多次元空間における角運動量の量子化について(第1報)

Abstract

In general quantum-mechanical investigations, the problem of the degeneracy of energy spectra plays an important role. This degeneracy is associated with the symmetry conditions under which the Hamiltonian is invariant, and is related not only to the 3-dimensional full rotationreflection group and the symmetric permutation group which produces the so-called permutation degeneracy, but also to the symmetry group in a broad sense. The latter type of degeneracy is based upon the fact that the Schrödinger equation for the hydrogen atom actually has the symmetry of the 4-dimensional rotation group for the negative energy (bound) states, and the symmetry of the Lorentz group for the positive energy (continuous) states. The degeneracy of the system with respect to the quantum number l is due to the invariance under these wider groups, of which the 3-dimensional rotations form a sub-group.1, 2) In this case there is an attempt of generalization of the degeneracy problem, which is related to the existence of groups of contact transformations, developing the correspondence between the transformations in classical and quantum theories.3) We have attempted to find a fundamental foundation of this problem, and it is a purpose of the present paper to give one of the preliminaries to them. In the first part the representations of the infinitesimal rotations in the multi-dimensional space have been obtained, and in the second part of the present title the energy levels of the 4- and 5-dimensional isotropic oscillators will be determined in relation to the problem of degeneracy, and it will be shown that the symmetry group of the Schrödinger equation for the n-dimensional oscillator is isomorphic to the n-dimensional unimodular unitary group with (n2--1)-parameters. The essential part of our investigations will successively appear after this second part.