The Elementary Dynamical Law for The Vibrations of an Ideal Oscillator. Hypothesis of Emission of Quanta

144. All that precedes has been by way of preparation. Before taking the final step, which will lead to the law of distribution of energy in the spectrum of black radiation, let us briefly put together the essentials of the problem still to be solved. As we have already seen in Sec. 93, the whole problem amounts to the determination of the temperature corresponding to a monochromatic radiation of given intensity. For among all conceivable distributions of energy the normal one, that is, the one peculiar to black radiation, is characterized by the fact that in it the rays of all frequencies have the same temperature. But the temperature of a radiation cannot be determined unless it be brought into thermodynamic equilibrium with a system of molecules or oscillators, the temperature of which is known from other sources. For if we did not consider any emitting and absorbing matter there would be no possibility of defining the entropy and temperature of the radiation, and the simple propagation of free radiation would be a reversible process, in which the entropy and temperature of the separate pencils would not undergo any change. (Compare below Sec. 166.)

Now we have deduced in the preceding section all the characteristic properties of the thermodynamic equilibrium of a system of ideal oscillators. Hence, if we succeed in indicating a state of radiation which is in thermodynamic equilibrium with the system of oscillators, the temperature of the radiation can be no other than that of the oscillators, and therewith the problem is solved.

145. Accordingly we now return to the considerations of Sec. 135 and assume a system of ideal linear oscillators in a stationary field of radiation. In order to make progress along the line proposed, it is necessary to know the elementary dynamical law, according to which the mutual action between an oscillator and the incident radiation takes place, and it is moreover easy to see that this law cannot be the same as the one which the classical electrodynamical theory postulates for the vibrations of a linear Hertzian oscillator. For, according to this law, all the oscillators, when placed in a stationary field of radiation, would, since their properties are exactly similar, assume the same energy of vibration, if we disregard certain irregular variations, which, however, will be smaller, the smaller we assume the damping constant of the oscillators, that is, the more pronounced their natural vibration is. This, however, is in direct contradiction to the definite discrete values of the distribution densities $w_1$, $w_2$, $w_3,\dots$ which we have found in Sec. 139 for the stationary state of the system of oscillators. The latter allows us to conclude with certainty that in the dynamical law to be established the quantity element of action $h$ must play a characteristic part. Of what nature this will be cannot be predicted a priori; this much, however, is certain, that the only type of dynamical law admissible is one that will give for the stationary state of the oscillators exactly the distribution densities $w$ calculated previously. It is in this problem that the question of the dynamical significance of the quantum of action $h$ stands for the first time in the foreground, a question the answer to which was unnecessary for the calculations of the preceding sections, and this is the principal reason why in our treatment the preceding section was taken up first.

146. In establishing the dynamical law, it will be rational to proceed in such a way as to make the deviation from the laws of classical electrodynamics, which was recognized as necessary, as slight as possible. Hence, as regards the influence of the field of radiation on an oscillator, we follow that theory closely. If the oscillator vibrates under the influence of any external electromagnetic field whatever, its energy $U$ will not in general remain constant, but the energy equation (205a) must be extended to include the work which the external electromagnetic field does on the oscillator, and, if the axis of the electric doublet coincides with the $z$-axis, this work is expressed by the term $E_{z}df=E_zḟdt$. Here $E_z$ denotes the $z$ component of the external electric field-strength at the position of the oscillator, that is, that electric field-strength which would exist at the position of the oscillator, if the latter were not there at all. The other components of the external field have no influence on the vibrations of the oscillator.

Hence the complete energy equation reads:

or:

and the energy absorbed by the oscillator during the time element $dt$ is:

147. While the oscillator is absorbing it must also be emitting, for otherwise a stationary state would be impossible. Now, since in the law of absorption just assumed the hypothesis of quanta has as yet found no room, it follows that it must come into play in some way or other in the emission of the oscillator, and this is provided for by the introduction of the hypothesis of emission of quanta. That is to say, we shall assume that the emission does not take place continuously, as does the absorption, but that it occurs only at certain definite times, suddenly, in pulses, and in particular we assume that an oscillator can emit energy only at the moment when its energy of vibration, $U$, is an integral multiple $n$ of the quantum of energy, $\epsilon =h\nu$. Whether it then really emits or whether its energy of vibration increases further by absorption will be regarded as a matter of chance. This will not be regarded as implying that there is no causality for emission; but the processes which cause the emission will be assumed to be of such a concealed nature that for the present their laws cannot be obtained by any but statistical methods. Such an assumption is not at all foreign to physics; it is, e.g., made in the atomistic theory of chemical reactions and the disintegration theory of radioactive substances.

It will be assumed, however, that if emission does take place, the entire energy of vibration, $U$, is emitted, so that the vibration of the oscillator decreases to zero and then increases again by further absorption of radiant energy.

It now remains to fix the law which gives the probability that an oscillator will or will not emit at an instant when its energy has reached an integral multiple of $\epsilon$. For it is evident that the statistical state of equilibrium, established in the system of oscillators by the assumed alternations of absorption and emission will depend on this law; and evidently the mean energy $U$ of the oscillators will be larger, the larger the probability that in such a critical state no emission takes place. On the other hand, since the mean energy $U$ will be larger, the larger the intensity of the field of radiation surrounding the oscillators, we shall state the law of emission as follows: The ratio of the probability that no emission takes place to the probability that emission does take place is proportional to the intensity $I$ of the vibration which excites the oscillator and which was defined in equation (158). The value of the constant of proportionality we shall determine later on by the application of the theory to the special case in which the energy of vibration is very large. For in this case, as we know, the familiar formulæ of the classical dynamics hold for any period of the oscillator whatever, since the quantity element of action $h$ may then, without any appreciable error, be regarded as infinitely small.

These statements define completely the way in which the radiation processes considered take place, as time goes on, and the properties of the stationary state. We shall now, in the first place, consider in the second chapter the absorption, and, then, in the third chapter the emission and the stationary distribution of energy, and, lastly, in the fourth chapter we shall compare the stationary state of the system of oscillators thus found with the thermodynamic state of equilibrium which was derived directly from the hypothesis of quanta in the preceding part. If we find them to agree, the hypothesis of emission of quanta may be regarded as admissible.

It is true that we shall not thereby prove that this hypothesis represents the only possible or even the most adequate expression of the elementary dynamical law of the vibrations of the oscillators. On the contrary I think it very probable that it may be greatly improved as regards form and contents. There is, however, no method of testing its admissibility except by the investigation of its consequences, and as long as no contradiction in itself or with experiment is discovered in it, and as long as no more adequate hypothesis can be advanced to replace it, it may justly claim a certain importance.