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Le Journal de Physique Colloques

ON SOME APPLICATIONS OF TIME-DEPENDENT INVARIANTS TO THE SEMICLASSICAL FORMULATION OF QUANTUM MECHANICS

R. M. Dreizler, H. Köhl

Published Jun 1, 1984• 1 citation
Publisher

Abstract

It is shown, that time-dependent classical and quantum mechanical invariants (TDI) may be of considerable value in semiclassical formulations of time-dependent quantum mechanical problems. We demonstrate, that the application of TDI allows the application of several ground-state technologies as for instance the Wigner-Kirkwood and WKB-expansion to the TD situation. Possible applications to semiclassical wave packet motion and to (low energy) nuclear hydrodynamics are outlined.

DOI
10.1051/jphyscol:1984605

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It is shown, that time-dependent classical and quantum mechanical invariants (TDI) may be of considerable value in semiclassical formulations of time-dependent quantum mechanical problems. We demonstrate, that the application of TDI allows the application of several ground-state technologies as for instance the Wigner-Kirkwood and WKB-expansion to the TD situation. Possible applications to semiclassical wave packet motion and to (low energy) nuclear hydrodynamics are outlined.
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Quantum states can be described equivalently by density matrices, Wigner functions, or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance, and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.
Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.
Transport and scattering phenomena in open quantum‐systems with a continuous energy spectrum are conveniently solved using the time‐dependent Schrödinger equation. In the time‐dependent picture, the evolution of an initially localized wave‐packet reveals the eigenstates and eigenvalues of the system under consideration. We discuss applications of the wave‐packet method in atomic, molecular, and mesoscopic systems and point out specific advantages of the time‐dependent approach. In connection with the familiar initial value formulation of classical mechanics, an intuitive interpretation of transport emerges. For interacting many‐particle systems, we discuss the efficient calculation of the self‐consistent classical transport in the presence of a magnetic field.