etc., fall away of themselves. There are no grounds for ascribing to light quanta trajectories in the spirit of classical kinematics.

This formulation, corresponding to modern points of view, of course coincides only in part with the formulation of Bohr, Kramers, and Slater. These authors, as is known, were compelled to make an attempt to dispute the strict applicability of the law of conservation of energy to radiation; the classically calculated wave field was supposed to

according to their point of view, give not the probability of finding, at a given place, a definite individual light quantum, as in our thought experiment, but the probability that at the given place on the screen a light quantum will suddenly appear, without the energy exactly corresponding to it having first been transferred to the ether. As is known, this hypothesis was experimentally tested by Bothe and Geiger, and also by Compton and Simon, and the experiments revealed, on the contrary, exact observance of the law of conservation of energy.

The assumption of only statistical conservation of energy was needed precisely because it had not proved possible to remove all the difficulties of the theory of light by denying them. Besides the apparent difficulties created by an incorrect formulation of the question, there were also real ones. For the case of a separate light quantum, with which we experiment only by means of macroscopic apparatus, we were able above to show, in the simplest form, how exact physical laws can be represented. But considerably more complex questions arise when we deal with interference effects in the reactions of radiation of individual atoms, whose reaction changes as a result of their own quantum jumps with recoil \(\frac{h\nu}{c}\). Bohr, Kramers, and Slater attempted to solve these questions too, but by means that were too primitive. A very instructive exposition and critique of this theory is given in Pauli’s article on the quantum theory in the Handbuch der Physik (Vol. XXIII); there, in particular, its relation to the problem of the natural width of spectral lines is also treated in detail. Here we shall note only that in this work Bohr for the first time recognized Einstein’s probability laws for the emission and absorption of radiation, as well as his recoil impulses in emission.

The assumption of the statistical character of the law of conservation of energy forced one to treat somewhat less seriously Einstein’s conclusions concerning fluctua-

radiation.¹ Indeed, these considerations of Einstein’s are not mentioned by Bohr, Kramers, and Slater even in a single word.

Let us now return again to our reasoning concerning the classical interference experiments, in order to clarify with all possible distinctness, by means of this simplest example, the point of view of the modern theory. We may set up two entirely different series of experiments: first, experiments with a large “classical” radiation density \(\left(\rho_\nu \gg \frac{8\pi h}{c^3}\nu^3 \text{ or } n_\nu \gg 1\right)\) and, secondly, experiments with a single separate light quantum. The theory of both series of experiments is based on one and the same mathematical basis (the wave equation), but its content in the two cases is entirely different. In experiments with a single separate light quantum, the wave intensity calculated classically represents the probability that the quantum will fall at a certain point. If such experiments are performed in series, or if the source is made to emit quanta in large numbers, but at sufficiently large intervals of time,² then their results can be predicted statistically from the wave theory. If now we raise the question of the fluctuations which appear in the results of a whole series of such experiments, then the answer will be given by the rules of probability, as in a game of dice;³ with classical calcul—

¹ It has been proposed, for example, instead of fluctuations in the “virtual” radiation field, which according to the theory of Bohr, Kramers, and Slater are in principle unobservable, to consider only fluctuations in the absorption and emission of material atoms. This idea, which later was developed with particular insistence by Smekal, apparently had already been intimated by the three authors mentioned. It must be noted, however, that this proposal can appear satisfactory only so long as the statistical conservation of energy is accepted. The same applies also to the recently proposed derivation by Bothe of the fluctuation formulae.

² Namely, such that the relation

\[ \rho_\nu \ll \frac{8\pi h}{c^3}\nu^3 \text{ or } n_\nu \ll 1 \]

still remains valid.

³ More precisely, as in a game of dice with loaded dice, in which the different results have different probabilities.

THE LIGHT-QUANTUM HYPOTHESIS

secret fluctuations of radiation this, of course, has nothing in common. But if instead we experiment with large radiation densities (\(n_\nu \gg 1\)), then we must relate the physical concepts in the usual way to the mathematical wave construction; the computed intensity of wave-like radiation fully corresponds to the intensity of the actual physical radiation—both with respect to the time-average value and with respect to all fluctuation effects.

Thus, for each of these, fundamentally different, cases we have a complete and satisfactory theory: the theory for cases of large radiation densities is the “classical” theory and, as such, it possesses only approximate applicability;¹ the theory for individual light quanta is a part of the general quantum theory, and it is exact. But with the means described above we cannot obtain a complete theory for the intermediate region, where neither of the inequalities \(n_\nu \gg 1\) and \(n_\nu \ll 1\) holds. True, we know enough to understand, in this case as well, all the usual interference experiments, for all the relative brightnesses observed on the screen are then completely independent of the absolute intensity of the light, i.e. in this case too they are the same as for \(n_\nu \gg 1\) and \(n_\nu \ll 1\). But we do not have a complete theory if, besides the time-average values of the intensity, we also wish correctly to calculate the fluctuations (without circumventing these difficulties by thermodynamic means). This could not be done so long as, for constructing a model of light radiation, we possessed only classical kinematics.

Alongside this problem there remained awaiting solution a group of problems touched upon above, connected with the interference effects of the radiation reactions of individual atoms; to these questions the theory of Bohr, Kramers, and Slater gave no satisfactory answer. In the already mentioned article Pauli noted that the in-

¹ More precisely: it would be valid only at infinitely large radiation density.

Advances in the Physical Sciences. Vol. X, issue 1
4

a necessary basis for approaching these questions is the development of the new quantum concept of the field. In the last section of this article I shall try to explain how and in what sense the newly formed concepts of the new quantum mechanics partly solve this problem.

III. MATTER WAVES

7. Bose–Einstein Statistics

We have seen that if one trusts Einstein’s arguments, then light, despite its wave character, which manifests itself in interference phenomena, on the other hand nevertheless has a corpuscular nature. At the present time it seems already almost entirely natural to suppose that in an atomic beam, whose purely corpuscular nature appeared indubitable, interference properties may also be manifested, so as thereby to make the analogy between light rays and material rays as complete as possible. But in order to venture to draw this conclusion, so much courage was required, and such deep faith in the correctness of the course of the ideas of the theory of light quanta, that apparently only two physicists—Einstein himself and L. de Broglie—dared to do so. Einstein had long been nurturing this idea, without publishing it, since he had obtained no results that would have made it possible to formulate it quantitatively. An approach to this opened up for him only after, in 1924, Bose proposed his new statistics of light quanta.

It has already been mentioned above that the statistics of corpuscular light quanta, carried out by classical methods, gives Wien’s law instead of Planck’s law. In this statistical derivation one proceeds as follows. The “phase space” of a light quantum1 is divided into unequal

large cells and imagine that the light quanta are successively and quite randomly scattered among these cells. The various equally probable microscopic distributions that arise in this way can evidently be described by specifying those individual light quanta which fall into one or another separate cell. The normal distribution, the most probable one for a given total energy, then turns out to be the one corresponding to Wien’s law. Bose observed that, instead, one can obtain Planck’s law if, first, one takes the volume of a cell to be constantly equal to $h^3$ (as had already been done earlier by other authors), and then introduces into the above line of reasoning the following change, which at first seems quite arbitrary and meaningless: in describing the equally probable cases it is not necessary to specify which light quanta fall into a given cell, but only how many of them are there.¹

Although at first the meaning of this condition may have seemed exceedingly obscure, nevertheless with its help it proved possible to develop the statistics of radiation, relying exclusively on corpuscular concepts, something that in classical statistics, without Bose’s changes, had been possible only for the Wien region. Einstein therefore immediately took up Bose’s statistics and expressed his conviction of the deep kinship of material rays with light by creating a theory of a monatomic ideal gas analogous to Bose statistics. In this case one indeed obtains a theory that coincides with the classical one within the limits in which experiment requires this: appreciable deviations from the classical gas laws can be expected only at very low

¹ Let us explain this with an example. Suppose that there are only two cells and two quanta. From the classical point of view there exist four equally probable cases: both quanta in the first cell; both quanta in the second; the first quantum in the first cell, the second in the second; the first quantum in the second, the second in the first cell. According to Bose, only three equally probable cases are to be counted: both quanta in the first cell, both in the second, one in each.

temperatures,^1 where, in one way or another, gases no longer behave as ideal gases. In exactly the same way one also obtains the correct value for the chemical constant of monatomic gases (and, with an appropriate construction of the theory, also for polyatomic gases). Connected with this, further, is the circumstance (Langevin, Jordan) that Einstein’s theory makes it possible to establish what quantity of matter in general can exist stationarily at a given temperature, if, in the spirit of Eddington’s hypothesis, one admits the possibility of the transformation of matter into radiation and poses the question of the thermodynamic equilibrium between radiant and material energy.^2 In doing so one obtains a formula which had already been derived earlier by another route by Stern. One may also study, by Einstein’s method, instead of individual gases, the equilibrium of systems of several gases, between which arbitrary chemical and similar transformations take place (Jordan); in this way one directly obtains all the known laws concerning chemical, ionization equilibrium, and the like, which can be derived with the aid of Nernst’s heat theorem and the known chemical constants of gases.^3

^1 Planck’s law differs from Wien’s law in that it contains the factor \(\dfrac{1}{e^{h\nu/kT}-1}\) instead of the factor \(e^{-h\nu/kT}\); correspondingly, the velocity distribution in an Einstein gas differs from the Maxwellian distribution by the factor \(\dfrac{1}{e^{a(T)+mv^{2}/2kT}-1}\) instead of \(e^{-mv^{2}/kT}\), where \(a(T)\) is a certain definite function of the temperature.

^2 In doing this it is necessary to seek the most probable distribution not for a given total energy and a given number of particles, but only for a given total energy. In complete analogy with a gas consisting of light quanta, instead of \(\dfrac{1}{e^{a(T)+mv^{2}/2kT}-1}\) we obtain \(\dfrac{1}{e^{E/kT}-1}\), where \(E\) is the total energy of the particles being sought, including their energy in the state of rest, \(mc^{2}\).

