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arXiv:nlin/0506009 · Physical Review A

Uniform approximation of barrier penetration in phase space

Christopher S. Drew, Stephen C. Creagh, Richard H. Tew

Published Dec 1, 2005• 15 citations
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Abstract

A method to approximate transmission probabilities for a nonseparable multidimensional barrier is applied to a waveguide model. The method uses complex barrier-crossing orbits to represent reaction probabilities in phase space and is uniform in the sense that it applies at and above a threshold energy at which classical reaction switches on. Above this threshold the geometry of the classically reacting region of phase space is clearly reflected in the quantum representation. Two versions of the approximation are applied. A harmonic version which uses dynamics linearized around an instanton orbit is valid only near threshold but is easy to use. A more accurate and more widely applicable version using nonlinear dynamics is also described.

DOI
10.1103/physreva.72.062501

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A method to approximate transmission probabilities for a nonseparable multidimensional barrier is applied to a waveguide model. The method uses complex barrier-crossing orbits to represent reaction probabilities in phase space and is uniform in the sense that it applies at and above a threshold energy at which classical reaction switches on. Above this threshold the geometry of the classically reacting region of phase space is clearly reflected in the quantum representation. Two versions of the approximation are applied. A harmonic version which uses dynamics linearized around an instanton orbit is valid only near threshold but is easy to use. A more accurate and more widely applicable version using nonlinear dynamics is also described.
The quantum Hamilton-Jacobi equation for the action function is approximately solved by propagating individual Lagrangian quantum trajectories in complex-valued phase space. Equations of motion for these trajectories are derived through use of the derivative propagation method (DPM), which leads to a hierarchy of coupled differential equations for the action function and its spatial derivatives along each trajectory. In this study, complex-valued classical trajectories (second order DPM), along which is transported quantum phase information, are used to study low energy barrier transmission for a model two-dimensional system involving either an Eckart or Gaussian barrier along the reaction coordinate coupled to a harmonic oscillator. The arrival time for trajectories to reach the transmitted (product) region is studied. Trajectories launched from an "equal arrival time surface," defined as an isochrone, all reach the real-valued subspace in the transmitted region at the same time. The Rutherford-type diffraction of trajectories around poles in the complex extended Eckart potential energy surface is described. For thin barriers, these poles are close to the real axis and present problems for computing the transmitted density. In contrast, for the Gaussian barrier or the thick Eckart barrier where the poles are further from the real axis, smooth transmitted densities are obtained. Results obtained using higher-order quantum trajectories (third order DPM) are described for both thick and thin barriers, and some issues that arise for thin barriers are examined.
We apply the quasilinearization method of Bellman and Kalaba [Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965)] to find approximate solutions for the multidimensional quantum tunneling for separable as well as nonseparable wave equations. By introducing the idea of the complex ``semiclassical trajectory'' which is valid for the motion over and under the barrier, and which, in the proper limit, reduces to the real classical trajectory in the allowed region, we obtain an eigenvalue equation for the characteristic wave numbers. This eigenvalue equation is similar to the corresponding equation obtained from the WKB approximation and yields complex eigenvalues with negative imaginary parts. When the barrier changes very rapidly as a function of the radial distance, we can replace the concept of the semiclassical trajectory, which may not be applicable in this case, by the concept of a complex ``quantum trajectory.'' The trajectory defined either way depends on a constant of integration, and by minimizing the action with respect to this constant we can obtain the minimum escape path. The case of two-dimensional tunneling is discussed as an example of this method.
Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one-dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semiclassical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem. \textcopyright{} 1996 The American Physical Society.
The reflection and transmission of a wave on a two-dimensional barrier is studied quantum mechanically and semiclassically. The simple model consists of two degrees of freedom. One of them contains a barrier (of the Eckart type); the second allows only a bound motion in discrete quantum states. These two degrees of freedom are coupled. The transition probabilities for transitions between the different quantum states can, in the model considered here, be evaluated analytically, both quantum mechanically and semiclassically. Several of the so called uniform semiclassical approximations and also the initial value integral representation for the semiclassical transition amplitudes are considered and compared to the exact quantum mechanical results. Some new insight on the semiclassical treatment of the tunneling process is gained, specially concerning the proper choice of the integration path for the initial value integral representation of the transition amplitudes.