The hydrogen atom via the four-dimensional spherical harmonics

Abstract

Using the fact that the Schrödinger equation for the stationary states of the hydrogen atom is equivalent to an integral equation on the unit sphere in a four-dimensional space, the eigenvalues, the eigenfunctions, and a dynamical symmetry group for this problem are obtained from the four-dimensional spherical harmonics and the group of rotations on the sphere. It is shown that the four-dimensional spherical harmonics separable in Euler angles correspond to solutions of the time-independent Schrödinger equation that are separable in parabolic coordinates.