Non-relativistic Effective Quantum Mechanics of the Coulomb Interaction

TL;DR

This paper introduces an effective quantum mechanics framework using boundary conditions to model short-range effects, accurately predicting bound states in Coulomb systems while maintaining gauge invariance and applicability to various physical systems.

Contribution

It develops a novel effective quantum mechanics approach employing boundary conditions to encode short-range effects, bridging quantum defect theory with effective field theory concepts.

Findings

Accurately predicts Coulomb bound state energies with few parameters.

Equivalent to quantum defect theory but derived from an effective framework.

Respects gauge invariance and models short-range decay processes.

Abstract

We apply the ideas of effective field theory to nonrelativistic quantum mechanics. Utilizing an artificial boundary of ignorance as a calculational tool, we develop the effective theory using boundary conditions to encode short-ranged effects that are deliberately not modeled; thus, the boundary conditions play a role similar to the effective action in field theory. Unitarity is temporarily violated in this method, but is preserved on average. As a demonstration of this approach, we consider the Coulomb interaction and find that this effective quantum mechanics can predict the bound state energies to very high accuracy with a small number of fitting parameters. It is also shown to be equivalent to the theory of quantum defects, but derived here using an effective framework. The method respects electromagnetic gauge invariance and also can describe decays due to short-ranged interactions, such as those found in positronium. Effective quantum mechanics appears applicable for systems that admit analytic long-range descriptions, but whose short-ranged effects are not reliably or efficiently modeled. Potential applications of this approach include atomic and condensed matter systems, but it may also provide a useful perspective for the study of blackholes.