states in the first row—are not simple, but that for each place in this row there are two stationary states. The energy values differ so little from one another that it is impossible to represent them in the figure as separate dots. The emission (and absorption) of the two components of the yellow line are, therefore, connected with two different processes. This was beautifully shown by some later researches of Wood and Dunoyer. They found that if sodium vapour is subjected to radiation from only one of the two components of the yellow line, the resonance radiation, at least at low pressures, consists only of this component. These experiments were later continued by Strutt, and were extended to the case where the exciting line corresponded to the second line in the principal series. Strutt found that the resonance radiation consisted apparently only to a small extent of light of the same frequency as the incident light, while the greater part consisted of the familiar yellow line. This result must appear very astonishing on the ordinary ideas of resonance, since, as Strutt pointed out, no rational connection exists between the frequencies of the first and second lines of the principal series. It is however easily explained from our point of view. From the figure it can be seen that when an atom has been transferred into the second state in the second row, in addition to the direct return to the normal state, there are still two other transitions which may give rise to radiation, namely the transitions to the second state in the first row and to the first state in the third row. The experiments seem to indicate that the second of these three transitions is most probable, and I shall show later that there is some theoretical justification for this conclusion. By this transition, which results in the emission of an infra-red line which could not be observed with the experimental arrangement, the atom is taken to the second state of the first row, and from this state only one transition is possible, which again gives an infra-red line. This transition takes the atom to the first state in the second row, and the subsequent transition to the normal state then gives rise to the yellow line. Strutt discovered another equally surprising result, that this yellow resonance radiation seemed to consist of both components of the first line of the principal series, even when the incident light consisted of only one component of the second line of the principal series. This is in beautiful agreement with our picture of the phenomenon. We must remember that the states in the first row are simple, so when the atom has arrived in one of these it has lost every possibility of later giving any indication from which of the two states in the second row it originally came.
Sodium vapour, in addition to the absorption corresponding to the lines of the principal series, exhibits a selective absorption in a continuous spectral region beginning at the limit of this series and extending into the ultra-violet. This confirms in a striking manner our assumption that the absorption of the lines of the principal series of sodium results in final states of the atom in which one of the electrons revolves in larger and larger orbits. For we must assume that this continuous absorption corresponds to transitions from the normal state to states in which the electron is in a position to remove itself infinitely far from the nucleus. This phenomenon exhibits a complete analogy with the photoelectric effect from an illuminated metal plate in which, by using light of a suitable frequency, electrons of any velocity can be obtained. The frequency, however, must always lie above a certain limit connected according to Einstein's theory in a simple manner with the energy necessary to bring an electron out of the metal.
This view of the origin of the emission and absorption spectra has been confirmed in a very interesting manner by experiments on the excitation of spectral lines and production of ionization by electron bombardment. The chief advance in this field is due to the well-known experiments of Franck and Hertz. These investigators obtained their first important results from their experiments on mercury vapour, whose properties particularly facilitate such experiments. On account of the great importance of the results, these experiments have been extended to most gases and metals that can be obtained in a gaseous state. With the aid of the figure I shall briefly illustrate the results for the case of sodium vapour. It was found that the electrons upon colliding with the atoms were thrown back with undiminished velocity when their energy was less than that required to transfer the atom from the normal state to the next succeeding stationary state of higher energy value. In the case of sodium vapour this means from the first state in the first row to the first state in the second row. As soon, however, as the energy of the electron reaches this critical value, a new type of collision takes place, in which the electron loses all its kinetic energy, while at the same time the vapour is excited and emits a radiation corresponding to the yellow line. This is what would be expected, if by the collision the atom was transferred from the normal state to the first one in the second row. For some time it was uncertain to what extent this explanation was correct, since in the experiments on mercury vapour it was found that, together with the occurrence of non-elastic impacts, ions were always formed in the vapour. From our figure, however, we would expect ions to be produced only when the kinetic energy of the electrons is sufficiently great to bring the atom out of the normal state to the common limit of the states. Later experiments, especially by Davis and Goucher, have settled this point. It has been shown that ions can only be directly produced by collisions when the kinetic energy of the electrons corresponds to the limit of the series, and that the ionization found at first was an indirect effect arising from the photoelectric effect produced at the metal walls of the apparatus by the radiation arising from the return of the mercury atoms to the normal state. These experiments provide a direct and independent proof of the reality of the distinctive stationary states, whose existence we were led to infer from the series spectra. At the same time we get a striking impression of the insufficiency of the ordinary electrodynamical and mechanical conceptions for the description of atomic processes, not only as regards the emission of radiation but also in such phenomena as the collision of free electrons with atoms.
