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arXiv:1202.3134 · Communications on Pure and Applied Mathematics

WKB Analysis of Bohmian Dynamics

Alessio Figalli, Christian Klein, Peter A. Markowich, Christof Sparber

Published Oct 18, 2013• 10 citations
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Abstract

We consider a semiclassically scaled Schrödinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer holds. In addition, we provide numerical simulations of the Bohmian trajectories in the semiclassical regime that illustrate the above results.© 2014 Wiley Periodicals, Inc.

DOI
10.1002/cpa.21487

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We consider a semiclassically scaled Schrödinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer holds. In addition, we provide numerical simulations of the Bohmian trajectories in the semiclassical regime that illustrate the above results.© 2014 Wiley Periodicals, Inc.
Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields.
We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.
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.
In previous articles (J. Chem. Phys. 2004, 121, 4501; 2006, 124, 034115; 2006, 124, 034116) a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schrödinger equation, such that the components Psi+/- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+ <--> -) transitions.
A solution ψψψ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field ℏℑ(▽ψ/mψ).\hbar \Im \left( \bigtriangledown ψ/mψ\right) .ℏℑ(▽ψ/mψ). In the case of one specific non-differentiable weak solution ΨΨΨ we show how Bohmian trajectories can be obtained for ΨΨΨ from the trajectories of a sequence Ψn→Ψ.Ψ_{n}\rightarrow Ψ.Ψn​→Ψ. (For any real ttt the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors ΨΨΨ in the Hilbert space on which the Schrödinger evolution % Ψ\mapsto e^{-iht}Ψ acts.
Abstract As a model for the semiclassical analysis of quantum-mechanical systems with both potentials and boundary conditions, we construct the WKB propagator for a linear potential sloping away from an impenetrable boundary. First, we find all classical paths from point y to point x in time t and calculate the corresponding action and amplitude functions. A large part of space-time turns out to be classically inaccessible, and the boundary of this region is a caustic of an unusual type, where the amplitude vanishes instead of diverging. We show that this curve is the limit of caustics in the usual sense when the reflecting boundary is approximated by steeply rising smooth potentials. Then, to improve the WKB approximation we construct the propagator for initial data in momentum space; this requires classifying the interesting variety of classical paths with initial momentum p arriving at x after time t . The two approximate propagators are compared by applying them to Gaussian initial packets by numerical integration; the results show physically expected behavior, with advantages to the momentum-based propagator in the classically forbidden regime (large t ).