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American Journal of Physics

The role of boundary conditions in separation of variables: Quantum oscillator in a box

Anthony C. Tanner

Published Apr 1, 1991• 15 citations
Publisher

Abstract

The Schrödinger equation for two (or more) particles can sometimes be separated into two (or more) one-particle equations. However, the boundary conditions can recouple the variables even though the Schrödinger equation itself is separable.
DOI
10.1119/1.16544

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Separation of the Schrödinger equation for molecular dynamics into sets of variables can sometimes be performed when separation into individual variables is neither possible nor for certain purposes necesary. Sufficient conditions for such a separation are derived. They are the same as those found by Stäckel for the corresponding Hamilton—Jacobi problem, with an additional one which is the analog of the Robertson condition for one-dimensional sets. Expressions are also derived for operators whose eigenvalues are the separation constants. They provide a variational property for these constants. For use in aperiodic problems an expression is obtained for the probability current in curvilinear coordinates in an invariant form. Application of these results to reaction rate theory is made elsewhere.
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