ON THE QUANTUM THEORY OF RADIATION.

N. Bohr, H. A. Kramers and J. C. Slater.— Ueber die Quantentheorie der Strahlung. Zeitschrift für Physik, 24, p. 69, 1924; Philosoph. Magazine, 47, p. 785, 1924.

The fundamental difficulty of modern optics consists in the fact that an essential group of phenomena has an evidently discrete character (for example, the emission of spectral lines, so happily interpreted by the quantum hypothesis), whereas another, no less important group requires for its understanding the classical conceptions of continuous absorption and emission (absorption, dispersion, etc.). In the article under review, Bohr and his collaborators make an attempt to facilitate the resolution of this difficulty by introducing a new conception concerning the character of the processes in the atom. Namely, while leaving in force the basic postulates of the quantum theory of the atom (the existence of discrete stationary states, the condition for the frequency \(h\nu = E_{n m_2\ldots} - E_{n' m'_2\ldots}\), and the correspondence principle), the authors introduce the following hypothesis, sharply different from earlier conceptions. They admit that a system is capable of emitting energy and passing from one stationary state into another, as was supposed up to now, while all the time remaining in a stationary state. However, the continuously occurring emission does not violate the stationarity of the state; the explanation for this lies in the supposition that the law of conservation of energy is not applicable to such an individual process: having lost some quantity of energy in the form of radiation, the system remains in the same stationary state. Only from time to time is the accumulated deficiency or excess of the system’s energy restored when the system jumps into a new stationary state, so that, on the average, the statistical law of conservation of energy remains in force ²). Thus radiation

¹) Should one not imagine that light quanta are emitted and absorbed by individual electrons (and not by atoms as a whole), being constituents or satellites of electrons in the same way as the latter are constituents of atoms or satellites of positive nuclei?

²) Not wishing to break so sharply with the principle of conservation of energy, one might admit that in the processes of radiation and absorption of energy, which take place continuously, a mechanism takes part resembling a reserve reservoir, concealing an excess of energy or making up its deficiency. The role of such a mechanism might be played by some unknown processes in the nucleus, for example. Thus, radiation would proceed at the expense of a store of energy, while absorption would increase this store, direct observation of which is inaccessible to us. Only from time to time, when the jump of the system into a new stationary state “smooths out” the accumulated change in the reservoir, do we obtain information about these suspected redistributions of energy. Of course, such a conception is not a salvation of the principle of energy in its classical form: the admission of “hidden” forms of energy, only from time to time becoming accessible to observation, is simply another expression of the idea of the statistical character of the principle of conservation of energy.

takes place during the motion of the electron in the stationary state; the moments of transition are merely the moments that end one cycle of emission and begin a new one, while the residence time in the stationary state (Verweilzeit) is the time of exhaustion of the coherent train of waves (it also determines the upper limit of the path difference at which interference is still possible). In this respect we take a step forward toward rapprochement with classical electrodynamics. Atoms, like classical electrodynamic systems, are surrounded by a field of radiation establishing a connection between individual atoms. But this field is not the field of classically radiating electrons: the frequency of the radiation is determined not by the character of the electron’s motion, but by the presence of separate possible stationary states, i.e. by the frequency conditions \((h\nu_1 = E_0 - E_1;\ h\nu_2 = E_0 - E_2\), etc.); the character of the motion determines in this case the relative intensity of the individual lines, according to the correspondence principle. Such a field might be called a virtual radiation field: this field corresponds to the classical field of virtual vibrators, whose frequency, intensity, and polarization of radiation are determined by the indicated conditions; the aggregate of such virtual vibrators constitutes the classical equivalent of the atom.

Thus, in Bohr’s new conceptions there is entirely absent the idea of a causal connection between the radiation (or absorption) of an atom and its transition from one stationary state to another. These transitions do not play the role of causes, but prove to be merely accompanying phenomena, the presence of which ensures the statistical applicability of the principle of conservation of energy. The transition from one stationary state to another therefore occurs in a disorderly fashion, like accidental phenomena; hence the natural application, to the processes of interaction between atoms and radiation, of those arguments based on the probability principle which led to Einstein’s well-known derivation of Planck’s formula (Einstein, Phys. ZS, 18, p. 121, 1917). Moreover, the possibility opens up of connecting the probability of those elementary processes from which, according to Einstein, the phenomenon of interaction between atoms and radiation is composed, with the character of the motion of the atoms. The probability of spontaneous radiation¹) (Spontane Ausstrahlung) depends on the proper motion of the atom, i.e. is determined by the amount of energy it emits: the more intense the radiation, the more often, generally speaking, transitions will occur that regulate the statistical validity of the principle of conservation of energy. The probabilities of positive and negative induced radiation (positive und negative Einstrahlung) are determined by the field of measurement, i.e. by the behavior of the surrounding atoms.

