GENERAL QUANTUM THEORY So far we have been concerned with systems possessing only one degree of freedom, actually or virtually, and consequently only one quantum number to fix a stationary state. In 1915 W. Wilson and A. Sommerfeld succeeded, independently of one another, in finding appropriate rules or conditions for fixing the stationary states of systems of more than one degree of freedom. These quantum conditions may be illustrated by the aid of the type of atom we have already studied. For the sake of simplicity we shall ignore the motion of the massive nucleus and regard the centre of mass of the atom as situated in the nucleus itself—an assumption which in any case is very near the truth. The electron has, of course, three co-ordinates. Since, however, its orbit lies in a plane, we can choose our system of reference so that one of the three is a constant and we need pay no further attention to it. We have then a system which has virtually two degrees of free dom. The principle of energy gives us the equation where the same convention as before is used in fixing the arbi trary constant in the energy and where the symbols have again the same meaning, except that r, the distance of the electron from the attracting nucleus, is not in general the radius of a circular orbit. In polar co-ordinates, r and 0, the equation becomes where e is the eccentricity. The integers and n2 are called the radial and azimuthal quantum numbers respectively. It is easy to show that for a given energy, or a given value of the total quantum number all the possible ellipses in which the electron may travel have a major axis of the same length.
In fig. 5 are illustrated the ellipses for which the total quantum number is 4. For instance, the ellipse 4.3 has a radial quantum number and an azimuthal quantum number Bohr prefers to characterize an orbit by specifying its total and azi muthal quantum numbers. He employs the letter k for the latter number and his general symbol for an orbit is nk. The orbit 4.3 for example would be described by Bohr as a 43 orbit. The orbit
for which n2 or k is zero is ob viously not one of those in which the electron is permitted to travel, since it is a straight line through the nucleus. It may be remarked that the quantum conditions (41) are in agreement with Planck's hypothesis for a simple harmonic oscillator, since the area of the closed curve in fig. 2 is given by pdq.
We are now in a position to give a more general description of the quantum theory, or at any rate that part of it which is based on the quantum conditions (41). We shall denote the coordinates of a dynamical system by qi, q2, q3, . . . and so, as is usual, the corresponding generalized momenta by pi, P2, p3, If it happens, as it does in many important cases, that the system is conservative from the point of view of classical dynamics, or sufficiently nearly so to be treated as such, and if the coordinates can be so chosen that p, is a function of only, a function of q2 only and so on, the variables then being said to be separable, it can be shown that each q librates between definite limits and associated with each libration is a definite fundamental frequency of libration of the coordinate ql, v2 of and so on. Such systems are said to be conditionally periodic. Each of the phase integrals where, in each case, the integration is extended over the complete range of values of the corresponding q and back to its original value again, is a constant and we can express the energy of the system E, and also the other constants of integration, as functions of the J's, just as we did in the simple example described above; where, in general, there are as many J's as the system has degrees of freedom. The stationary states are now fixed by equations (41). We thus obtain an expression for the energy of a station ary state in terms of quantum integers of the constant Is and of constants which are inherent in the system, such as electronic charge and mass, etc., and we are in a position to calculate fre quencies of radiation by applying Bohr's postulate.