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Physical Review A

Diatomic-molecular spectrum in view of quantum group theory

Zhe Chang, Hong Yan

Published Dec 1, 1991• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.44.7405

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We develop a model with Hq{\mathit{H}}_{\mathit{q}}Hq​(4) and SUq{\mathrm{SU}}_{\mathit{q}}SUq​(2) quantum group symmetries to describe the vibrational and rotational spectra of diatomic molecules. The representations of Hq{\mathit{H}}_{\mathit{q}}Hq​(4) and SUq{\mathrm{SU}}_{\mathit{q}}SUq​(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 Hq{\mathit{H}}_{\mathit{q}}Hq​(4) \ensuremath{\bigotimes}SUq{\mathrm{SU}}_{\mathit{q}}SUq​(2) is an essential symmetry of vibrations and rotations in diatomic molecules.
We suggest a model of q-deformed rotating oscillator which possesses the symmetry of quantum group and point out that this model can be used to describe the vibrational and rotational structures of many microscopic systems. We construct the representation of q-oscilla or algebra and give energy values of q-oscillaor in this representation. It is shown that the q-oscillator spectra can be used to describe infrared spectra, Raman spectra and vibrational structure of electronic transitions of diatomic molecules.
q-deformation of the U(1) dynamical symmetry of U(2) group is introduced for the first time for vibrations of diatomic molecules. The question of pure real and pure imaginary value of q-deformation parameter, often raised in the literature, is found to be related to the particular choice of the constants of the model. The complex q-deformation parameter is allowed, with pure real and pure imaginary values as its special cases. Compared to the other dynamical symmetry Oq(2) of Uq(2), the result of Uq(1) of Uq(2) remains better even when the vibron number in Oq(2) is varied. The role of the vibron number in the two dynamical symmetries of U(2) is shown to be different. In Uq(1) symmetry, the vibron number is fixed by the number of observed states of the system since Uq(1) represents the q-deformation of a truncated anharmonic oscillator. A possible physical meaning to the q-deformation parameter is suggested in terms of higher order anharmonicities of the truncated anharmonic oscillator.
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.