New Quantum Theory

We cannot, in this article, deal with the manner in which problems connected with polarization, intensity, etc., are dealt Now it is clear that if q is increased by 2.7rR, i.e., if the ring is turned once round, the original condition of things is reproduced and we must, therefore, demand that the value of IA or more precisely the part of it depending on q, is unaltered. Therefore the increase of the argument of the cosine, namely, p • 27rR numerically, must be equal to 27rit where n is an integer, or p • nh. We recognize in this equation a case of the familiar quantizing conditions (40 which enter wave mechanics quite naturally and have not to be imposed in the arbitrary and ad hoc manner of the older theory. Since the kinetic energy 7- we have, just as in the earlier theory, where K = is the moment of inertia of the ring about the axis of spin.

In the case of a rigid body which is not in a field of force and can turn freely about a fixed point, we have two coordinates, qi and and their corresponding momenta, and the problem is much more complicated. It will suffice to say here that the de mand that tk must be one-valued now leads to the formula which, as we have pointed out before, is actually required by spectroscopists to explain the phenomena of bands and which the older quantum theory is wholly unable to produce.

In order to utilize the wave theory of mechanics to the fullest extent, we must resort, as we do in optics, to the differential equa tion of the state of undulation. Turning again to the example of the rigid body spinning about a fixed axis, the appropriate differ ential equation is This is the type of equation which represents plane waves travel ling in the direction q with the constant velocity is, or the type of equation which represents the vibrations of a cord. The appro priate solution, in the case of a cord fixed at both ends, for ex ample, consists of a sum of particular solutions so chosen as to satisfy the given initial or boundary conditions. The solution represents the motion of the cord as a superposition of vibrations consisting of a fundamental one and harmonic overtones, the fre quencies of which are integral multiples of the fundamental one. The difference between this old-fashioned problem and the one we have in hand lies, not in the differential equation or its solution, but in the character of the "boundary" conditions which have to be satisfied and which, in this case, lie in the requirement that 4, must be a one-valued function of q. Since u =E/p and = 2m (E —V), where V is the potential energy—the part of the energy not de pendent on the velocity and which in this example is constant, we have This is called the Schroedinger equation of the problem and the "boundary" conditions require that in the particular solution shall have one of the values : with by the wave mechanics. It will suffice to say that the new

theory accomplishes all that the older quantum theory does, and in a manner which is much more coherent and complete, and that it goes far beyond it. In quite recent times it has received remarkable confirmation from observations of Davison and Germer, G. P. Thomson and others, on the scattering and diffraction of electrons. These observations are, in the main, just what the theory predicts. A stream of electrons, each having a definite momentum p, constitutes, according to the theory, a beam of waves with the wave length X = h/p, and the investiga tions mentioned entirely bear out this prediction.

Matrix Mechanics.

This originated independently of wave mechanics, but its character can be described most conveniently with the help of the relationships between the two. We shall confine our attention to systems with one degree of freedom for which the classical equations of motion are: provided the energy H is expressed as a function of p, q and t. We are concerned with the case where the system is conservative, so that H is a function of p and q only. For example, in the case of a particle in simple harmonic motion where m is the mass of the particle andµ is the restoring force per unit displacement. These equations of Hamilton (59) are retained in matrix mechanics ; but the quantities involved in them, q, p, H are replaced by matrices. A matrix is a set of num bers arranged like the constituents of a determinant (q.v.). The product of two matrices is, by definition, the matrix 'obtained by applying the rule for multiplying determinants. This rule gives two products which are not, in general, identical, i.e., the con stituent in the mth row and nth column of one product is not generally the same as that in the corresponding place in the other product. The very essence of matrix mechanics is bound up with this ambiguity. (See DYNAMICS.) We have next to explain what sense is to be attached to the operations a a applied to matrices. The mn consti tuent of a typical matrix in the new mechanics is represented in the following way : where i=1,/— I and v,„„ is that frequency which, in the language of the older theory, is associated with a transition from the state d .