^3 Of course always with those modifications which correspond to the degeneracy of gases.

The circumstance that the true significance of Einstein’s theory of gases consists in attributing to material rays certain properties of wave radiation becomes clearly noticeable when one studies the law of probability for collisions of the atoms of an Einstein gas and their interactions with light quanta (Jordan). Indeed, these elementary processes would violate the Einstein velocity distribution (and would gradually transform it into the Maxwellian one), if the law of mass action still remained valid for them. Instead of a simple proportionality between the probability of a reaction and the concentration of the reacting material particles, it is necessary, it would seem, to introduce a modified law of probability; and indeed, we obtain a statistical equilibrium corresponding to the Einstein velocity distribution if, for these reaction frequencies, we adopt a complete analogy between material particles and light quanta.

Einstein further discovered the wave character acquired by material rays in his theory of gases by studying density fluctuations. However, before touching upon this point, we must turn to de Broglie’s discovery.

8. De Broglie Waves

L. de Broglie also addressed the idea of the hypothesis of light quanta, putting forward the proposition that moving material particles too have a wave character. As is known, to him belongs the merit of establishing a quantitative relation between the mass and velocity of particles, on the one hand, and the frequency and phase velocity of waves, on the other (1922). His relations read:

\[ E=h\nu;\quad \Gamma=\frac{h\nu}{v} \tag{13} \]

Here \(E,\Gamma\) are the energy and momentum of the particle; \(\nu, v\) are the frequency and phase velocity of the corresponding plane wave, propagating in the same direction as the particle. In the case of light \(v=c\). The most essential thing in de Broglie’s views,

by which he justified equality (13), was the following: these equalities must be invariant in the sense of the special theory of relativity, i.e., they must not depend on the observer’s coordinate system. With respect to another, moving coordinate system, the particle has other values of energy and momentum \(E', \Gamma'\), and the wave—other \(\nu'\) and \(v'\), and between these four quantities the relations (13) must again be established automatically. That such invariance indeed holds for the special case of light quanta was first noted, several years before this, by Bothe.^1

But the general de Broglie relations (13) do indeed correspond to the principle of relativity.^2

As is known, the interference effects corresponding to these waves have recently been the subject of a large number of experiments. For free electrons, which, by virtue of the small value of their rest mass, are associated with the comparatively long-wavelength de Broglie radiation, the corresponding diffraction phenomena could be detected very clearly and convincingly.^3

A theoretical justification of de Broglie’s theory can also be obtained from the arguments of Duane (W. Duane, 1923), which are very instructive in themselves. In the form in which they were presented by A. Compton, they reduce to the following.

^1 This assertion was also implicit in one paper by Schrödinger, concerning the relativistic invariance of Bohr’s frequency condition under the assumption of a recoil of \(\dfrac{h\nu}{c}\).

^2 If, for example, \(\cos 2\pi (k_1x + k_2y + k_3z - k_4ct)\) is an invariant wave amplitude, while \(x, y, z, ict\) form a four-dimensional vector, then \(k_1, k_2, k_3, k_4\) also form a four-dimensional vector; since, moreover, \(\Gamma_x, \Gamma_y, \Gamma_z, \dfrac{iE}{c}\) is a four-dimensional vector (\(\Gamma_x\), etc.—components of the momentum), the equalities \(\Gamma_x = hk_1\), \(\Gamma_y = hk_2\), \(\Gamma_z = hk_3\), \(E = hk_4c\), identical with the equalities (13), remain invariant.

^3 See the articles by P. Tartakovsky (Uspekhi fizich. nauk 8, 338, 1928), C. Davisson (ibid. 8, 483, 1928), G. P. Thomson (8, 570, 1928), V. L. Granovsky (9, 508, 1929).

Ed.

If an infinitely thin and perfectly regular diffraction grating moves in its own plane perpendicular to the lines of the grating, then obviously we have, in a certain sense, a periodic motion, which can be quantized according to the rules known from atomic theory. If we now direct a corpuscular light quantum onto the grating, then a quantum jump of the grating will result. In this process, however, the quantized grating can emit only certain discrete changes of momentum, and correspondingly, by the theorem of conservation of momentum, there exists only a series of discrete different directions in which the light quantum can be reflected. Further, it can be shown that from the assumption of a momentum \(\frac{h\nu}{c}\) for the light quantum there follow exactly the same laws of reflection as from the classical wave theory. From these considerations one sees how the gulf that yawned in the world of classical ideas between two such different things as corpuscles and waves gradually begins to be filled, if only the characteristic concepts of quantum theory are introduced. Ehrenfest and Epstein were able further to show what an extraordinarily great productivity and wide field of application Duane’s point of view possesses.

These views can also be applied to establishing a connection between de Broglie waves and material particles. Indeed, if one takes seriously in general the basic idea of Duane’s reasoning, then also in the reflection of material particles, instead of light quanta, the grating can experience only those changes of momentum which are permitted by quantum theory. In this connection it turns out (Jordan) that a material particle, too, behaves with respect to reflection exactly like a classical wave, and moreover like a wave of precisely such a wavelength as is required according to de Broglie.

However, if one has once resolved, at least for one special case, to connect energy with frequency by the equality \(E=h\nu\), then there is no longer any possibility of evading the conclusion that this relation must have a uni-

universal character. It is precisely the assumption of a universal connection between corpuscles and waves in the form of de Broglie—and only this assumption—that makes it possible to bring into agreement the conclusions that are obtained, on the one hand, from the law of conservation of momentum if one adheres to corpuscular conceptions, and, on the other, from the conceptions of interfering waves.

As an example, let us note here the ordinary law of refraction. Suppose that in a two-dimensional plane the \(x\)-axis is the boundary of two homogeneous media with different refractive indices \(n_1\) and \(n_2\). Then in the two media the wavelengths \(\lambda_1\) and \(\lambda_2\) and the phase velocities \(v_1\) and \(v_2\) of a light wave of frequency \(\nu\) (which remains constant in both media) are equal to

\[ \lambda_1=\frac{c}{n_1\nu},\qquad \lambda_2=\frac{c}{n_2\nu},\qquad v_1=\frac{c}{n_1},\qquad v_2=\frac{c}{n_2} \tag{14} \]

If now the corresponding light quantum \(h\nu\), flying through both media, has momentum components respectively \(p_{1x}, p_{1y}\) and \(p_{2x}, p_{2y}\), then (according to de Broglie):

\[ \sqrt{p_{1x}^{2}+p_{1y}^{2}}=\frac{h\nu}{v_1},\qquad \sqrt{p_{2x}^{2}+p_{2y}^{2}}=\frac{h\nu}{v_2}. \tag{15} \]

and, moreover, in passing through the boundary surface the tangential component remains unchanged: \(p_{1x}=p_{2x}\). Hence it follows that the proportion must hold

\[ \frac{p_{1x}}{\sqrt{p_{1x}^{2}+p_{1y}^{2}}}:\frac{p_{2x}}{\sqrt{p_{2x}^{2}+p_{2y}^{2}}}=v_1:v_2=n_2:n_1 \tag{16} \]

This is precisely Snellius’ law of refraction. In the example of this, long-known derivation,\(^1\) it is already clear how one may cast into corpuscular form explanations that are usually associated with Huygens’ principle. If, however, these two different derivations of the law of refraction are compared more exactly, it is easy to see that there is nothing strange in the coincidence of the results to which they lead, for, despite the great difference in the intuitive basis of both

\(^1\) Unfortunately, I cannot indicate by whom these considerations, known even before de Broglie, were first adduced.

of reasoning, in essence in both cases one has to carry out one and the same geometrical construction.

Many other examples can be indicated in which the same relations occur. The most remarkable of them is probably the Compton effect, for which Schrödinger explained in detail how the changes of frequency can be understood without explicitly introducing corpuscular concepts, while allowing for the possibility of an interference interaction between light and de Broglie waves. In general one may say that there exists a completely general and unified analogy between the simple rules of addition of momentum (conservation laws) and the elementary laws of combinations of frequencies and wavelengths related to interference. The premise of this analogy, however, is the assumption of the universality of the de Broglie relation after it has been accepted for the special case of light quanta.

De Broglie’s work, besides the quantitative formulation of the connection between waves and corpuscles, contains yet another proposition which, it is true, had long before been expressed by Hamilton, but which now acquires great importance for quantum mechanics. Let us consider the propagation of light waves whose wavelength is very small in comparison with the obstacle standing in their path; under these conditions, quite independently of the wave structure of light, we may use geometrical optics instead of wave optics. But the mathematical method of higher geometrical optics formally coincides completely with the so-called Hamilton–Jacobi theory in the mechanics of a system of material points. Historically the latter theory too was developed by Hamilton on the basis of this analogy, which, however, for a long time remained almost forgotten and was rediscovered independently by de Broglie.

It was natural to connect these facts with the problem of matter waves, and de Broglie, and especially Schrödinger, posed the question as follows: if one adopts the standpoint of a radical wave theory of matter, then one can nevertheless conclude directly that the “trajectories” of mater—

real particles, within well-known limits, can be calculated by the methods of classical mechanics, without any connection with their wave nature. Here we have in fact simply a “geometrical” optics of de Broglie waves; “trajectories” are “rays,” if one speaks optically. This, however, is another step along the path toward the well-known proposition that the exact quantum mechanics of the atom is related to classical mechanics in precisely the same way as wave optics is to geometrical optics. This point of view, which led Schrödinger to his important discovery, lends visual clarity to a vast domain of quantum-mechanical laws. We shall dwell on this in greater detail shortly, but already here, on the basis of what has been said earlier, we shall try to clarify within what limits this proposition is valid. It is known in advance that it can in no way contain the solution of the proper quantum problem, for the whole hypothesis of light quanta was developed precisely because classical wave mechanics proved insufficient for the description of light radiation. If we now see that in material rays, as in light, interference phenomena are observed, then we cannot doubt that classical wave theory is essentially just as insufficient for the description of material radiation as it is for the description of light.

