THE QUANTUM THEORY Rydberg in his famous investigation of line spectra more than 3o years ago was able to analyse in a similar way many spectra of other elements. Just as in the case of hydrogen he found that the frequencies of a line-spectrum (such as that of sodium) could be represented by a formula of the type where T". T' can be approximately represented by aK is a constant for any one series, but takes different values
a, ... for the different series, while n takes a set of succes sive integral values. R is constant throughout for all spectra, and is the same constant as that appearing in (1) ; it is generally called the "Rydberg number." In many spectra the terms of most series are multiple, i.e., the terms which we consider as forming a series do actually form two, three or more series cor responding to two, three or more slightly different values of aK. Rydberg also discovered that the spectra of elements occupying homologous positions in the periodic table were very similar to each other, a similarity which is especially pronounced as regards the multiplicity of the terms.
The study of X-ray spectra made pos sible by the work of Laue and Bragg brought out relations of a still simpler kind between different elements. Thus Moseley (q.v.) in 1913 made the fundamental discovery that the X-ray spectra of all elements show a striking similarity in their struc ture, and that the frequencies of corresponding lines depend in a very simple way on the ordinal number of the element in the periodic table. Moreover the structure of these spectra was very like that of the hydrogen spectrum. The frequency of one of the strongest X-ray lines for the various elements could, for instance, be given approximately by The discovery of the electron and of the nucleus was based on experiments, the interpretation of which rested on applica tions of the classical laws of electrodynamics. As soon, how ever, as an attempt is made to apply these laws to the interaction of the particles within the atom, in order to account for the phys ical and chemical properties of the elements, we are• confronted with serious difficulties. Consider the case of an atom containing one electron : it is evident that an electrodynamical system con sisting of a positive nucleus and a single electron will not exhibit the peculiar stability of an actual atom. Even if the electron might be assumed to describe an elliptical orbit with the nucleus in one of the foci, there would be nothing to fix the dimensions of the orbit, so that the magnitude of the atom would be an un determined quantity. Moreover, according to the classical theory the revolving electron would continually radiate energy in the form of electromagnetic waves of changing frequency and the elec tron would finally fall into the nucleus. In short, all the promising results of the classical electronic theory of matter would seem at first sight to have become illusory. It has nevertheless been possible to develop a coherent atomic theory based on this pic ture of the atom by the introduction of the concepts which formed the basis of the famous theory of temperature radiation developed by Planck in 190o.
This theory marked a complete departure from the ideas which had hitherto been applied to the explanation of natural phenomena, in that it ascribed to the atomic processes a certain element of discontinuity of a kind quite foreign to the laws of
physics. One of its outstanding features is the appear ance in the formulation of physical laws of a new universal con stant, the so-called Planck's constant, which has the dimen sions of energy multiplied by time, and which is often called the "elementary quantum of action." We shall not enter upon the form which the quantum theory exhibited in Planck's original investigations, or on the important theories developed by Ein stein in 1905, in which the fertility of Planck's ideas in explaining various physical phenomena was shown in an ingenious way. We shall proceed at once to explain the form in which it has been possible to apply the quantum theory to the problem of atomic constitution. This rests upon the following two postulates: 1. An atomic system is stable only in a certain set of states, the "stationary states," which in general corresponds to a dis crete sequence of values of the energy of the atom. Every change in this energy is associated with a complete "transition" of the atom from one stationary state to another.
The power of the atom to absorb and emit radiation is gov erned by the law that the radiation associated with a transition must be monochromatic and of frequency v such that by =
E2 (6) where h is Planck's constant and
and E2 are the energies in the two stationary states concerned.
The first of these postulates aims at a definition of the in herent stability of atomic structures, manifested so clearly in a great number of chemical and physical phenomena. The second postulate, which is closely related to Einstein's law of the photo electric effect, offers a basis for the interpretation of line spec tra; it explains directly the fundamental spectral law expressed by relation (2) . We see in fact that the spectral terms appearing in this relation can be identified with the energy values of the stationary states divided by h. This view of the origin of spectra has been found to agree with the experimental results obtained in the excitation of radiation. This is shown especially in the dis covery of Franck and Hertz relating to impacts between free electrons and atoms. They found that an energy transfer from the electron to the atom can take place only in amounts which correspond with the energy differences of the stationary states as computed from the spectral terms.
From the Balmer formula (r) and the quantum theory postulates, it follows that the hydrogen atom has a single sequence of stationary states, the numerical value of the energy in the nth state being
Applying this result to the nuclear model of the hydrogen atom, we may as sume that this expression represents the work necessary to re move the electron from the nth state to an infinite distance from the nucleus. If the interaction of the atomic particles is to be explained upon the laws of classical mechanics, the electron in any one of the stationary states must move in an elliptical orbit about the nucleus as focus, with a major axis whose length is pro portional to
The state for which n is equal to i may be con sidered as the normal state of the atom, the energy then being a minimum. For this state the major axis is found to be approxi mately
centimetres. It is satisfactory that this is of the same order of magnitude as the atomic dimensions derived from experiments of various kinds. It is clear, however, from the na ture of the postulates, that such a mechanical picture of the sta tionary states can have only a symbolic character. This is per haps most clearly manifested by the fact that the frequencies of the orbital revolution in these pictures have no direct connection with the frequencies of the radiation emitted by the atom. Nevertheless, the attempts at visualizing the stationary states by mechanical pictures have brought to light a far-reaching analogy between the quantum theory and the classical theory. This an alogy was traced by examining the radiation processes in the limit where successive stationary states differ comparatively little from each other. Here it was found that the frequencies associated with the transition from any state to the next succeeding one tend to coincide with the frequencies of revolution in these states, if the Rydberg constant appearing in the Balmer formula (I) is given by the following expression: '(7) where e and m are the charge and mass of the electron and h is Planck's constant. This relation is actually found to be fulfilled within the limits of the experimental errors involved in the meas urements of e, m and li, and seems to establish a definite relation between the spectrum and the atomic model of hydrogen.
The considerations just men tioned constitute an example of the application of the so-called "correspondence principle" which has played an important part in the development of the theory. This principle gives expression to the endeavour, in the laws of the atom, to trace the analogy with classical electrodynamics as far as the peculiar character of the quantum theory postulates permits. On this line much work has been done in the last few years, and quite recently in the hands of Heisenberg has resulted in the formulation of a rational quan tum kinematics and mechanics. In this theory the concepts of the classical theories are from the outset transcribed in a way appropriate to the fundamental postulates and every direct refer ence to mechanical pictures is discarded. Heisenberg's theory constitutes a bold departure from the classical way of describing natural phenomena but may count as a merit that it deals only with quantities open to direct observation. This theory has already given rise to various interesting and important results, and it has in particular allowed the Balmer formula to be derived without any arbitrary assumptions as to the nature of the station ary states. However, the methods of quantum mechanics have not yet been applied to the problem of the constitution of atoms con taining several electrons, and in what follows we are reduced to a discussion of results which have been derived by using mechani cal pictures of the stationary states. Although in this way a .rig orous quantitative treatment is not obtainable it has nevertheless been possible, with the guidance of the correspondence principle, to obtain a general insight into the problem of atomic constitution.
