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Chemical Physics Letters

Dynamics of wave packets and the time-dependent variational principle

Frans Arickx, J. Broeckhove, E. Kesteloot, Luc Lathouwers, Piet Van Leuven

Published Jul 1, 1986• 38 citations
Publisher
DOI
10.1016/0009-2614(86)80346-3

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The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well behaved. The power of the method is demonstrated for a nonintegrable system with two degrees of freedom.
Based on the variational principle in a general manifold in the Hilbert space and the principle of gauge invariance, the general gauge invariant periodic quantization method is formulated. This method, which is a direct generalization of the one previously obtained in the case of time-dependent Hartree-Fock, leads to a quantization rule which takes the Bohr-Sommerfeld form when the variational parameters are transformed to canonical variables. The method is illustrated by applying it to the manifold of minimum wave packets for a simple harmonic oscillator. It leads to the exact energy spectrum and a discrete set of wave packets which have maximum overlap with the exact eigenstates. The method is also applied to the exact Schr\"odinger dynamics where it exposes some open questions of the method.[NUCLEAR STRUCTURE Time-dependent description of many-body bound states. Gauge invariant periodic quantization method, time-dependent variational principle and their canonical formulation.]