A general comparison theorem

TL;DR

This paper establishes a general comparison theorem for eigenvalues of quantum systems using the Hellmann-Feynman theorem, relating potential and kinetic energy bounds to eigenvalue ordering.

Contribution

It introduces a broad comparison theorem for eigenvalues of Hamiltonians with position-dependent potentials and momentum-dependent kinetic terms, extending prior specific results.

Findings

Eigenvalues are ordered if potentials are ordered pointwise.

Eigenvalues are ordered if kinetic terms are ordered pointwise.

Analytical applications demonstrate the theorem's utility.

Abstract

Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form $(T+V)|\psi> = E|\psi>$, where $T$ is a kinetic part which depends only on momentums and $V$ is a potential which depends only on positions. We assume that $H^{(1)}=T+V^{(1)}$ and $H^{(2)}=T+V^{(2)}$ ($H^{(1)}=T^{(1)}+V$ and $H^{(2)}=T^{(2)}+V$) support both discrete eigenvalues $E^{(1)}_{\{\alpha\}}$ and $E^{(2)}_{\{\alpha\}}$, where ${\{\alpha\}}$ represents a set of quantum numbers. We prove that, if $V^{(1)} \le V^{(2)}$ ($T^{(1)} \le T^{(2)}$) for all position (momentum) variables, then the corresponding eigenvalues are ordered $E^{(1)}_{\{\alpha\}} \le E^{(2)}_{\{\alpha\}}$. Some analytical applications are given.