9. De Broglie Waves and Einstein’s Gas Theory

Einstein showed that the wave relations of de Broglie give precisely an exact formulation of the wave character which his gas theory implicitly ascribed to material particles. This became apparent in the study of fluctuations of the density of the Einstein gas, which we discussed above. Taking the change, following from Einstein’s theory, of the Maxwellian velocity distribution, one can, for example, again derive thermodynamically the expression for the square of the mean fluctuation of the amount of gas in some part of a large

THE HYPOTHESIS OF LIGHT QUANTA

of the volume of the gas and obtain a formula entirely analogous to formula (5) for radiation in an empty space. The first term, which alone has significance for a non-degenerate gas, is the same as the entire square of the fluctuation for a classical ideal gas; the second term, which acquires decisive importance at very large “densities” of material radiation, is completely identical with the full square of the fluctuation for the empty space in which there is a classical field of oscillations with frequencies and wavelengths corresponding to de Broglie’s formulas.

Thus, if the Einstein theory of the gas is indeed correct (later we shall still discuss within what limits the present level of our knowledge permits us to accept it), then for a material gas we have exactly the same situation as for radiation in an empty space: neither the classical corpuscular conceptions nor the classical wave conceptions give a correct picture of them.

As was shown independently by Schrödinger and Born–Heisenberg–Jordan, Einstein’s theory of the gas can be presented in a form analogous to the Debye presentation of the theory of radiation in an empty space;¹ at the same time Boze–Einstein’s remarkable statistical hypothesis is also clarified. Indeed, first of all, both for radiation in an empty space and for Einstein’s gas, one can show on the basis of equalities (13) that the number of “cells” of phase space belonging to the frequency interval \(\nu, \nu+\Delta\nu\) is equal to the number of natural oscillations belonging to this frequency interval.

Thus one can establish, in a certain sense, the translation of corpuscular statistics into the statistics of quantized natural oscillations by means of the following key:

\[ \begin{array}{rcl} \text{“Cell”} & \longrightarrow & \text{“natural oscillation”,}\\[0.8em] \left. \begin{array}{c} \text{Number of particles in a cell} \end{array} \right\} & \longrightarrow & \left\{ \begin{array}{l} \text{Quantum number of quan-}\\ \text{tized natural}\\ \text{oscillations} \end{array} \right. \end{array} \]

If we now consider a quantized wave field, then it is quite natural to characterize each equally probable state by specifying the quantum numbers of all the individual proper oscillations. But this is precisely the Bose–Einstein proposition, expressed in the translated terminology.

Of course, here one must clearly understand that, for the time being, this establishes only an entirely formal dependence between Debye’s representations (more precisely, Debye’s representations generalized to de Broglie waves), on the one hand, and Bose–Einstein’s corpuscular theory, on the other.

The mutual connection of these two points of view, manifested in the possibility of translating one theory into the other by simply changing a few words, cannot, however, conceal the profound differences in the model representations that underlie them.

Only the application of the concepts of quantum mechanics makes it possible to uncover, in these formal analogies, an internal equivalence and to recognize in these images equivalent aspects of one and the same integral thing.

IV. THE INFLUENCE OF THE LIGHT-QUANTUM HYPOTHESIS ON THE DEVELOPMENT OF QUANTUM MECHANICS

10. Schrödinger Waves

The exceptional importance acquired by de Broglie’s ideas, owing to the fact that they served as the starting point for Schrödinger’s discovery, is at present so well known that in what follows we shall bring to the fore the factual connection of Schrödinger’s theory with the hypothesis of light quanta, touching upon the historical interrelations only insofar as this is necessary for understanding.

Let us first return to our previous arguments concerning the quantum dynamics of an individual light quantum. We saw that a purely classical wave

THE HYPOTHESIS OF LIGHT QUANTA

theory provided the formal mathematical apparatus for an exact treatment of this problem; only the interpretation of the formulas was in this case entirely different from that in the ordinary theory of light, which gave the first impulse to the development of mathematical wave theory.¹

Einstein’s axiom concerning the analogy between light and material radiation must therefore lead to the following supposition: for the quantum mechanics of a single isolated material point one can obtain an exact formulation by introducing a suitable wave field functioning according to the classical laws. De Broglie showed how these waves should be chosen in the case of a particle moving without external forces. Schrödinger set himself the more difficult task—to carry out, for a material point located in a force field, the exact construction of the waves determining its behavior. The analogy, indicated by Hamilton and de Broglie, between the classical mechanics of a system and geometrical optics gave Schrödinger the starting points for mastering this problem: the wave theory had to be formulated precisely so that, in the limiting case of geometrical optics, the classical trajectories of the particles would be obtained.

Such were (in broad outline) the foundations from which Schrödinger proceeded, and, as is known, he did indeed succeed in finding the solution of this important problem. His result reduces to the following: let \(U(x,y,z)\) be the potential energy, and \(m\) the mass of the particle; then the corresponding—

¹ Even in classical physics there are examples of the fact that one and the same formal mathematical apparatus can be applied in theories that are essentially quite different. Thus, for example, the equation

\[ \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}=0 \]

occurs both in the theory of the potential and in the theory of heat conduction, and in both cases the same mathematical consequences are extracted from it. However, the physical interpretation of the mathematical theory is in the two cases entirely different.

where the field of vibrations has a (complex) amplitude \(\psi\), which satisfies the equation:¹

\[ -\frac{h^2}{4\pi^2}\cdot\frac{1}{2m}\Delta\psi+U(x,y,z)\cdot\psi+\frac{h}{2\pi i}\frac{\partial\psi}{\partial t}=0 \tag{17} \]

The most general solution of equation (17) can be represented as a sum of particular solutions, each of which has the form

\[ \varphi(x,y,z,t)=\varphi(x,y,z)\cdot e^{-2\pi i\nu t} \tag{18} \]

i.e. in time it oscillates purely harmonically (in the complex sense). For the spatial factor of the amplitude we obtain the equation

\[ -\frac{h^2}{4\pi^2}\cdot\frac{1}{2m}\Delta\psi+U\varphi=W\varphi;\qquad W=h\nu \tag{19} \]

Those (always real) values \(W=h\nu\) for which there exists a solution \(\psi\) of equation (19) corresponding to the nature of the problem (i.e. having no special infinite values and satisfying the known “boundary conditions”) are called, in mathematics, the characteristic numbers of equation (19), and the corresponding solutions \(\psi\) are called “fundamental functions.”

Just as for de Broglie waves, these characteristic numbers, i.e. the Schrödinger wave frequencies multiplied by \(h\), have a simple and important physical meaning: they are the energies of the various “stationary states” which, according to quantum theory, may be assumed by a material point moving in a force field.

These propositions, as is well known, have been confirmed in the best possible way by experiment (within the limits in which this could be expected). Obviously, they represent nothing other than the legitimate development of the ideas implicitly contained in the hypothesis of light quanta. In retrospective con-

¹ By \(\Delta\) is denoted the operator \(\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}\).

THE LIGHT-QUANTUM HYPOTHESIS

one can discern, in the entire development of these ideas, from the first proposition of Einstein’s light-quantum hypothesis to Schrödinger’s wave mechanics, a complete logical necessity.

With the same necessity there also follows the physical interpretation of the functions \(\psi\) or \(\varphi\). Equation (17), for an individual moving material point, has, as was said, the same significance as Maxwell’s equations for an individual light quantum. From Maxwell’s equations we could find the probability that an individual light quantum with which we are experimenting will be perceived at a definite place; correspondingly, from the function \(\psi\) we can obtain the probability that the material point with which we are carrying out experiments will be found at a definite place. (This probability is proportional to the square of the absolute value \(|\psi|^2\) of the function \(\psi\) at the given place). This significance of the function \(\psi\), which follows with complete necessity from the analogy with the dynamics of light quanta, was especially clarified by Born. Closely connected with this is the interpretation, first put forward by Pauli, of the separate fundamental functions \(\varphi\). If we know that an atom is precisely in the \(n\)-th quantum state,—the supposition of a definite energy of the atom corresponds to its representation by means of a purely harmonic wave, i.e. a separate proper oscillation,—then \(|\varphi_n|^2\,dx\,dy\,dz\) is the probability that in the corresponding experiment the moving particle will be found in the volume element \(dx\,dy\,dz\). (Here \(\varphi_n\) is the fundamental function corresponding to the \(n\)-th state). What sort of experiments are meant here we shall still have to discuss below.

It has already been noted above that the possibility of an exact derivation of the known quantum-mechanical properties of material particles from the representation in terms of classical waves can in no way be regarded as proof that, with the aid of wave representations, one can return to classical concepts. The analysis carried out above of the basic ideas of Schrödinger’s theory should only reveal—

to explain that Schrödinger’s discovery in reality means nothing other than the establishment of a profound analogy between light quanta and material particles, and that therefore the applicability of classical waves to the description of matter has been proved only within those limits in which it applies to light. But we know—if it is necessary to say this once more—that with respect to light the applicability of the classical wave theory is, generally speaking, limited (indeed, in essence, limited within very narrow bounds). Therefore, if we wished to suppose that waves of matter function in a purely classical way, this would mean that from now on we resolve categorically to deny the analogy between light and matter.

However, there is hardly any possibility of overcoming in this way the difficulties that confront us. If we recall the analogy between classical mechanics and geometrical optics—mechanical trajectories coincide mathematically exactly with the rays of a wave field—then we must not forget also that, despite this coincidence of mathematical laws, in essence the two theories are entirely different. (We have already encountered relations of this kind in other examples as well.) In the mechanics of a material point there exists not only the trajectory, but also the material point itself, and there exists the possibility of detecting this point on its trajectory. There is no such analogy to this in geometrical optics. Only with the aid of special additional assumptions could one try to construct a purely classical theory of matter adjoining Schrödinger’s wave mechanics, but such an attempt would have no more chance of success than the numerous futile attempts at an analogous formulation of the theory of light quanta. Therefore, when in the case of waves of matter one considers a statistical relation between Schrödinger waves and the position of a particle in space as a law that is not explained but simply accepted, this is again an expression of confidence in Einstein’s position on the analogy between light and matter.

