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Theoretical Chemistry Accounts

A Bohmian total potential view to quantum effects. II: decay of temporarily trapped states

María F. González, Antoni Aguilar‐Mogas, Javier González, Ramón Crehuet, Josep Maria Anglada, Josep Maria Anglada, Xavier Giménez

Published Mar 5, 2009• 2 citations
Publisher
DOI
10.1007/s00214-009-0548-6

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The tunnel effect in one-dimensional collision problems is analyzed in terms of numerical wavepacket time propagations, in position, momentum, and phase space representations of the problem. Bohm's total potential is used to provide a classical-like description of tunnelling quantum dynamics, under a time- and density-dependent potential. Wigner, Husimi, as well as Bohm phase space distributions are used in the present study, with their relative performances for the present problem having been analyzed.
This paper presents a number of numerical investigations of orbits in the de Broglie–Bohm version of quantum mechanics. We first clarify how the notion of chaos should be implemented in the case of Bohmian orbits. Then, we investigate the Bohmian orbits in three different characteristic quantum systems: (a) superposition of three stationary states in the Hamiltonian of two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets in a Hénon–Heiles-type Hamiltonian and (c) a modified two-slit experiment. In these examples, we identify regular or chaotic orbits and also orbits exhibiting a temporarily regular and then chaotic behaviour. Then, we focus on a numerical investigation of the Bohm–Vigier (Bohm and Vigier 1954 Phys. Rev. 26 208) theory, that an arbitrary initial particle distribution P should asymptotically tend to |ψ| 2 , by considering the role of chaotic mixing in causing irregularity of Madelung's flow, a necessary condition for P to tend to |ψ| 2 . We find that the degree of chaos of a particular system correlates with the speed of convergence of P to |ψ| 2 . In the case of wave-packet dynamics, our numerical data show that the time of convergence scales exponentially with the inverse of the effective perturbation from the harmonic oscillator Hamiltonian. The latter result can be viewed as a quantum analogue of Nekhoroshev's (Nekhoroshev 1977 Russ. Math. Surveys 32 1) theorem of exponential stability in classical nonlinear Hamiltonian dynamics.