Supersymmetry identifies molecular Stark states whose eigenproperties can be obtained analytically

TL;DR

This paper uses supersymmetric quantum mechanics to identify conditions under which the Stark effect problem for diatomic molecules is exactly solvable, providing explicit solutions and formulas for molecular properties under combined electric fields.

Contribution

It introduces a supersymmetry-based condition for exact solvability of molecular Stark states and derives analytic solutions and expectation values for these states.

Findings

Exact solutions for specific molecular states under combined fields.

Analytic formulas for molecular orientation and alignment properties.

Identification of supersymmetry and shape-invariance in strong-field limits.

Abstract

We made use of supersymmetric (SUSY) quantum mechanics to find a condition under which the Stark effect problem for a polar and polarizable closed-shell diatomic molecule subject to collinear electrostatic and nonresonant radiative fields becomes exactly solvable. The condition, $\Delta \omega = \frac{\omega^2}{4 (m+1)^2}$, connects values of the dimensionless parameters $\omega$ and $\Delta \omega$ that characterize the strengths of the permanent and induced dipole interactions of the molecule with the respective fields. The exact solutions are obtained for the $|\tilde{J}=m,m;\omega,\Delta \omega>$ family of "stretched" states. The field-free and strong-field limits of the combined-fields problem were found to exhibit supersymmetry and shape-invariance, which is indeed the reason why they are analytically solvable. By making use of the analytic form of the $|\tilde{J}=m,m;\omega,\Delta \omega>$ wavefunctions, we obtained simple formulae for the expectation values of the space-fixed electric dipole moment, the alignment cosine, the angular momentum squared, and derived a "sum rule" which combines the above expectation values into a formula for the eigenenergy. The analytic expressions for the characteristics of the strongly oriented and aligned states provide a direct access to the values of the interaction parameters required for creating such states in the laboratory.