11. Wave Mechanics and Matrix Theory

The physical quantities obtained directly from Schrödinger’s theory—the values of the energy \(W\) and the probabilities of being at a given place \(|\psi|^2\)—are only part of what we want to know. If we first consider the example of the atom, then what interests us above all are the probabilities of various transition processes associated with radiation.

These questions can be solved in direct connection with Schrödinger’s propositions, while applying at the same time, in the proper way, the characteristic concepts of quantum mechanics: discontinuous transitions, etc. (Klein, Gordon); in this way one obtains a derivation of the law of emission and absorption given by Bohr and Einstein (as well as of the dispersion formulas and the formulas for the intensity of Compton scattering, which are obtained from the correspondence principle).1

In fact, however, the answer to these questions was given not by a further development of Schrödinger’s concepts, but by the simultaneous development of another independent theory. Later it proved possible, quite simply, to connect wave mechanics with this theory, so that the two theories turned out to be merely different expositions of the same facts. All this is now so well known that here we may confine ourselves to only a few brief remarks on the matter.

A classical periodic motion may, according to Fourier, be represented as the result of the superposition of a series of purely harmonic motions. The electromagnetic radiation of a point charge moving according to the classical laws makes it possible, from the intensities of the individual harmonic components of the radiation, to judge the amplitudes

of the separate harmonic components of the motion of a material point. Further, according to the classical theory, on the basis of dispersion experiments we could determine the phase relations between the various harmonic components of the motion of a point charge. In this way we would be able to trace exactly the course of the motion of a particle from observations of its radiation. The situation is quite the same for the motion of an electron in a real atom. Here, in just the same way, the intensities of emission or absorption of individual atomic frequencies correspond to the harmonic components of the “motion” of the electron in the atom, and again conclusions can be drawn from dispersion experiments about the phase relations between these partial harmonic oscillations (whereas the emission intensities make it possible to judge only the absolute values of the harmonic components of the motion). Of course, it is not possible to combine these partial harmonic oscillations with the “trajectories” of electrons in the sense of classical kinematics; this cannot be done because, unlike the classical limiting case, they cannot be regarded as components of motion in one definite atomic state, but can be connected, by means of the frequency condition \(h\nu\), only with the different possibilities of transition in the atom. Despite this difference, as Bohr established in his correspondence principle, there nevertheless exists a formal similarity between the classical and quantum-mechanical Fourier components of motion, and after this similarity had been investigated and quantitatively formulated for various examples, Heisenberg and the authors associated with him finally succeeded in showing how this closeness can be used in order to obtain such a formulation of the quantum-mechanical laws of motion by which one can completely determine the energy states and the probabilities of transitions in radiation.

It turned out that, from the mathematical point of view, at the basis of the laws formulated in this way lies the problem of charac-

of characteristic numbers (Born–Heisenberg–Jordan), and it was then clarified that this problem is entirely equivalent to the same problem in Schrödinger’s wave mechanics (Schrödinger, Eckart, Pauli). We obtain the same energy values from both wave and matrix mechanics, and those quantities which in matrix mechanics determine transition probabilities can also be calculated from Schrödinger’s fundamental functions.

It is important, however, to make clear to oneself that although the “Fourier components” or “matrix elements” \(q(nm)\) in matrix mechanics were historically introduced in connection with transition probabilities, and although this connection makes them especially intuitive, nevertheless in essence these \(q(nm)\) do not at all have the direct and necessary significance of transition probabilities. On the contrary, in them one must first of all see purely kinematic quantities containing the necessary generalizations of the classical concepts of motion. Therefore they can also be defined in those cases where there can be no question of electromagnetic radiation or of anything of the kind.\(^1\)

Independently of the reaction of radiation, a physical interpretation of the quantities \(q(nm)\) may be given, for example, as follows. The time average of the quantity \(q\) with the matrix scheme \(\left(q^{(nm)}\right)\) for an atom in the \(n\)-th quantum state is the \(n\)-th diagonal element of the matrix \(q^{(nm)}\). If, from the matrix for \(q\), one forms the matrices for \(q^2, q^3, \ldots q^s\), then one can correspondingly determine the time averages of all these powers of \(q\). Conversely, by these time averages the whole matrix \(q\) is essentially determined. Here we find a new connection between Schrödinger’s theory and matrix theory, and again it turns out that both theories lead to concordant results. Indeed, if the atom is in the \(n\)-th state, then according to wave mechanics the probability that \(q\) has a value in the interval \(q'\), \(q' + dq'\), is equal to

\(^1\) We shall make use of this again later.

Pauli \(\int |\varphi_n(q')|^2 dq'\); whence for the mean value of \(q^s\) in the \(n\)-th state we obtain the expression\(^1\)

\[ \int_{-\infty}^{+\infty} q^s |\varphi_n(q')|^2 dq' \tag{20} \]

and the mathematical connection between wave and matrix mechanics is such that in fact this expression (20) is exactly equal to the \(n\)-th diagonal number of the matrix for \(q^s\).

12. Further development

The logical consistency and necessity in the evolution from the hypothesis of light quanta to wave mechanics and from the correspondence principle to matrix theory, and then the agreement of the results obtained by both routes, should in themselves have inspired great confidence in the new theory; still more effective was the fact that, as the applications of the theory developed, the latter received more and more confirmations. On the other hand, however, it could not fail to be seen that the system of quantum mechanics was still to a considerable extent imperfect and unfinished, both from the point of view of the totality of its concepts and from the mathematical side. Despite the broad range of concepts discussed in the preceding sections, they were still by no means sufficient for the interpretation of all the varied experiments that can be performed with atoms. In a concealed form, the formal apparatus of elementary wave or quantum mechanics does indeed already contain the answer to every possible question, but it was necessary precisely to develop what was present in concealed form, in order to answer all conceivable particular questions and to clarify the theory’s fundamental relation to such problems as causality. Approaches to this (apart from those indicated above) were obtained from investigations of the exchange of energy between atoms in resonance (Heisenberg; Jordan). The known completeness on the formal side was

\(^1\) For example, the mean value of \(q^2\) for the \(n\)-th state is equal to \(\sum q(nk)q(kn)\).

achieved by the statistical formulation of quantum mechanics in the transformation theory of Dirac–Jordan.^1

From the standpoint of intuitive physics and theoretical cognition, complete clarity was brought by the work of Heisenberg and then by Bohr’s report.^2

Without entering here into the details of this development of ideas, which would take us too far afield, we wish briefly to touch upon at least several points in which the old problematic of the theory of light quanta again emerges.

Here one must first of all mention the uncertainty rules investigated by Heisenberg and Bohr.

According to quantum mechanics, a corpuscular particle cannot simultaneously be assigned precisely determined momenta \(p_x, p_y, p_z\) and precisely determined coordinates of position \(q_x, q_y, q_z\). The unavoidable errors \(\Delta p_x,\ldots,\ \Delta q_x\ldots\) can never be made so small that the inequalities

\[ \Delta p_x \cdot \Delta q_x > h;\qquad \Delta p_y \cdot \Delta q_y > h;\qquad \Delta p_z \cdot \Delta q_z > h \tag{21} \]

are not satisfied.

This circumstance, which within the framework of transformation theory is closely connected with the so-called rules for the permutation of matrices for \(p_x q_x\ldots\) \(\left(p_x q_x - q_x p_x = \dfrac{h}{2\pi i}\ \text{etc.}\right)\), becomes directly evident in connection with de Broglie waves.

Indeed, if we imagine a freely moving particle, then the waves associated with it have the property that they must necessarily fill all space if their frequency and wavelength are strictly determined (i.e. if the wave is purely harmonic). If, however—

^1 This theory was supplemented in a most remarkable way by H. Weyl and Neumann, who, alongside “particular cases,” investigated the quantum-mechanical “mixture.” Weyl also attempted, in addition, to give a very interesting construction of the mathematical side of the theory; however, it is still difficult now to decide which of Weyl’s mathematical concepts may have a genuine physical meaning.

^2 A report read at the International Congress in Como in 1927 in honor of Volta’s jubilee. For a Russian translation see Uspekhi fizicheskikh nauk 8, 306, 1928.

Ed.

if we wish to confine a wave in space, we must inevitably define its frequency imprecisely. One may even pass to another limiting case, when the amplitude of the wave (during a very short time) is different from zero only in a region as close as desired to a given point of space, but then the wave image in general has no definite frequency whatever. The “wave packets” encountered here were introduced into quantum mechanics (originally for somewhat different purposes) by Schrödinger.

We have emphasized many times that the establishment of a correspondence between waves and particles must be regarded as something primary, not subject to any explanation by the facts given to us. In this connection it remains only for us to consider this impossibility of simultaneously sharply determining the position and the frequency of a wave packet as the expression of analogous properties of particles; the energy of a particle, determined from the de Broglie equations, must itself be defined imprecisely if the corresponding wave image has no precisely determined frequency. Starting from this point of view, by simple geometrical investigations of waves, we arrive at the uncertainty rules (21).

From this point of view we also understand the formal agreement of Debye’s and Einstein’s radiation statistics in full space as a real internal unity. The proposition that the volume of a cell of phase space is equal to \(h^3\) corresponds to the circumstance that, in the best case, we can transform the three inequalities (21) into three equalities; then we do indeed obtain cells with volume \(h^3\). But at the same time, in the product \(\Delta p_x \cdot \Delta q_x = h\), we can still choose one of the factors arbitrarily, and Debye’s exposition corresponds precisely to the limiting case in which the momenta \(p_x, p_y, p_z\) are determined with complete precision, while the positions of the light quanta remain completely indeterminate.

Of decisive importance is the fact that the uncertainty of positions or of momentum is, from the point of view of modern quantum mechanics just presented, by no means

not a consequence of the incompleteness of our present knowledge, but, on the contrary, an indeterminacy connected with the facts themselves, or, better said, with the limited applicability of classical concepts to reality, corresponding to the nature of things. Indeed, even a classical wave, since it is not propagated uniformly throughout all space, possesses a certain indeterminacy with respect to its frequency; but here the indeterminacy is not in the physical picture itself, but only an inadequacy of the concept of frequency for the exact transmission of these pictures. Experimentally, however, one can establish a definite position of a particle even in the case when the momentum of the particle is determined and its position remains indeterminate. One can, for example, at least in a thought experiment, determine the position of an electron by illuminating it with light of very short wavelength (e.g. with \(\gamma\)-rays) and then observing in a suitable “microscope” the flash of light that arises when the electron collides with a light quantum. But in doing so one must imagine that precisely as a result of this observation the coordinates of the particle’s position acquire definite values; by preceding experiments only the probability laws for these position measurements were determined. Simultaneously with this establishment of the particle’s position there also occurs, in essence, an uncontrollable, i.e. essentially indeterminate, change of momentum.

These considerations bring us to the problem of causality; modern quantum mechanics contains a categorical denial of classical determinism. In substance, here too there is only a consistent carrying through of those ideas toward which the hypothesis of light quanta was directed from the very beginning. Our previous formulation, according to which the statistical connection between waves and particles must be regarded as the basic principle of the theory, in essence already contains a rejection of the principle of causality. Further development has only strengthened this rejection.^1

^1 Modern quantum mechanics has shown the inadequacy of mechanical determinism. The role which in the new quantum mechanics

It is very remarkable that, despite this renunciation of classical causality, quantum mechanics has nevertheless managed, on this question as well, to preserve a certain agreement with classical conceptions. Classical theory makes it possible to precompute the motion of a microscopic “material point” under the influence of known forces, if at some initial instant its position and velocity are known. Quantum mechanics does not simply assert the incorrectness of this statement; on the contrary, its indeterminism manifests itself in the fact that the presuppositions of this classical statement are not fulfilled in the microphysical domain (Heisenberg).

Let us note one more point in which, at the same time, one can see how the fundamental difficulties of the former quantum-theoretical formulations are softened by the new conceptions. For the same reasons that place and momentum are never exactly determined, the time and the energetic transformations in quantum jumps are also never exactly determined. This removes the earlier strange idea that, at some definite instant of time, a definite finite change of energy takes place by a jump; at the same time, all questions about the “mechanism” of these quantum jumps also disappear.

These examples shed light on the astonishing relations between classical and quantum mechanics, which, in a somewhat paradoxical form, may be formulated as follows: between the two theories there exists, on the one hand, the greatest possible opposition, and on the other—the greatest possible similarity. The greatest opposition in the basic premises, and nevertheless a similarity everywhere manifested in all regularities. The basic ideas of the hypothesis of light quanta: the dualism of waves and

a statistical regularity plays, compelled many physicists, including Jordan, to speak of indeterminism in the world of the microcosm.

Such an assertion is incorrect: statistical regularity, although it does not fit within the framework of mechanical determinism, nevertheless in no way violates the principle of causality. For more detail on this problem, see B. Hessen’s article in Haas’s book Matter Waves and Quantum Mechanics, GIZ, 1930. —Ed.

particles, the correspondence between frequency and energy characterize, in a twofold manner, on the one hand the fundamental antithesis, and on the other the close kinship of the two theories.

V. QUANTUM THEORY OF RADIATION

13. QUANTUM MECHANICS OF RADIATION FLUCTUATIONS

Even before the ideas that had grown out of the hypothesis of light quanta entered into the development of quantum mechanics and there showed their fruitfulness—even before this, matrix mechanics, for its part, had taken the first successful step toward a complete clarification of the problem of light quanta. In now beginning to acquaint ourselves with these circumstances, we approach the newest and most difficult investigations in the field of quantum theory, and therefore must ask the reader not only to be especially attentive to the reasoning of this last chapter, but also, if possible, to recall more distinctly everything that has been said earlier.

We left the hypothesis of light quanta at the moment when it became clear to us that, on the basis of classical waves, we can construct an exact theory for the special case of a single light quantum (in interaction with microscopic optical instruments). After that we established in detail that the same is possible also for a single (isolated) material particle. However, we still have no answer to the question of how one can construct an exact theory for a hollow space containing more than one light quantum.

To clarify for ourselves the meaning of the answer that must be given to this question, let us again imagine the picture of the quantum mechanics of a material point, of which we spoke in the preceding section. This theory found a clear foundation for itself in de Broglie’s correspondence between waves and particles and in the analogy between classical mechanics and geometrical optics. However, one can also give this theory another form, renouncing these clear foundations in favor of more abstract points of view;

this is also done by the theory of transformations, for which the waves propagating in space of Schrödinger are only a special case of general “transformation functions” (“probability amplitudes”), which determine the statistical relations between the results of individual experiments (measurements) performed with a given material point. Within this theory of transformations one can give manifold different matrix representations of quantum-mechanically measurable quantities, and the matrices introduced by Heisenberg again prove to be only a special case of them.

The disadvantage that we have now abandoned the visualizable foundations of the de Broglie–Schrödinger theory is offset by the gain consisting in the fact that we acquire a general abstract quantum-mechanical formal apparatus, which can also be applied to the solution of such problems for which de Broglie’s visualizable ideas by themselves provide nothing. We have already clarified earlier that the problem of light radiation with a number of light quanta greater than unity, or the problem of an ideal gas (with several atoms), in the sense of Einstein’s theory, cannot obtain a satisfactory solution on the basis of de Broglie’s ideas. Such problems lie beyond the reach of these ideas (de Broglie’s investigations and Einstein’s gas theory have shown us that analogous difficulties arise in both of these problems).

There are, however, no obstacles whatsoever to applying the formal apparatus of the theory of transformations, instead of to a single material point, to a completely different mechanical system—namely, to the electromagnetic medium within some hollow space; in doing so, as Born, Heisenberg, and Jordan have shown, even by means of matrix theory in its former form one can obtain substantially important results here. In exactly the same way as in the case of a material point, we represent the coordinates \(q_s \ldots\) or momenta \(p_s \ldots\), or the kinetic energy, or in general an arbitrary function of the coordinates and momenta, in the form of matrices with elements \(q(nm)\), where \(n\) and \(m\) denote two quantum

states of the system; likewise they may be represented by means of matrices for the full space, for example, the electric-field strength \(\mathfrak{E}(x,y,z)\) at the point \(x,y,z\) of our space \(V\), or the energy \(E_0\) in a partial volume \(V_0\),

\[ \begin{aligned} \mathfrak{E}(x,y,z) &= \left(\mathfrak{E}^{(x,y,z)}(n,m)\right) \\ E_0 &= \left(E_0(n,m)\right) \end{aligned} \tag{22} \]

where \(n,m\) denote two stationary states of the whole system (of the oscillating full space). In essence this means the same thing that Debye had already tried to do in 1910, carrying out the quantization of the oscillations of the full space; but this time the attempt is made not with the weak means of the former theory, which represented a mixture of classical and quantum-theoretical elements, but in an exact quantum-mechanical form. Therefore our premises yield not only Planck’s law instead of the Rayleigh–Jeans law, as was also the case with Debye, but also remove the deeper difficulties of the theory of radiation which Debye did not touch.

Above we have already established quite clearly that the matrix elements \(q(n,m)\) by no means necessarily have the original significance of probabilities of transitions connected with the emission or absorption of radiation; therefore there are no fundamental difficulties in our representing, by means of matrices, the quantities \(\mathfrak{E}(x,y,z)\) and \(E_0\), although the concept of radiation probabilities is inapplicable here. The significance of this transition to the matrices (22) consists precisely in the fact that we ascribe a changed kinematics both to the quantities characterizing the field \(\mathfrak{E}, \mathfrak{H}\) and their functions (for example, \(E_0\)), and to the coordinates of the oscillating electron. This change of kinematics becomes imperceptible when we ask about any time averages of the intensity of light; in these cases we always again obtain the empirically confirmed propositions of classical optics. However, it makes itself felt as soon as we again turn to the question of fluctuations. We

we can, for example, repeat Lorentz’s previous calculations—this time, to be sure, with the wave amplitudes $\mathfrak{C}$, $\mathfrak{S}$ represented in the form of matrices—and compute, allowing for interference, the time average of the square $E_0^2$, or of the square of the fluctuation $(\Delta E_0)^2=(E_0-\bar E_0)^2$.

In doing so we obtain the result that precisely on the basis of those deviations from the classical theory which are described by the matrix relations, there do indeed result the formulas referred to above as the “second Einstein law of fluctuations.” This was discovered by Born, Heisenberg, and Jordan, who investigated a somewhat simpler system than a field space, namely a one-dimensional perfectly elastic string (here, in principle, the same relations hold).

In two respects these achievements remained, of course, not entirely satisfactory.^1 First, the method used at that time still did not make it possible to obtain deductively also the first Einstein law of fluctuations, in which the probability of certain extremely improbable distributions of energy in field space is determined. True, above we encountered one route which in principle could shed light on this point as well. By successively computing the time average of all powers, $E_0^2$, $E_0^3,\ldots E_0^s\ldots$, we can in the end compute from this the probability of the realization of a definite value $E_0^1$ of the quantity $E_0$, but, of course, in practice this is unrealizable. Therefore only later, with the aid of more refined means, was it possible to show that the first Einstein law of fluctuations also follows of itself when we quantize the waves in field space in the manner indicated above.

Second, a well-known lack of elegance in the theory consists in the appearance of a certain (even infinitely large) zero-point energy. Indeed, according to quantum mechanics the energy of a harmonic oscillator of frequency $\nu$, located—

^1 Einstein drew my attention to both of these points at the time in his letter.

