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Journal of Mathematical Physics

Separable Schrödinger Equations for Two Interacting Particles in External Fields

Ruedi Seiler, Richard H. Pratt

Published Jul 1, 1970• 0 citations
Publisher

Abstract

We derive the most general form of the Schrödinger equation for two interacting particles in an external field which can be separated by a linear transformation of the coordinates. In particular, the transformation to center-of-mass and relative coordinates is shown to separate only those Hamiltonians with external potentials of the 3-dimensional harmonic oscillator type. The most general allowed form contains in addition specified cubic terms in the potentials.

DOI
10.1063/1.1665381

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We derive the most general form of the Schrödinger equation for two interacting particles in an external field which can be separated by a linear transformation of the coordinates. In particular, the transformation to center-of-mass and relative coordinates is shown to separate only those Hamiltonians with external potentials of the 3-dimensional harmonic oscillator type. The most general allowed form contains in addition specified cubic terms in the potentials.
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The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.
For pt.II see Proc. R. Soc. London, Ser. A, vol.384, no.1786, p.89-105 (1982). A procedure for solving the few-particle Schrodinger equation exactly is applied to a model system consisting of two identical particles and a massive third particle. The type of interaction potential is not specified except that it should not diverge more rapidly than r -2 at the particle positions. Allowable interactions include the Coulomb and the harmonic oscillator potentials. The principles are illustrated by reference to the spatially symmetric states of the system. The solution has the form of a multipole expansion in spherical polar coordinates. The radial dependence for each multipole component is defined by a power series including logarithmic terms. Explicit expressions are given for the coefficients in the expansion. Exact coefficients are obtained for low-order terms in the wavefunction of a two-electron atom.
We present simple explicit coordinate transformations which serve to decouple the Schrödinger equation for a pair of (not necessarily identical) harmonic oscillators in the presence of bilinear perturbing potentials. We derive general conditions for the decoupling, and give some examples of physical interest. These include the much studied example with just a static perturbation, the parallel problem with a dynamic coupling term, and the classic example of an isotropic two-dimensional oscillator in a transverse magnetic field, first solved by Fock (1928) by separation of variables.