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International Journal of Quantum Chemistry

Nonlinear response of a deformed oscillator

J. Récamier, Wolf Luis Mochán, Jesús A. Maytorena

Published Jan 1, 2005• 2 citations
Publisher

Abstract

Abstract We analyze the response of a single anharmonic diatomic molecule to a monochromatic time dependent electric field ω(t) and evaluate the temporal evolution of the dipole moment, phase space trajectories, and several transition probabilities as a function of the intensity of the field. We define deformed boson operators A† = f( )a†, A = af( ) in terms of the usual harmonic oscillator creation and annihilation operators a, a†, = a†a and choose the number operator function f( ) such that the energy spectrum of a harmonic oscillator‐type Hamiltonian written in terms of the deformed operators yields an energy spectrum similar to that of a Morse potential. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

DOI
10.1002/qua.20663

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arXiv:1204.5391•Europhysics Letters (EPL)•Jan 1, 2013•98 citations
Abstract We analyze the response of a single anharmonic diatomic molecule to a monochromatic time dependent electric field ω(t) and evaluate the temporal evolution of the dipole moment, phase space trajectories, and several transition probabilities as a function of the intensity of the field. We define deformed boson operators A† = f( )a†, A = af( ) in terms of the usual harmonic oscillator creation and annihilation operators a, a†, = a†a and choose the number operator function f( ) such that the energy spectrum of a harmonic oscillator‐type Hamiltonian written in terms of the deformed operators yields an energy spectrum similar to that of a Morse potential. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
We model an anharmonic diatomic molecule using deformed creation and annihilation operators such that the energy spectrum generated by a Hamiltonian of the harmonic oscillator's form written in terms of deformed operators is similar to that of a Morse potential. We construct an approximate time evolution operator and evaluate transition probabilities which are compared with those obtained by an expansion in a basis of Morse eigenfunctions. The algebraic results compare favorably with the numerical results.
We construct a deformed oscillator whose energy spectra is similar to that of a Morse potential. We obtain a convenient algebraic representation of the displacement and the momentum of a Morse oscillator by expanding them in terms of deformed creation and annihilation operators and we compute their average values between approximate coherent states of the deformed oscillator, and we compare them to the results obtained using the exact Morse coordinate and momenta. Finally we evaluate the temporal evolution of the dispersion (Δx)(Δp) and show that these states are not minimum uncertainty states.
We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.
We show that a direct connection can be drawn, based on fundamental quantum principles, between the Morse potential, extensively used as an empirical description for the atomic interaction in diatomic molecules, and the harmonic potential. This is conceptually achieved here through a non-additive translation operator, whose action leads to a perfect equivalence between the quantum harmonic oscillator in deformed space and the quantum Morse oscillator in regular space. In this way, our theoretical approach provides a distinctive first principle rationale for anharmonicity, therefore revealing a possible quantum origin for several related properties as, for example, the dissociation energy of diatomic molecules and the deformation of cubic metals.