The Hypothesis of Light Quanta

situated in the \(n\)-th quantum state (for the ground state \(n=0\)) is equal not to \(nh\nu\), but to \(\left(n+\dfrac{1}{2}\right)h\nu\); thus, even in the ground state there still remains a certain energy \(\dfrac{1}{2}h\nu\). The quantum field therefore receives, for each individual proper oscillation, the corresponding energy, and the sum of these energies becomes infinitely large. Let us note here as well that this unpleasant point has so far not been removed. It must be said, however, that the matter here is more one of a formal complication than of a real difficulty. In fact, for empty space we can observe only differences of the energies of various stationary states,\(^1\) and for them the zero-point energy is of no significance.

Leaving aside the still unresolved problems and turning to what has already been accomplished, we see first of all that there is no need to return again to the hypothesis of light quanta in explicit form. Indeed, we have throughout adhered to the wave theory of light and have altered, in the quantum-mechanical direction, only the kinematics of waves in empty space. As a result of this, effects characteristic of light quanta have emerged of themselves. Quite independently of the fact that here for the first time we have a theory applicable to all problems connected with empty space, the problem of light quanta in general receives an entirely new formulation. On the premises of this theory there is no need to introduce conceptions of light quanta. On the contrary, one may—and this, apparently, is the most natural course—start from wave conceptions; if they are formulated in the terms of quantum mechanics, then the corpuscular effects of light quanta arise by themselves, as necessary consequences of the general laws of quantum theory. It is clear what prospects open up here if one again thinks of matter waves instead of light waves. We gain the hope of understanding when—

\(^1\) In this respect, an oscillating empty space behaves quite differently from, for example, an oscillating crystal lattice.

...in some analogous way and to encompass the atomic structure of matter and of electricity, which had presented insurmountable difficulties for the classical theory, as a particular consequence of the general quantum-theoretical laws.

A substantial result of the investigations of Born, Heisenberg, and Jordan for the theory of light itself is the proposition that one can carry out the development of a new concept of the field (which Pauli had demanded) by applying the concepts of quantum mechanics to the oscillating field. But this proposition, in a certain sense, shared the fate of those arguments of Einstein for whose clarification its justification had been sought: for a long time it was judged with reserve or negatively even by adherents of quantum mechanics.^1 Its recognition came only when, a year later, Dirac showed that Einstein’s laws of emission and absorption for an atom situated in a radiation field follow from these assumptions naturally and exactly.

14. Interaction of atoms and radiation

Dirac also undertook investigations of the reactions of an atom’s radiation, proceeding from the conception of a quantized field space. Of course, until the interaction of the oscillating dipole moment of atoms with the space of radiation was considered from this point of view, it was possible only to give a derivation of Einstein’s probability laws, half connected with the correspondence principle and, in any case, it was impossible to obtain a deductively natural grounding for it. With the aid of the newly created tools of general quantum mechanics, Dirac succeeded in obtaining a clear solution of this problem. Indeed, by this route we arrive at Einstein’s formulas—

^1 This general non-recognition for some time even aroused in the author himself certain doubts, which are set forth in the corresponding remarks to the article “The Development of Quantum Mechanics” in Naturwissenschaften.

s h t e i n, and then to the corresponding formulas for dispersion, etc.¹ In addition, important new results were obtained in the most subtle and complex questions connected with the problem of the natural width of spectral lines. Other authors (Oppenheimer, Landau, Bloch) further developed and applied Dirac’s method.

For us here, of primary importance are those arguments in Dirac’s investigations by means of which he made an important contribution to clarifying the dualism between quantized waves, on the one hand, and corpuscular particles (in any number), on the other. But we shall return to this later. Here we shall note only one problem which, in a certain sense, is a natural continuation of the questions developed by Dirac. According to Dirac, it is now possible without difficulty to calculate the interaction of two distant atoms effected by radiation; but in doing so it is essentially necessary to regard the distance between them as so great that the crude electrostatic interaction of the electrons on both sides is excluded. On the other hand, this crude interaction of electrons can be mastered quantitatively in those approximations in which it is described by the Coulomb force, acting instantaneously at a distance. But in reality there exist only field-mediated, retarded interactions between electrons. The classical electron theory of Lorentz, as is well known, was able to give comparatively simple statements about these retarded interactions. Quantum theory, on the contrary, has not yet gone so far. We shall see below how much effort and thoughtful reflection had to be expended in order to achieve at least a certain preparation for mastering this problem. Nevertheless, the situation up to now is such that this preparation still proves insufficient. Only one thing may evidently be considered firmly established: namely, that the conse-

¹ In this connection, conclusions based on quasi-classical premises—such as, for example, those of Joos—also fall away and are set aside.

...a satisfactory theory of delayed interactions of electrons must include, as a special case, also the interactions of radiation in the form in which they are described by Dirac. Thus, there is obviously no possibility of solving this problem without resorting to the quantization of the electromagnetic field.

15. The Many-Body Problem and Quantum Mechanics

Now we return once again from the quanta of light to material particles. Above we have emphasized often enough that de Broglie’s ideas and Schrödinger’s results based upon them initially gave an exact theory only for the problem of the quantum mechanics of a single isolated particle. But at the same time we found that it is possible to abstract from the quantum mechanics of a single isolated particle a formal quantum-mechanical theory which can also be applied to other problems, although here it does not have the same vivid foundations. This was most easily done for the many-body problem in quantum mechanics; elementary matrix mechanics, not bound in advance to so narrow a problem as de Broglie’s ideas, indicated the only path for this. If, for example, we wish to describe, according to Schrödinger, the mechanics of the hydrogen atom, consisting of a nucleus and one electron, then for this we need waves in an abstract six-dimensional space. (Correspondingly, the limiting case of classical mechanics for this problem is analogous to geometrical optics in a certain six-dimensional extended medium.) From the formally mathematical side, this is a simple generalization of Schrödinger’s theory of the one-body problem; but the intuitive starting point—the idea that matter, like light, must be described by means of certain waves in real three-dimensional space—is here completely abandoned and has given way to abstract quantum-mechanical representations.

Further progress along this path led Dirac and Heisenberg to very remarkable results. We shall confine ourselves to recalling these things, which are at present well known. In solving the many-body problem for several bodies with completely homogeneous, essentially indistinguishable particles (an atom with several electrons, a gas of homogeneous atoms), new relations arise from the appearance of systems of terms that are essentially non-combining. In nature, in each definite case, only one of them can be realized, but the theory does not make it possible to determine which one. It became necessary, therefore, to supplement the theory here with certain additional assertions based on experiment. Experiment shows that only two of these systems of terms have physical existence; mathematically they are characterized respectively by “symmetric” and “antisymmetric” fundamental functions.

If a gas of homogeneous atoms possesses states that correspond to the “symmetric” system of terms, then it behaves like a gas in Einstein’s theory. We thus obtain the astonishing possibility of understanding the enigmatic Bose–Einstein statistics also on the basis of corpuscular representations. Conversely, in the case of an “antisymmetric” system of terms, Pauli’s famous “exclusion principle” is fulfilled. In each “cell” of phase space there can be no more than one particle. According to experimental data this holds for negative electrons (Pauli), and also for positive particles (Dennison). It is also easy to calculate (Fermi, Dirac) how, in the absence of energetic interaction of the particles, an ideal gas subject to the exclusion principle behaves;1 this is important for the theory of metallic electrical conductivity (Pauli, Nordheim,

Finally, it is also easy to see how the laws of probability for collisions of gas atoms, and of all the other elementary processes, must be modified so that they correspond to the distribution of velocities in such a gas, which differs from the Maxwellian one (Jordan, Kramers, and Ornstein).

From the experimental fact that Pauli’s exclusion principle is fulfilled for both kinds of charged particles, one may theoretically draw the following conclusion (Elsasser, Wigner): for a gas whose particles carry charges \(\pm e, \pm 3e, \pm 5e\), etc. (\(e\) being the charge of the electron), Pauli’s exclusion principle is valid; for charges \(0, \pm 2e, \pm 4e\), etc., on the contrary, Bose–Einstein statistics holds. As Heisenberg observed, this proposition can also be applied to light quanta, if one accepts, with Eddington, the hypothesis of the neutralization of electrons, accompanied by transformation into light quanta; and this, for the first time, gives a theoretical foundation for such an elementary fact as the applicability of Bose–Einstein statistics to light quanta (instead of Pauli’s exclusion principle).

16. Waves and Corpuscles

The exceptional successes achieved by the formal theory—briefly outlined above—of the problem of several bodies with indistinguishable parts, including the derivation of the basic elements of a qualitative and quantitative theory of atoms with several electrons, could not, however, conceal the essentially unsatisfactory character of this theory. Apart from the circumstance that, in this connection, it proved necessary, alongside the quantum-mechanical equations of motion, to give additional instructions concerning the choice of one of the possible solutions, the very abandonment of the intuitive core of de Broglie’s ideas remained a serious indication that the theory had not yet found its most appropriate form.

The results obtained for the problem of light quanta, however, gave definite indications for eliminating these defects. Above we showed in detail how one may understand

from the standpoint of wave theory, Einstein’s theory of the gas—on it we shall now concentrate our attention—if one accepts that the de Broglie natural oscillations of the whole space must be quantized in the same way as the light wave. We can now assert that, if this quantization is carried out with the aid of the concepts of quantum mechanics, then not only does Einstein’s statistics become intelligible, but the fluctuation effects in the ideal gas are also described exactly. With the aid of Schrödinger waves one can develop, in this form, the theory also for the case when the atoms of the gas (not interacting with one another) are subjected to the action of some force field within the space occupied by the gas. In this way the intuitive core of de Broglie’s theory is again reinstated in its rights, and a possibly complete analogy is established between light and material radiation for the case of an arbitrary number of corpuscles. By introducing the quantization of waves, we satisfy the necessity, justified in detail above, of considering de Broglie waves in Einstein’s gas theory not from a purely classical standpoint. As in the case of light, so now also in the ideal Einstein gas, the fact that there always exists a whole number of “indivisible” corpuscles is a consequence of the fact that, according to quantum theory, the harmonic oscillator can have only discrete, equally spaced energy levels; the atomic structure of matter is reduced to the general laws of quantum theory.

