Force, quantum mechanics and approximate energy eigenstates

TL;DR

This paper explores the use of force concepts in quantum mechanics to improve approximate calculations of stationary states, offering a force-based variational principle and applications to semiclassical mechanics.

Contribution

It introduces a novel force-based approach to approximate energy eigenstates, including a variational principle applicable beyond traditional energy extremization methods.

Findings

Force differentiation helps identify nodal structures quickly.

Force-based minimization provides a new variational principle for bound states.

Applicable to semiclassical and Siegert states where traditional methods fail.

Abstract

The prevalent role of force in traditional quantum mechanics is outlined, with special reference to approximate calculations for stationary states. It will be explored how far this force concept can be made useful in the concerned area. The basic idea is to differentiate the Schroedinger stationary equation once. Thus, one can eliminate the unknown energy as well, and then examine how a force-based approach can be beneficial in providing quickly the nodal information and in assessing the quality of an approximate function. Further, it will be demonstrated how the minimization of a suitable quantity derived from force may constitute a variational principle for bound states. The strategy applies also to Siegert states where traditional energy extremization principle ceases to work. Additionally, the utility of the force concept in semiclassical mechanics will be investigated.