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Proceedings of the Physical Society

Problem of Perturbed Boundary Condition in Quantum Mechanics

S Sen Gupta, H Mukhopadhyay

Published Feb 1, 1958• 2 citations
Publisher

Abstract

Problem of perturbed boundary condition is reduced to an equivalent problem of perturbation potential with the introduction of a deformed coordinate system. The method is applied to determine the energy of a nucleon in a rigid ellipsoidal box.
DOI
10.1088/0370-1328/71/2/304

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A simple formula is presented for the change in energy of a quantum system in a box when the box is infinitesimally deformed. The result is applied to rederive a formula given by Barton et al., [Am. J. Phys. 58, 751–755 (1990)] for the difference in energy of a bound state calculated with and without box boundary conditions. The formula can also be used to discuss pressure in a quantum system. A generalization of the result has been used for calculating scattering amplitudes for waves scattered by a rough extended surface. Here, a similar manifestly reciprocal result is derived for scattering by bounded objects.
The authors propose a new method of solving the Schrodinger equation for a quantum system enclosed in a box with infinite potential walls. The method combines the variational technique with boundary perturbation theory and can be applied to a general non-separable case. When the Schrodinger equation for an enclosed system separates into an ordinary differential equation, the method gives the exact energy and wavefunction. As an application of the method the authors obtain the ground-state energy of the hydrogen atom placed off-centre in a spherical cavity. A generalisation of the variational boundary perturbation technique for finite potential walls is suggested.
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