This program of theoretical development of the problem of several bodies, which (in writing and orally) was formulated, on the one hand, in direct connection with the fluctuation investigations in the work of Born–Heisenberg–Jordan, and, on the other, in connection with Schrödinger’s first work, required much time for its completion. The first steps in this direction were taken by the development of the more formal theory of Dirac–Heisenberg, resting on different fundamental principles, of which we spoke briefly above. After it turned out that this theory is, in any case, correct *

mathematically, at every attempt to give it another formulation it was necessary to take care that it be mathematically equivalent to the formulations already available.

In the already mentioned investigation of the reactions of atomic radiation, Dirac succeeded at first, for the special case of an ideal gas obeying Einstein statistics, in understanding more precisely the mathematical connection existing between the corpuscular theory, which works with multidimensional fundamental functions, and the theory of quantized waves. This connection, which was later further substantially clarified (Jordan), may be explained, without entering into the mathematical theory, as follows. If initially we have in a box with gas only one particle, then Schrödinger’s wave function \(\psi(x, y, z; t)\) and its complex conjugate \(\psi^*(x, y, z; t)\) determine the probability of finding the particle at the given place \(|\psi_n|^2=\psi_n^*\cdot\psi\). Developing Einstein’s wave-form theory of gases for an arbitrary number of atoms, we obtain instead a matrix \(\varphi=\varphi(x, y, z)\), which is again a function of the point \(x, y, z\); from this matrix amplitude and its complex conjugate \(\varphi^+(x, y, z)\)^1 we form the mass density:^2

\[ N(x, y, z)=\varphi^+(x, y, z)\cdot\varphi(x, y, z) \tag{23} \]

These quantities: the de Broglie amplitude \(\varphi\) and the density \(N=\varphi^+\cdot\varphi\), must be interpreted in the same sense as, for example, the coordinates of the matrix \(q\) for a quantum-kinematically oscillating material point. Here one cannot speak of wave amplitudes or of the distribution of masses in one definite state of the whole gas; rather, all those physical properties which these quantities can in general reveal in any state of the system are gathered here into one matrix scheme. By spatial integration we can, from the quantity \(N(x, y, z)\), form the total mass in some partic-

^1 More precisely, “adjointed.”

^2 To avoid introducing a normalizing factor, we suppose that the mass of an individual atom of the gas is equal to 1.

of volume \(V_0\) of the volume \(V\) (it is also a certain matrix). We can raise these quantities to the square or to another power and then calculate mean values; in this way we find fluctuations of the density of the gas in exactly the same way as for the hollow electromagnetic space. The results obtained thereby will be completely determined if we make the following two assumptions (again analogous to the conditions that we have for light):

1. For the matrix \(\varphi(x,y,z)\) the same differential equation must be satisfied as for Schrödinger’s function \(\psi(x,y,z)\).1

2. For the matrices \(\varphi(x,y,z)\), \(\varphi^\dagger(x,y,z)\) there must be established (as is usually accepted in matrix theory) certain non-commutative rules of multiplication.2

In these formulations3 everything is already contained. One need only apply the general mathematical formal apparatus of quantum mechanics in order to obtain the whole theory by a deductive route. The quantization of the individual proper oscillations is obtained in the desired manner. Further, one obtains—and here too we encounter corpuscles in the most vivid form—that for the total mass in some part \(V_0\) of the volume \(V\) (with the normalization adopted here), if it is subjected to measurement, there can be obtained only

integer (of course, zero may also result).¹ All these characteristic quantum-mechanical laws are naturally derived from postulates 1 and 2, although these postulates, apparently, specify not particles, but only a wave continuum.

In these remarkable quantized waves we can carry out the most varied experiments and measurements, calculating for these experiments probability functions according to the rules of quantum mechanics. In doing so, the former theory of the many-body problem, due to Dirac–Heisenberg, is incorporated into the new theory: the fundamental functions calculated according to the former theory can very simply be interpreted as known probability functions referring to quantized waves. Thus, for the special case of an ideal Einstein gas, the synthesis of the wave and corpuscular theories is completely accomplished, and it is natural that the first law of Einstein’s fluctuations, whose deductive derivation from wave quantization has until now remained our task, no longer presents any difficulty.

These results posed two new problems before us. For the case of Bose–Einstein statistics it was necessary to investigate whether, and if so how, one can treat from this point of view also a non-ideal gas with energetically interacting particles. And then it was necessary to see how one could achieve the same in the case of the many-body problem under Pauli conditions. The first question was answered by Jordan and Klein, the second by Jordan and Wigner. We shall briefly outline the course of ideas in these investigations.²

In the problem of interaction the issue is essentially to find a suitable expression for the interaction energy, with intuitive notions again being taken as the basis—

¹ In mathematical formulation this means that the characteristic numbers of the matrix representing the integral of \(N(x, y, z)\) over \(V_0\) are integers.

² See on the following page.

notions of quantized waves of matter. But, of course, at first it remained unclear whether anything similar was possible from the mathematical point of view. As is well known, Schrödinger examines in detail the possibility of interpreting the quantity \(\psi^2\), formed from his wave function \(\psi\), not only as a probability, but also as a real density of electricity or mass (with the corresponding factor). On the basis of these ideas, for the electrostatic energy of interaction of Schrödinger waves the expression

\[ E_w=\iint dV\cdot dV'\,\frac{|\psi(\mathbf r)|^2\cdot|\psi(\mathbf r')|^2}{|\mathbf r-\mathbf r'|} \tag{24} \]

was obtained. Here \(dV\) and \(dV'\) are volume differentials, \(\mathbf r\) and \(\mathbf r'\) are written respectively instead of \(x,y,z\) and \(x',y',z'\), and \((\mathbf r-\mathbf r')\) is the distance between the points \(\mathbf r\) and \(\mathbf r'\). But, as Schrödinger himself clearly established and sharply emphasized, this is impossible. This can be seen quite distinctly, for example, in the hydrogen problem; in fact, the potential of interaction of a discrete nucleus and a discrete electron should have entered into Schrödinger’s differential equation, and not the electrostatic interaction of the continuously distributed density \(|\psi|^2\), as follows from equations (24). However, the situation changes if we consider the matrix density \(N=\varphi^\dagger\varphi\). For here, as we already know from the theory of the ideal gas, discrete corpuscles are hidden in a dual quantum-mechanical form. If, for the problem of several bodies, one substitutes, as the interaction-energy matrix of the quantized \(\varphi\)-waves, the expression

\[ E_w=\iint dV\,dV'\,\frac{\varphi^\dagger(\mathbf r)\,\varphi^\dagger(\mathbf r')\,\varphi(\mathbf r')\varphi(\mathbf r)}{|\mathbf r-\mathbf r'|} \tag{25}, \]

then for the corpuscles contained in the waves one obtains precisely the desired Coulomb interaction energy. Obviously, this expression represents almost the same thing as

\[ \iint dV\cdot dV'\,\frac{N(\mathbf r)N(\mathbf r')}{|\mathbf r-\mathbf r'|} \tag{26} \]

but not quite the same, because the factors \(\varphi\) and \(\varphi^\dagger\) are not all-

where one may interchange them. This is connected with the fact that, for several electric point charges, there is an interaction $e^2/r$ only between two different ones, while the corresponding energy of a charge acting on itself should not be included in the calculation. Here it is precisely the noncommutativity of multiplication in quantum mechanics that is responsible for this effect. We see how the famous and much-discussed problem of the strange back-action of the electron upon itself (“why do electrons not fly apart?”) receives a new form in the light of quantum theory.

Pauli’s exclusion principle and the statistics based on it may be referred, just like Einstein’s, to de Broglie’s proper oscillations instead of to “cells” in the phase space of particles. Therefore, in Pauli’s case each of these proper oscillations must behave not as a quantum-mechanical harmonic oscillator, but rather as an object that can have only two different stationary states (for this proper oscillation may account either for one atom or for not a single one). Such a remarkable object has already been encountered by us: it is the magnetic electron. Indeed, a rotating electron (if one leaves aside its translational motion) can occupy, in a magnetic field, only two different positions. After it became possible to include the theory of the “resting” magnetic electron (Darwin, Pauli, Jordan) within the framework of general quantum mechanics, it was possible also to approach the quantization of de Broglie’s proper oscillations corresponding to Pauli’s principle. For quantum mechanics asserts: if anywhere, in otherwise arbitrary relations, we encounter a quantum-mechanical system for which, under some arbitrary conditions, only two stationary states or only two different reactions are possible, then this system, from the formal-abstract point of view, must be equivalent to a (resting) magnetic electron. As another example we may point to the polarized light quantum (Jordan). If, for example, we investigate a light quantum with the aid of a Nicol prism set in a definite way, then there are possible

only two results: either the light quantum passes through, or it is reflected. It follows from this that all experiments that can be performed on a light quantum with the aid of a Nicol prism or other polarization analyzers can be described statistically with the aid of a formal theory abstracted from the magnetic electron. With this, no deeper model analogies between the light quantum and the rotating electron should be associated. The relativistic theory of the electron given by Dirac implicitly includes the same formalism with respect to the distinction of the electron charges (there exist two values of the charge, \(+\) and \(-e\)).

The construction on these foundations of the theory of the many-body problem under Pauli conditions proved, from the mathematical side, to be more difficult and complex than in the case of Einstein, but the final result again proved to be very simple. The theory of the ideal Fermi gas can be obtained deductively from premises that fully correspond to requirements 1 and 2 for the Einstein gas. Only the noncommutative properties of the multiplication of amplitudes \(\varphi \cdot \varphi^\dagger\) now turn out to be somewhat different (the point here is a difference in sign. ^1 From these premises all the rest again follows: the appearance of particles and another quantization of the natural oscillations (Pauli exclusion). Finally, the energetic interactions of particles can be treated almost in the same way as in the case of Einstein.

