Supersymmetry and eigensurface topology of the spherical quantum pendulum

TL;DR

This paper combines analytic and computational methods, including supersymmetric quantum mechanics, to study the eigenenergy surfaces and topology of a spherical quantum pendulum under combined interactions, revealing conditions for exact solvability and implications for molecular spectra.

Contribution

It introduces conditions for the analytical solvability of the spherical quantum pendulum using SUSY QM and explores the topological features of its eigenenergy surfaces.

Findings

Identified loci where the quantum pendulum problem is analytically solvable.

Discovered that the topological index is independent of the eigenstate.

Implications for molecular rotational spectra and dynamics.

Abstract

We undertook a mutually complementary analytic and computational study of the full-fledged spherical (3D) quantum rotor subject to combined orienting and aligning interactions characterized, respectively, by dimensionless parameters $\eta$ and $\zeta$. By making use of supersymmetric quantum mechanics (SUSY QM), we found two sets of conditions under which the problem of a spherical quantum pendulum becomes analytically solvable. These conditions coincide with the loci $\zeta=\frac{\eta^2}{4k^2}$ of the intersections of the eigenenergy surfaces spanned by the $\eta$ and $\zeta$ parameters. The integer topological index $k$ is independent of the eigenstate and thus of the projection quantum number $m$. These findings have repercussions for rotational spectra and dynamics of molecules subject to combined permanent and induced dipole interactions.