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Scientific Herald of Uzhhorod University Series Physics

The two Coulomb centers problem in quantum mechanics: influence of dimension

Denys I. Bondar, В. Ю. Лазур, І. М. Шваб

Published Dec 31, 2004• 0 citations
Publisher
DOI
10.24144/2415-8038.2004.16.38-53

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The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are: Nuclear separation 1.0 1.3 1.6 1.8 2.0 2.2 2.4 2.95 Electronic energy 2.896 2.648 2.436 2.309 2.204 2.109 2.025 1.836 The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. This simple problem of a particle in a quadratic potential energy is one of the typical problems of quantization. In two dimensions, the problem is more interesting, but not particularly more complicated. In three dimensions, people could extend the treatment to a three-dimensional harmonic oscillator, but a more important problem has spherical symmetry in the potential, and the latter is not the quadratic potential of the harmonic oscillator. It is of more interest to turn our attention to the motion of an electron about the nucleus of an atom.