Physical Review A

Spectra of symmetric-top molecules and the quantum-deformed-symmetric-top model

Zhe Chang

Published Aug 1, 1992• 3 citations

Abstract

By means of geometric quantization, we discuss the quantum-deformed-symmetric-top model. We show that the quantum-deformed symmetric top and the quantum-standard symmetric top obey the same Heisenberg equations of motion. The quantum-deformed-symmetric-top model possesses quantum group symmetry; thus the quantum group can be dynamical symmetries of the symmetric-top system. The energy levels and wave functions of stationary states of the quantum-deformed-symmetric-top model are obtained exactly by a quantum-group-theoretic approach. Using the exactly solved model, we investigate the rotational spectra of symmetric-top molecules and diatomic molecules when the moment of inertia about the line joining the nuclei is not exactly zero.

DOI: 10.1103/physreva.46.1400

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Physical Review A

Spectra of symmetric-top molecules and the quantum-deformed-symmetric-top model

Zhe Chang

Published Aug 1, 1992• 3 citations

Publisher

Abstract

DOI: 10.1103/physreva.46.1400

Related papers

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We suggest the quantum Heisenberg group

{\mathit{H}}_{\mathit{q}(\mathit{J})}

(4) as the dynamical symmetry group of a diatomic-molecule system. The deformation parameter q of the

{\mathit{H}}_{\mathit{q}(\mathit{J})}

(4) symmetry group is a function of J (the quantum number of the angular momentum of the rigid rotation) and characterizes the interaction between vibration and rotation. The vibrational and rotational structures of diatomic molecules are compactly expressed using the quantum-group-theoretic approach. Taylor expansions of the analytic formulas from this approach reproduce the results of the anharmonic-oscillator and the nonrigid-rotator models. For some randomly selected molecules in particular states, the parameters are computed to fit the phenomenological formulas to high accuracy.

We develop a model with

{\mathit{H}}_{\mathit{q}}

(4) and

{\mathrm{SU}}_{\mathit{q}}

(2) quantum group symmetries to describe the vibrational and rotational spectra of diatomic molecules. The representations of

{\mathit{H}}_{\mathit{q}}

(4) and

{\mathrm{SU}}_{\mathit{q}}

(2) and solutions to the quantum systems of q oscillators and q rotators are provided. We emphasize that the energy spectra of the q oscillator and q rotator exhibit remarkably the properties of the infrared, vibrational, and rotational Raman spectra as well as vibrational and rotational structures of electronic transitions of the diatomic molecule. When the parameters are properly selected, the spectra of this model coincide with the observed spectra of diatomic molecules. It is suggested that

{\mathit{H}}_{\mathit{q}}

(4) \ensuremath{\bigotimes}

{\mathrm{SU}}_{\mathit{q}}

(2) is an essential symmetry of vibrations and rotations in diatomic molecules.

We show that the quantum group

{\mathrm{SU}}_{\mathit{q}}

(2) can be realized in a classical symmetric-top system as long as a suitable Hamiltonian and symplectic structure are chosen. The usual Lie group can be recovered when the deformation parameter q equals 1. The Hamiltonian system in which the quantum group is realized obeys the same equations of motion with the standard Hamiltonian system, but these two Hamiltonian systems have different constants of motion that reflect that the symmetries are different.