The molecular asymmetric rigid rotor Hamiltonian as an exactly solvable model

TL;DR

This paper demonstrates that the molecular asymmetric rigid rotor Hamiltonian can be exactly solved by linking it to algebraic structures like the $su_q(1,1)$ Yang-Baxter algebra, revealing new insights into its spectrum and eigenstates.

Contribution

It shows that the general asymmetric rigid rotor Hamiltonian is an exactly solvable model through its connection to $su_q(1,1)$ algebraic structures.

Findings

The Hamiltonian is quadratic in Gaudin operators.

The model's spectrum and eigenstates can be explicitly characterized.

The approach provides new analytical tools for molecular rotation analysis.

Abstract

Representations of the rotation group may be formulated in second-quantised language via Schwinger's transcription of angular momentum states onto states of an effective two-dimensional oscillator. In the case of the molecular asymmetric rigid rotor, by projecting onto the state space of rigid body rotations, the standard Ray Hamiltonian $H(1,\kappa,-1)$ (with asymmetry parameter $1 \ge \kappa \ge -1$), becomes a quadratic polynomial in the generators of the associated dynamical $su(1,1)$ algebra. We point out that $H(1,\kappa,-1)$ is in fact quadratic in the Gaudin operators arising from the quasiclassical limit of an associated $su_q(1,1)$ Yang-Baxter algebra. The general asymmetric rigid rotor Hamiltonian is thus an exactly solvable model. This fact has important implications for the structure of the spectrum, as well as for the eigenstates and correlation functions of the model.