The Dance of the Sheared Eigenfunctions

TL;DR

This paper investigates how shearing potentials in quantum mechanics affects eigenfunctions and spectra, providing new insights into spectral changes and the underlying physical processes involved.

Contribution

It introduces a detailed analysis of sheared potentials in quantum systems, focusing on eigenfunctions and spectra, which is rarely addressed in existing literature.

Findings

Sheared eigenfunctions offer deeper spectral insights.

Spectral changes relate to external work required.

Analysis applied to harmonic oscillator and |x| potential.

Abstract

In this work, we delve into the theory of sheared potentials in non-relativistic quantum mechanics. After defining what we mean by a family of sheared potentials, we consider these families in two particular but emblematic cases, the harmonic oscillator and the symmetric potential well proportional to $|x|$. In both cases, besides determining the spectra, we analyse the impact of the shearing process on the respective eigenfunctions. The latter analysis is typically left aside in the literature, but here we show that the sheared eigenfunctions yield insights that allow for a deeper understanding of the main features exhibited by the spectra. Finally, we make a few comments about the connection between the change in the spectra of a given sheared family and the necessary work that must be made by an external agent to implement such a change.