Bounds for Hamiltonians with arbitrary kinetic parts

TL;DR

This paper introduces a method to approximate eigenvalues of quantum Hamiltonians with arbitrary kinetic terms, providing bounds and a semiclassical interpretation that enhances understanding in solid state physics.

Contribution

It presents a novel approach to compute bounds for eigenvalues in quantum systems with arbitrary kinetic parts, including a semiclassical interpretation and validation through a toy model.

Findings

Approximate eigenvalues can be analytically determined as bounds.

The method offers a semiclassical interpretation of eigenvalues.

Validation with a Gaussian momentum dependence toy model confirms the approach.

Abstract

A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or upper bounds. A semiclassical interpretation of the generic formula obtained for the eigenvalues supports a new definition of the effective particle mass used in solid state physics. An analytical toy model with a Gaussian dependence in the momentum is studied in order to check the validity of the method.