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Equal partial revivals of wavepackets in long time scales

Jie Jie, QLWang, Sjdescription, Wave-Packets, Wigner Function, Term Evolution, States, Oscillator

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

In terms of the action-angle Wigner function, we analyse the effects of In terms of the action-angle Wigner function, we analyse the effects of

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In terms of the action-angle Wigner function, we analyse the effects of energy spectrum anharmonicity on the evolutions of wavepackets in long time scales. The formulation shows a close connection between classical periodic motions of phase space points and the quantum (full and partial) revivals of wavepackets. This enables us to find the conditions of (full and partial) revivals in an intuitive way. For one-dimensional cases, we obtain analytic solutions for times of equal partial revivals, i.e. an initial wavepacket splits equally into several small parts in long time scales. Numerical results for the JC model confirm the theoretical predictions.
Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.
The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigner function is shown to exist in the semi-discrete limit of this quantization scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an action-angle phase space distribution respects are derived. The dynamical properties of the action-angle Wigner function are analysed for discrete and finite-dimensional model Hamiltonians. The limit of the discrete and finite-dimensional formalism including a discrete analogue of the Gaussian wavefunction spread, viz. the binomial wavepacket, is examined and shown by examples that standard (continuum) quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity.