Comparing energy levels in isotropic and anisotropic potentials

TL;DR

This paper investigates the relationship between energy levels in isotropic and anisotropic potentials, establishing inequalities that connect their bound-state energies, which provides qualitative insights into quantum systems.

Contribution

It introduces inequalities relating the ground and first excited state energies of anisotropic potentials to their spherical averages, offering new qualitative tools.

Findings

Ground-state energies satisfy specific inequalities.

First excited states also satisfy similar inequalities.

Provides qualitative bounds for energy levels in quantum systems.

Abstract

Qualitative information about the quantized energy levels of a system can be of great value. We study the relationship between the bound-state energies of an anisotropic potential and those of its spherical average. It is shown that the two ground-state energies satisfy an inequality, and there is a similar inequality for the first excited states.