Thus, in the new model there are preserved, on the one hand, the fundamental quantum features of Bohr’s atom, necessary for the interpretation of spectral regularities, while on the other hand—thanks to the assumption of continuous emission—the framework of the correspondence principle is expanded, so that it becomes possible to judge the interaction between atom and radiation. Indeed, since the results of the action of the virtual field on the atom coincide with the classically calculated action of the field on the set of virtual vibrators equivalent to the atom, all conclusions of the classical theory concerning the passage of radiation through a medium with resonators remain in force, i.e. all the conclusions of the theory of refraction, reflection, and dispersion. (Kramers has already published a short letter in Nature, no. 2845, May 10, 1924, p. 673, devoted to the exposition of the theory of dispersion.)

In accordance with the new conceptions, the processes of absorption must also be set forth in different terms. The old expression said: absorption occurs as a result of transitions that increase the energy of the atom at the expense of the energy that has come from the field. The new view, in agreement with the classical one, sees in the weakening of light during absorption the result of interference between the incoming waves and the waves of corresponding frequency continuously emitted by the atoms. The transitions, however, are simply secondary phenomena—

¹) More precisely: the probability of a transition compensating this arbitrary radiation.

occurring in such a way that, in the aggregate and on the average, the law of conservation of energy is satisfied. The explanation of absorption by the presence of secondary coherent waves makes comprehensible the phenomenon, connected with absorption, of anomalous dispersion, as well as of selective “metallic” reflection (first observed in vapors by Wood).

What has been set forth shows how the new views are attempting to fill the gap between “discrete” and “continuous” optics. In this respect, of course, the theory is bearing such rich fruit that, probably, the price paid for them—the peculiar, statistical interpretation of the principle of conservation of energy—will not seem too high to many.

Unfortunately, in the field of optical phenomena there still remains a group of facts whose classical interpretation presented even greater difficulties. These are the phenomena that called into being and strengthened the hypothesis of light quanta: the photoelectric effect, the short-wavelength limit of the continuous X-ray spectrum, the Compton effect, and so on—phenomena justifying Einstein’s relation \(h\nu=\varepsilon V\).

The new theory of Bohr and his collaborators attempts to embrace these facts as well. However, their interpretation is associated with still greater difficulties. The interpretation of the Compton effect, for example, leads to the necessity of admitting that, alongside the principle of conservation of energy, the principle of conservation of momentum also has a statistical character (which, incidentally, follows quite naturally in view of the complete equivalence of these principles in a number of questions; cf., for example, Pauli Jr., Z. f. Ph. XVIII, p. 272, 1923). However, this assumption is insufficient. In attempting to explain the change in wavelength of scattered X-ray light by the Doppler phenomenon (as Compton and Debye have already done), Bohr finds it necessary to recognize that the velocity of the virtual vibrators does not coincide with the velocities of the illuminated electrons: the velocities of the vibrators are determined by the condition of the change in the frequency of the radiation, while the velocities of the electrons are determined by the requirement of the principle of conservation of momentum (understood statistically) (that in this case the virtual oscillator moves with a velocity different from that of the irradiated electrons themselves means, of course, a feature especially alien to classical ideas).

Thus, in the new conceptions, alongside a number of new ideas and perspectives, new difficulties and complications also reveal themselves. It is natural, therefore, that the new ideas have already found both supporters and opponents.¹

Thus the foundations of the new theory reduce to the following:

1. The fundamental quantum postulates remain in force (the existence of discrete stationary states, the condition for the frequency).

2. Radiation is regarded as a continuous process accompanying the atom’s stay in a stationary state (the moment of transition interrupts the radiation, but does not condition it, as was the case in the old model). Continuous radiation is reconciled with the existence of stationary states at the cost of reducing the principle of conservation of energy to the level of a statistical principle.

3. The correspondence principle is extended in the sense that, with its help, not only is the virtual field produced around the atom described, but the possibility also opens up of taking into account the interaction between the field and the atom, regarding the latter as equivalent to a set of virtual oscillators classically interacting with the virtual field.

Gr. Landsberg.

¹ According to a report by Academician A. F. Ioffe, who has just returned from abroad, in Germany Sommerfeld and Born have adopted Bohr’s point of view; the Berlin physicists, headed by Einstein, do not share it.