Calculating energy shifts in terms of phase shifts

TL;DR

This paper explores the relationship between energy shifts and scattering phase shifts in various quantum systems, deriving general formulas and extending them to many-body problems and different potential configurations.

Contribution

It provides a unified derivation of energy shifts in terms of phase shifts for both single-particle and many-body quantum systems, including complex geometries and interactions.

Findings

Energy shifts in spherical containers are proportional to phase shifts.

Cubic containers exhibit more complex energy shift behavior.

Many-body free energy changes relate to averaged phase shifts.

Abstract

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.