III. DEVELOPMENT OF THE QUANTUM THEORY OF SPECTRA
We see that it is possible by making use of a few simple ideas to obtain a certain insight into the origin of the series spectra. But when we attempt to penetrate more deeply, difficulties arise. In fact, for systems which are not simply periodic it is not possible to obtain sufficient information about the motions of these systems in the stationary states from the numerical values of the energy alone; more determining factors are required for the fixation of the motion. We meet the same difficulties when we try to explain in detail the characteristic effect of external forces upon the spectrum of hydrogen. A foundation for further advances in this field has been made in recent years through a development of the quantum theory, which allows a fixation of the stationary states not only in the case of simple periodic systems, but also for certain classes of non-periodic systems. These are the conditionally periodic systems whose equations of motion can be solved by a "separation of the variables." If generalized coordinates are used the description of the motion of these systems can be reduced to the consideration of a number of generalized "components of motion." Each of these corresponds to the change of only one of the coordinates and may therefore in a certain sense be regarded as "independent." The method for the fixation of the stationary states consists in fixing the motion of each of these components by a condition, which can be considered as a direct generalization of condition (1) for a Planck oscillator, so that the stationary states are in general characterized by as many whole numbers as the number of the degrees of freedom which the system possesses. A considerable number of physicists have taken part in this development of the quantum theory, including Planck himself. I also wish to mention the important contribution made by Ehrenfest to this subject on the limitations of the applicability of the laws of mechanics to atomic processes. The decisive advance in the application of the quantum theory to spectra, however, is due to Sommerfeld and his followers. However, I shall not further discuss the systematic form in which these authors have presented their results. In a paper which appeared some time ago in the Transactions of the Copenhagen Academy, I have shown that the spectra, calculated with the aid of this method for the fixation of the stationary states, exhibit a correspondence with the spectra which should correspond to the motion of the system similar to that which we have already considered in the case of hydrogen. With the aid of this general correspondence I shall try in the remainder of this lecture to show how it is possible to present the theory of series spectra and the effects produced by external fields of force upon these spectra in a form which may be considered as the natural generalization of the foregoing considerations. This form appears to me to be especially suited for future work in the theory of spectra, since it allows of an immediate insight into problems for which the methods mentioned above fail on account of the complexity of the motions in the atom.
Effect of external forces on the hydrogen spectrum. We shall now proceed to investigate the effect of small perturbing forces upon the spectrum of the simple system consisting of a single electron revolving about a nucleus. For the sake of simplicity we shall for the moment disregard the variation of the mass of the electron with its velocity. The consideration of the small changes in the motion due to this variation has been of great importance in the development of Sommerfeld's theory which originated in the explanation of the fine structure of the hydrogen lines. This fine structure is due to the fact, that taking into account the variation of mass with velocity the orbit of the electron deviates a little from a simple ellipse and is no longer exactly periodic. This deviation from a Keplerian motion is, however, very small compared with the perturbations due to the presence of external forces, such as occur in experiments on the Zeeman and Stark effects. In atoms of higher atomic number it is also negligible compared with the disturbing effect of the inner electrons on the motion of the outer electron. The neglect of the change in mass will therefore have no important influence upon the explanation of the Zeeman and Stark effects, or upon the explanation of the difference between the hydrogen spectrum and the spectra of other elements.
We shall therefore as before consider the motion of the unperturbed hydrogen atom as simply periodic and inquire in the first place about the stationary states corresponding to this motion. The energy in these states will then be determined by expression (7) which was derived from the spectrum of hydrogen. The energy of the system being given, the major axis of the elliptical orbit of the electron and its frequency of revolution are also determined. Substituting in formulae (7) and (8) the expression for
given in (12), we obtain for the energy, major axis and frequency of revolution in the
