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.
Highlights
The problem of the spherical quantum pendulum, i.e., that of a three-dimensional (3D) rigid rotor under a cosine potential and/or its variants, belongs to prototypical problems in quantum mechanics
In our previous work [57,58], we showed that the pendular eigenproblem can be solved analytically, but only for a particular ratio of η to ζ. We found this ratio—which represents a condition for analytic solvability—as well as the analytic solution itself by invoking the apparatus of supersymmetric quantum mechanics (SUSY QM) [59,60,61,62,63,64]
IV, we present two sets of conditions that lead to analytic solutions of the quantum pendulum eigenproblem and find that these sets pertain to a particular topological index
Summary
The problem of the spherical quantum pendulum, i.e., that of a three-dimensional (3D) rigid rotor under a cosine potential and/or its variants, belongs to prototypical problems in quantum mechanics. The matrix elements are a function of dimensionless parameters η and ζ that characterize, respectively, the strengths of the pendulum’s orienting (∝η cos θ ) and aligning (∝ζ cos θ ) interactions (where θ is a polar angle) These interactions correspond to those of a polar (permanent electric dipole moment) and polarizable (induced electric dipole moment) molecule subject to either an electrostatic field or a combination of an electric and a far-off-resonant optical field or a single-cycle nonresonant optical field [3,17,19,21].
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