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Physical Review A

Continuum effects on the temporal evolution of anharmonic coherent molecular vibrations

J. A. Castañeda, M. A. Hernández, Rocío Jáuregui

Published Aug 6, 2008• 6 citations

Abstract

The generalization of the concept of coherent states for a Morse potential based on the real or complex displacements of the equilibrium position of the ground-state wave function yields a natural framework for their generation via Franck-Condon transitions. We study the relevance of including continuum states in the proper treatment of these displaced states, particularly for the description of their time evolution. It is shown that displaced Morse states may exhibit mathematical and physical temporal stability manifested through the presence of revivals in spite of the unbound character of the included continuum states. The Wigner function of these states is presented as well as relations to previous constructions. The time evolution of particular phase space properties such as mean values of position and momentum as well as their uncertainties are also analyzed.

DOI: 10.1103/physreva.78.023809

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The generalization of the concept of coherent states for a Morse potential based on the real or complex displacements of the equilibrium position of the ground-state wave function yields a natural framework for their generation via Franck-Condon transitions. We study the relevance of including continuum states in the proper treatment of these displaced states, particularly for the description of their time evolution. It is shown that displaced Morse states may exhibit mathematical and physical temporal stability manifested through the presence of revivals in spite of the unbound character of the included continuum states. The Wigner function of these states is presented as well as relations to previous constructions. The time evolution of particular phase space properties such as mean values of position and momentum as well as their uncertainties are also analyzed.

We investigate the time evolution of Morse coherent states in the potential of the NO molecule. We present animated wave functions and Wigner functions of the system exhibiting spontaneous formation of Schrödinger-cat states at certain stages of the time evolution. These nonclassical states are coherent superpositions of two localized states corresponding to two di.erent positions of the center of mass. We analyze the degree of nonclassicality as the function of the expectation value of the position in the initial state. Our numerical calculations are based on a novel, essentially algebraic treatment of the Morse potential.

The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s . The representations of the central extension of the affine group denoted by G 0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.

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.