How Quantum Theory Connects to Radiation Theory

The ordinary theory of radiation predicts that we should expect the system to emit a spectrum consisting of a series of lines of frequencies equal to τω.

But this is just equal to the series of frequencies which we obtain from (13) by introducing different values for n 0 − n 00.

As far as the frequencies are concerned, we see therefore that in the limit where n is large there exists a close relation between the ordinary theory of radiation and the theory of spectra based on (1) and (10).

In the ordinary theory of radiation, radiations of the different frequencies τω corresponding to different values of τ are emitted or absorbed at the same time.

These frequencies, in my theory based on the assumption I and II, are connected with entirely different processes of emission or absorption. They correspond to the transition of the system from a given state to different neighbouring stationary states.

To connect it to the the ordinary theory of radiation in the limit of slow vibrations, I claim that a relation, as that just proved for the frequencies, will, in the limit of large n, hold also for the intensities of the different lines in the spectrum.

In ordinary electrodynamics, the intensities of the radiations corresponding to different values of τ are directly determined from the coefficients C|tau in (14).

We must expect that for large values of n these coefficients will on the quantum theory determine the probability of spontaneous transition from a given stationary state for which n = n 0 to a neighbouring state for which is

n = n
00 = n
0 − τ .

This connection between the amplitudes of the different harmonic vibrations into which the motion can be resolved, characterised by different values of τ , and the probabilities of transition from a given stationary state to the different neighbouring stationary states, characterised by different values of n 0 − n 00, may clearly be expected to be of a general nature.

Without a detailed theory of the mechanism of transition, we cannot obtain an exact calculation of the latter probabilities, unless n is large.

We may expect that also for small values of n the amplitude of the harmonic vibrations corresponding to a given value of τ will in some way give a measure for the probability of a transition between two states for which n 0 − n 00 is equal to τ.

Thus in general there will be a certain probability of an atomic system in a stationary state to pass spontaneously to any other state of smaller energy, but if for all motions of a given system the coefficients C in (14) are zero for certain values of τ , we are led to expect that no transition will be possible, for which n 0 − n 00 is equal to one of these values.

A simple illustration of these considerations is offered by the linear harmonic vibrator mentioned above in connection with Planck’s theory.

Since in this case Cτ is equal to zero for any τ different from 1, we shall expect that for this system only such transitions are possible in which n alters by one unit.

From (1) and (9) we obtain therefore the simple result that the frequency of any radiation emitted or absorbed by a linear harmonic vibrator is equal to the constant frequency ω0.

This result seems to be supported by observations on the absorption-spectra of diatomic gases.

It shows that certain strong absorption-lines, which according to general evidence may be ascribed to vibrations of the two atoms in the molecule relative to each other, are not accompanied by lines of the same order of intensity and corresponding to entire multipla of the frequency, such as it should be expected from (1) if the system had any considerable tendency to pass between non-successive states.

In the absorption spectra of some diatomic gases, faint lines occur corresponding to the double frequency of the main lines,1).

This is explained by assuming that for finite amplitudes, the vibrations are not exactly harmonic. Therefore, the molecules possess a small probability of passing also between non-successive states.