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Journal of Mathematical Chemistry

The construction of ladder operators and coherent states for the Wei Hua anharmonic oscillator using the supersymmetric quantum mechanics

Damian Mikulski, Marcin Molski, Jerzy Konarski, Krzysztof Eder

Published Aug 27, 2013• 3 citations
Publisher

Abstract

The unknown ladder operators for the Wei Hua potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The annihilation and creation operators have been obtained with the use of the factorization method. The coherent states for the Wei Hua anharmonic oscillator, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed in the framework of the supersymmetric quantum mechanics. The constructed ladder operators can be applied to compute the matrix elements in theoretical spectroscopy due to the fact that this potential is widely used to a proper description the large-amplitude vibrations of diatomic molecules.

DOI
10.1007/s10910-013-0251-7

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The unknown ladder operators for the Wei Hua potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The annihilation and creation operators have been obtained with the use of the factorization method. The coherent states for the Wei Hua anharmonic oscillator, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed in the framework of the supersymmetric quantum mechanics. The constructed ladder operators can be applied to compute the matrix elements in theoretical spectroscopy due to the fact that this potential is widely used to a proper description the large-amplitude vibrations of diatomic molecules.
The ladder operators for the Goldman and Krivchenkov anharmonic potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The coherent states for the Goldman and Krivchenkov oscillator, which are the eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed within the framework of supersymmetric quantum mechanics. The constructed ladder operators can be a useful tool in quantum chemistry computations of non-trivial matrix elements. In particular, they can be employed in molecular vibrational–rotational spectroscopy of diatomic molecules to compute transition energies and dipole matrix elements.
A mixed supersymmetric-algebraic approach to construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua, Kratzer-Fues and generalized Morse and Kratzer-Fues oscillators. The method can be applied also to generate superpotentials indispensable for deriving the Schrödinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.
A mixed supersymmetric–algebraic approach to the construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua and generalized Morse and Kratzer–Fues oscillators. The method can be applied to generate superpotentials indispensable for deriving the Schrödinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.
During the century of existence of the notion of coherent states, either linear or nonlinear, several schemes for their construction, theoretical or experimental, have been developed. Generally, the mathematical structure of coherent states depends on the choice of ladder operators, and consequently on the structure constants. In this paper, we propose a way to construct generalized coherent states for anharmonic oscillators that is based on a diagonal operator ordering technique (DOOT) applied to generalized hypergeometric functions, that is, on some of the most general special functions. These states are generated by the action of a pair of ladder operators, the creation and the annihilation, whose ordered normal product is equal to the dimensionless Hamiltonian of the quantum system. In addition, the action of these operators is easy to find if the expression for the dimensionless energy eigenvalues is known.
The coherent states for the Kratzer-Fues potential, which are eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed using the algebraic approach. The method is extended to include the rotating Kratzer-Fues oscillator. Construction of such states is important in the theory of molecular interactions with coherent electromagnetic fields, for example, in calculation of the dynamic alternation of the refractive index due to the interactions of the laser pulse with the coherent vibration-rotational states of the diatomic molecule.
Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wavefunctions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.