17. Relativistic construction of the theory

Thus, there exists the possibility of replacing the Dirac–Heisenberg theory of the many-body problem, connected with a multidimensional space, by another formulation which is based on representations of quantized

^1 Instead of the equalities given in note 2 on p. 85 for the case of Einstein, in the case of Pauli we have the equalities:

\[ \varphi(r)\varphi(r')+\varphi(r')\varphi(r)=0 \]

\[ \varphi^\dagger(r)\varphi(r')+\varphi(r')\varphi^\dagger(r)=\delta(r-r'). \]

waves in ordinary space. A consequence of the mathematical equivalence of the two theories is, naturally, that the new theory by no means intends to displace the old one in practical computational problems, especially after Wigner’s work has shown the ever greater fruitfulness of the old theory.¹ On the contrary, the advantage of the new theory lies above all in its closer connection with the intuitive foundations of de Broglie’s and Einstein’s ideas.

There are, however, deeper problems whose solution can hardly be obtained on the basis of a description of electronic systems in a multidimensional coordinate space. These include all questions connected with relativity, in particular the problem touched upon above of the retarded interaction between electrons. The inadequacy of the theory of coordinate space becomes still more obvious if one thinks of such problems as the transformation of matter into radiation.² Thanks to its closer connection with the intuitive foundations of Einstein–de Broglie theory, the new theory of the many-body problem is in a more favorable position with respect to these problems.

Indeed, it seems so obviously natural to construct quantum-theoretical electrodynamics on the direct interaction of the waves of light and matter; evidently this is the most natural and simple way to take into account the wave nature of matter. This idea was developed in detail by Schrödinger, who from the very beginning recommended interpreting the square of the amplitude \(|\psi|^2\) not only as the probability of finding an electron at a given place, but also as a real electric density, and, as such,

¹ Nevertheless, apparently, many regularities of atomic structure, etc., which until now have been derived only with the aid of the difficult methods of group theory, can be more easily justified by using the method of Jordan and Wigner; however, these interrelations have not been studied in detail.

² If such elementary processes are admitted, the very number of dimensions of the coordinate space we use would have to be made variable.

the quantity \(\psi^2\) should, according to Maxwell’s theory, produce the corresponding electric field. Schrödinger also showed how, in this sense, one may describe the interaction of the \(\psi\)-field and the electromagnetic field \(\mathfrak E,\mathfrak H\) in a relativistically invariant theory. However, this line of ideas naturally led to serious contradictions so long as we regarded \(\psi,\mathfrak E,\mathfrak H\) as classical, unquantized field functions; above (in Section 6) we have already noted that this difficulty was clearly emphasized by Schrödinger himself. These conceptions can be developed only after we accept that \(\psi,\mathfrak E,\mathfrak H\) are numerical or matrix \(q\)-functions in the sense explained in detail above. But the propositions which we set forth above at first applied only to the case of classical nonrelativistic mechanics with non-retarded forces of interaction between particles, acting instantaneously at a distance; and even in the investigations mentioned concerning the electromagnetic field of radiation, the relativistic symmetry of space and time was not at first taken into account. The problem, therefore, was to determine whether, and in what precise way, this theory could be generalized in a relativistic sense, and whether in doing so it would be possible to formulate satisfactorily the interaction of the quantized fields \(\psi,\mathfrak E,\mathfrak H\) so as to obtain, as a result, the laws of the retarded interaction of individual electrons.

First of all, it is in fact possible to construct an invariant theory of the electromagnetic field free from charges (Jordan–Pauli). Characteristic of this formulation is the introduction of the “relativistic \(\Delta\)-function,” which is analogous to Dirac’s special \(\delta\)-function; whereas Dirac’s function \(\delta(x,y,z)\) has a particular infinite point only at \(x=0,\ y=z=0\), for the function \(\Delta(x,y,z,ct)\) this occurs on the entire “light cone” \((x^2+y^2+z^2-c^2t^2=0)\). This formulation has the same drawback of elegance consisting in the presence of zero energy; but so long as we are dealing with a field free from charges, this causes no real difficulties. It is necessary to note the following circumstance—

testimony (not stipulated in the work named above). A relativistic presentation of the theory makes it possible to establish a logical connection between the form of Maxwell’s equations and the fact that light quanta obey Bose statistics, and not the Pauli exclusion principle. It would have been possible, to be sure, to try to apply to light waves a different quantization as well (according to the Jordan–Wigner scheme), but in doing so we arrive at mathematically unsatisfactory results; only Bose quantization is mathematically compatible with the form of Maxwell’s field equations.

The next step after this is the relativistic quantization solely of the waves of matter, without taking into account their interaction with the light field. As the basis for this one must have a relativistic mechanics of a single individual electron; such a mechanics was provided by Dirac’s theory of the spinning electron. At first it seems as if there is no possibility of expressing the multiplication properties of the quantized Dirac field amplitudes \(\psi_1, \psi_2, \psi_3, \psi_4\) through the \(\Delta\)-function. However, Heisenberg and Pauli showed that this is nevertheless possible under one restriction, fully corresponding to the meaning of things. In this connection it turns out, moreover, that just as for the Maxwell field in empty space—mathematically there is no possibility of carrying out quantization at will either in the Bose direction or in the Pauli direction; only the latter case is mathematically possible. The fact that electrons obey the Pauli principle and not Bose statistics is, consequently, in the closest mathematical connection with Dirac’s theory of the spinning electron. Finally, it turns out that for matter waves there is no zero-point energy.

Despite their reassuring results, these last considerations have only doubtful value, since their basis, the Dirac theory of a single electron, is subject to severe attacks and is certainly not completely correct, although it undoubtedly contains a number of major achievements. We have no possibility here of considering in greater detail those circumstances,

which speak against Dirac’s theory: let us note only that it has become entangled in difficulties not yet resolved, connected with the dark problem of the asymmetry of the two kinds of electricity, the inequality of the masses of positive and negative elementary charges. The uncertainty of our present knowledge in the problem of the individual electron is naturally carried over also to the related arguments concerning the problem of several electrons.

If, however, one first abstracts from these difficulties, one may ask whether it will be possible not only to carry out an invariant quantization of the Dirac quantities $\psi_1, \psi_2, \psi_3, \psi_4$ in themselves and of the quantities $\mathfrak{E}, \mathfrak{H}$ in themselves, but also to formulate satisfactorily, in invariant form, the interaction of light and matter. This would mean carrying out Schrödinger’s program and giving a certain completion to the whole cycle of ideas explained above. Unfortunately, it seems that here, so close to the goal and having overcome so many difficulties, we nevertheless suffer a failure. The considerations adduced by Heisenberg and Pauli apparently prove this with complete definiteness. It was explained above what an important role, in the nonrelativistic theory, was played by the fact that one could leave out of consideration the peculiar reaction of the electron upon itself. This does not succeed, however, for the relativistic interaction of electrons effected by the electromagnetic field; moreover, the mathematical sources of these difficulties lie in the same properties of the quantized field $\mathfrak{E}, \mathfrak{H}$ which give rise to the zero-point energy.

Thus we encounter here a serious failure, and it is necessary to consider what it means. The arguments of Heisenberg–Pauli clearly show that all these difficulties lie only in the electromagnetic field; with the waves themselves all is in good order. This compels one to suppose that perhaps there are no grounds for placing the $\psi$-field and the field $\mathfrak{E}, \mathfrak{H}$ on an equal footing, and that perhaps, in accordance with Eddington’s hypothesis, only the field $\psi$ should be regarded as a primary quantity,

while the electromagnetic field, with its light quanta, should be regarded as a secondary “product of neutralization” of positive and negative charges, taking into account the possible interconversions of matter and radiation. At present this is, of course, still an unresolved problem, but the conclusions of Geisenberg–Pauli by no means deprive us of the possibility of an approximate solution of the problem in more or less close connection with the ideas set forth here; such approximate solutions we must seek first of all, in order slowly to move further ahead. But the final solution will probably, in the sense indicated, be connected with Eddington’s problem of the transformation of matter into radiation.

Thus the present state of quantum theory is characterized by the fact that the problems, still unresolved at the present time, concerning the problem of one and of several electrons are apparently closely connected, on the one hand, with the question of the significance of the asymmetry of the two kinds of electricity, and, on the other hand, with the question of the Eddington transformation of matter into radiation, i.e., with two questions which, to a certain extent, surpass by an order of magnitude the problems solved hitherto, and in the solution of which we cannot rely either on sufficiently extensive experimental material or on the aid of classical theory afforded by the correspondence principle.

However, it is now timely to recall that the actual subject of this article is only the hypothesis of light quanta, and that we had to enter into the more difficult questions of the problem of material particles only in order to show the further development and fruitfulness of the ideas of light quanta in this broader domain. If, in questions relating to matter, we have had to stop before deeper, still insurmountable difficulties, then what we have achieved in our understanding of light quanta gives us full confidence that we are on the right path.

1. There we had
   \[ -\frac{h^2}{4\pi^2}\Delta\psi + U(x,y,z)\cdot\psi + \frac{h}{2\pi i}\dot{\psi} = 0. \] ↩↩↩↩↩

2. Indeed, if for brevity we write \(r\) and \(r'\) respectively instead of \(x,y,z\) and \(x',y',z'\), then
   \[ \varphi(r)\varphi(r')-\varphi(r')\varphi(r)=0 \]
   \[ \varphi^\dagger(r)\varphi(r')-\varphi(r')\varphi^\dagger(r)=\delta(r-r'); \]
   here \(\delta\) is the so-called Dirac \(\delta\)-function. ↩

3. Requirement 1 can also be expressed thus: the energy density must be equal to
   \[ -\frac{h^2}{4\pi^2}\psi^*\cdot\Delta\psi+\psi U^*(x,y,z)\psi. \]
   The potential energy \(U(x,y,z)\) is—as in the ordinary Schrödinger equation—not a matrix, but a simple numerical function of \(x,y,z\). ↩