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arXiv:math-ph/0505074 · Journal of Physics A Mathematical and General

Asymptotic behaviour of Bohmian trajectories in scattering situations

Sarah Römer, D. Dürr, T. Moser

Published Sep 14, 2005• 7 citations
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Abstract

We study the asymptotic behaviour of Bohmian trajectories in a scattering situation with short-range potential V and for wavefunctions with a scattering and a bound part. It is shown that the set of possible trajectories splits into trajectories whose long-time behaviour is governed by the scattering part of the wavefunction (scattering trajectories) and trajectories whose long time behaviour is governed by the bound part of the wavefunction (bound trajectories). Furthermore, the scattering trajectories behave like trajectories in classical mechanics in the limit t → ∞. As an intermediate step, we show that the asymptotic velocity exists almost surely and is randomly distributed with the density , where is the outgoing asymptote of the scattering part of the wavefunction.

DOI
10.1088/0305-4470/38/39/009

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We study the asymptotic behaviour of Bohmian trajectories in a scattering situation with short-range potential V and for wavefunctions with a scattering and a bound part. It is shown that the set of possible trajectories splits into trajectories whose long-time behaviour is governed by the scattering part of the wavefunction (scattering trajectories) and trajectories whose long time behaviour is governed by the bound part of the wavefunction (bound trajectories). Furthermore, the scattering trajectories behave like trajectories in classical mechanics in the limit t → ∞. As an intermediate step, we show that the asymptotic velocity exists almost surely and is randomly distributed with the density , where is the outgoing asymptote of the scattering part of the wavefunction.
In this paper the relations between the asymptotic velocity operators of a quantum system and the asymptotic velocities of the associated Bohmian trajectories are studied. In particular it is proved that, under suitable conditions of asymptotic regularity, the probability distribution of the asymptotic velocities of the Bohmian trajectories is equal to the one derived from the asymptotic velocity operators of the associated quantum system. It is also shown that in the relativistic case the distribution of the asymptotic velocities of the Bohmian trajectories is covariant, or equivalently, it does not depend on a preferred foliation (it is well known that this is not the case for the structure of the Bohmian trajectories or for their spatial distribution at a finite time). This result allows us to develop a covariant formulation of relativistic Bohmian mechanics; such a formulation is proposed here merely as a mathematical possibility, while its empirical adequacy will be discussed elsewhere.
In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition Ψ=Ψ1+Ψ2 was presented for stationary bound states Ψ of the one-dimensional Schrödinger equation, such that the components Ψ1 and Ψ2 approach their semiclassical WKB analogs in the large-action limit. The corresponding bipolar quantum trajectories, as defined in the usual Bohmian mechanical formulation, are classical-like and well behaved, even when Ψ has many nodes or is wildly oscillatory. A modification for discontinuous potential stationary scattering states was presented in a second, companion paper [C. Trahan and B. Poirier, J. Chem. Phys.124, 034115 (2006), previous paper], whose generalization for continuous potentials is given here. The result is an exact quantum scattering methodology using classical trajectories. For additional convenience in handling the tunneling case, a constant-velocity-trajectory version is also developed.
We prove an almost sure existence of the modified wave operators for a class of Schrödinger operators with random long range potentials. The assumed decay of potential at infinity places it beyond the threshold of the standard class of long range potentials as described in the work of Buslaev-Matveev, Alsholm-Kato, and Hörmander. We develop an approach to quantum scattering which relies on averaging of potential over classical random trajectories. We also establish classical scattering for corresponding classical hamiltonians.
Beginning with the conventional asymptotic description of quantum scattering, we propose a simple wave function that is valid at all distances and compute the corresponding Bohmian trajectories. This 'simulated experiment' allows a concrete visualization of the process throughout the scattering interaction. Not only is this of pedagogic value at the undergraduate level, it also illustrates several important themes in time-dependent scattering theory, such as the importance of wave packets, the role of cross sections in describing measurement and the wave and particle pictures of quantum mechanics.
Conventional scattering theory is incomplete in that it does not adequately describe the behaviour of the wave function at macroscopic distances from the scattering reaction volume. In scattering experiments particles are incident from sources at macroscopic distance and measured at macroscopic distance from the microscopic reaction volume. Standard experimental procedure is to describe asymptotic particle motion by classical mechanics. However, this transition to classical mechanics is not accounted for in standard quantum scattering theory. Hence, we revisit conventional scattering theory with the introduction of semiclassical propagators to describe the wave function in both incident and outgoing asymptotic scattering channels. This description leads to the manifestation of classical trajectories in the quantum wave function both before and after the collision. It includes the case of extraction with external fields as in modern multi-particle detectors. We derive the asymptotic Kemble imaging theorem (IT) limit to demonstrate how formal quantum scattering theory contains implicit classical mechanics threaded throughout the asymptotic wave functions.