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American Journal of Physics

Boundary effects in quantum mechanics

David H. Berman

Published Oct 1, 1991• 29 citations
Publisher

Abstract

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.

DOI
10.1119/1.16676

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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.
A particle moves in one spatial dimension in an attractive symmetric negative potential vf(x), and obeys non-relativistic quantum mechanics. If the system is enclosed in a finite box then, unlike the situation in , the ground-state energy is negative only when the coupling v is sufficiently large. General upper- and lower-bound formulae are derived for the critical coupling which corresponds to . Some generalizations are derived for potentials which change sign and for unsymmetric unimodal potentials.
To clarify the relation of energy shifts to scattering phase shifts in one-body and many-body problems, we examine their relation in a number of different situations. We derive, for a particle in a container of arbitrary shape with a short-range scattering center, a general result for the energy eigenvalues in terms of the s-wave scattering phase shift and the eigenstates in the absence of the scatterer. We show that, while the energy shifts for a spherical container are proportional to the phase shift over large ranges, those for a cubic container have a more complicated behavior. We connect our result to the description of energy shifts in terms of the scattering T-matrix. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. We then consider the many-body problem for particles with a two-body interaction and show that the free energy change due to the interaction is proportional to an average of a generalized phase shift that includes the effects of the medium. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energy of a many-body system is not.