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American Journal of Physics

Accidental Degeneracy of the Coulomb Interaction

R. Aldrovandi, J. A. C. Alcaras, Paulo Leal Ferreira

Published Jun 1, 1967• 2 citations
Publisher

Abstract

A treatment of the bound states for the hydrogen atom is given, based on the Schrödinger equation in momentum representation. The work runs parallel to Fock's treatment of the integral equation in momentum space. The problem is reduced to an eigenvalue equation for the Laplace-Beltrami operator in a momentum space of constant curvature, from which Balmer formula and the eigenfunctions are easily derived.

DOI
10.1119/1.1974161

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A treatment of the bound states for the hydrogen atom is given, based on the Schrödinger equation in momentum representation. The work runs parallel to Fock's treatment of the integral equation in momentum space. The problem is reduced to an eigenvalue equation for the Laplace-Beltrami operator in a momentum space of constant curvature, from which Balmer formula and the eigenfunctions are easily derived.
A differential equation in momentum space is derived for the case of an attractive Coulomb potential. The bound-state energies and the momentum eigenfunctions are shown to arise from this differential equation in a simple fashion. The zero-energy partial-wave momentum eigenfunctions are also derived. The s-wave bound states and their momentum eigenfunctions in a three-dimensional linear potential and an attractive 1/(r+ beta ) potential are derived by similar techniques. The connection between the quantisation formulae in these potentials and the classical action integral in momentum space is explored.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The Schrödinger equation does not account for the 2n2 degeneracy of the hydrogen atom, which it dismisses as an “accidental” degeneracy. The factor of “2” in the 2n2 degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n2 degeneracy is nevertheless not quite an “accident”; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the n2 degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the n2 degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrödinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrödinger equation.
We consider Fock's fundamental theory of the hydrogen atom in momentum space which allows a realization of the previously predicted rotation group of a three-dimensional (3D) sphere in four-dimensional (4D) space. We then modify Fock's theory and abandon the momentum space description. To transform and simplify the theory, we use invariant tensor methods of electrostatics in 3D and 4D spaces. We find a coordinate 4D space where the Schrodinger equation becomes the 4D Laplace equation. The transition from harmonic 4D polynomials to original 3D physical space is algebraic and involves derivatives with respect to a coordinate that is interpreted as time. We obtain a differential equation for eigenfunctions in the momentum space and find its solutions. A concise calculation of the quadratic Stark effect is given. The Schwinger resolvent is derived by the method of harmonic polynomials. Vector ladder operators are also considered.