th state of the unperturbed atom the expressions
We must further assume that in the stationary states of the unperturbed system the form of the orbit is so far undetermined that the eccentricity can vary continuously. This is not only immediately indicated by the principle of correspondence,—since the frequency of revolution is determined only by the energy and not by the eccentricity,—but also by the fact that the presence of any small external forces will in general, in the course of time, produce a finite change in the position as well as in the eccentricity of the periodic orbit, while in the major axis it can produce only small changes proportional to the intensity of the perturbing forces.
In order to fix the stationary states of systems in the presence of a given conservative external field of force, we shall have to investigate, on the basis of the principle of correspondence, how these forces affect the decomposition of the motion into harmonic oscillations. Owing to the external forces the form and position of the orbit will vary continuously. In the general case these changes will be so complicated that it will not be possible to decompose the perturbed motion into discrete harmonic oscillations. In such a case we must expect that the perturbed system will not possess any sharply separated stationary states. Although each emission of radiation must be assumed to be monochromatic and to proceed according to the general frequency condition we shall therefore expect the final effect to be a broadening of the sharp spectral lines of the unperturbed system. In certain cases, however, the perturbations will be of such a regular character that the perturbed system can be decomposed into harmonic oscillations, although the ensemble of these oscillations will naturally be of a more complicated kind than in the unperturbed system. This happens, for example, when the variations of the orbit with respect to time are periodic. In this case harmonic oscillations will appear in the motion of the system the frequencies of which are equal to whole multiples of the period of the orbital perturbations, and in the spectrum to be expected on the basis of the ordinary theory of radiation we would expect components corresponding to these frequencies. According to the principle of correspondence we are therefore immediately led to the conclusion, that to each stationary state in the unperturbed system there corresponds a number of stationary states in the perturbed system in such a manner, that for a transition between two of these states a radiation is emitted, whose frequency stands in the same relationship to the periodic course of the variations in the orbit, as the spectrum of a simple periodic system does to its motion in the stationary states.
The Stark effect. An instructive example of the appearance of periodic perturbations is obtained when hydrogen is subjected to the effect of a homogeneous electric field. The eccentricity and the position of the orbit vary continuously under the influence of the field. During these changes, however, it is found that the centre of the orbit remains in a plane perpendicular to the direction of the electric force and that its motion in this plane is simply periodic. When the centre has returned to its starting point, the orbit will resume its original eccentricity and position, and from this moment the entire cycle of orbits will be repeated. In this case the determination of the energy of the stationary states of the disturbed system is extremely simple, since it is found that the period of the disturbance does not depend upon the original configuration of the orbits nor therefore upon the position of the plane in which the centre of the orbit moves, but only upon the major axis and the frequency of revolution. From a simple calculation it is found that the period a is given by the following formula
where
is the intensity of the external electric field. From analogy with the fixation of the distinctive energy values of a Planck oscillator we must therefore expect that the energy difference between two different states, corresponding to the same stationary state of the unperturbed system, will simply be equal to a whole multiple of the product of
