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Physical review. E

Multidimensional WKB approximation for particle tunneling

Jaroslav Zamastil

Published Jul 19, 2018• 5 citations
Publisher

Abstract

In this paper we present a method for obtaining the WKB approximation for non-separable multidimensional potentials. At its leading order one has to solve the classical Hamilton-Jacobi equation with zero energy for which a very efficient method is proposed. The essence of this method lies in the recognition that in the vicinity to the potential minimum the solution of Schrödinger equation has to approach that of coupled harmonic oscillators. Quantum corrections to the semiclassical result are then obtained very easily and to an arbitrarily high order. The method is applied to the calculation of the tunneling splitting and tunneling lifetime. We show that classical turning points are part of the dynamical problem; they could not be determined simply from looking at the form of the potential, but are obtained from the solution of the pertinent equations of motion.

DOI
10.1103/physreve.98.012211

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In this paper we present a method for obtaining the WKB approximation for non-separable multidimensional potentials. At its leading order one has to solve the classical Hamilton-Jacobi equation with zero energy for which a very efficient method is proposed. The essence of this method lies in the recognition that in the vicinity to the potential minimum the solution of Schrödinger equation has to approach that of coupled harmonic oscillators. Quantum corrections to the semiclassical result are then obtained very easily and to an arbitrarily high order. The method is applied to the calculation of the tunneling splitting and tunneling lifetime. We show that classical turning points are part of the dynamical problem; they could not be determined simply from looking at the form of the potential, but are obtained from the solution of the pertinent equations of motion.
A method for obtaining the WKB wave function describing the particle tunneling outside of a two-dimensional potential well is suggested. The Cartesian coordinates (x,y)(x,y)(x,y) are chosen in such a way that the xxx axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by simultaneous expansion of the wave function in the coordinate yyy and the parameter determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. It is shown that the method provides systematic approximation to the outgoing probability flux. Both the technical and conceptual advantages of this approach in comparison with the usual approach based on the solution of classical equations of motion are pointed out. The method is applied to the problem of the coupled anharmonic oscillators and verified through the dispersion relations.
The WKB method is commonly used in semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of a one-dimensional potential. In a multidimensional space, the method can be adapted to construct wave functions in an ``allowed'' region from classical trajectories or wave normals. However, in the ``forbidden'' region the WKB wave function is in general specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces or equivalently by two sets of paths defined to be normal to these surfaces, respectively. We present a Huygens-type construction for obtaining these wave fronts and paths, which reveals that for non-normal incidence the paths are coupled to each other. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. A special and important case occurs when the incident wave is normal to the turning surface. We show that for normal incidence the path equations are decoupled and are equivalent to Newton's equations of motion for the inverted potential and energy.
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.
In this paper, we develop quantum mechanics of quasi-one-dimensional systems upon the framework of the quantum-mechanical Hamilton–Jacobi theory. We will show that the Schrödinger point of view and the Hamilton–Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result of this, a wavefunction in Hamilton–Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. Using the quasi-linearization technique, we derive quantum correction functions in every order of . The quantum correction functions will remove the turning-point singularity that plagues the WKB-series expansion already in zeroth order and thus provide an extremely good approximation to the full solution of the Schrödinger equation. In the language of quantum action it is also possible to elegantly solve the connection problem without asymptotic approximations. The use of quantum action further allows us to derive an equation by which the Maslov index is directly calculable without any approximations. Stationary quantum trajectories will also be considered and thoroughly discussed.
A general Wentzel–Kramers–Brillouin (WKB) theory of multidimensional tunneling is formulated and an illuminating physical picture of the effects of multidimensionality is provided. Two basic problems are solved: (i) Maslov’s semiclassical wave function in the classically accessible region is connected to the wave function in the classically inaccessible region and (ii) the latter is propagated into the deep tunneling region. It is found that there exist two distinct types of tunneling: pure tunneling and mixed tunneling. The former is the usual one in which the tunneling path can be defined by a certain classical trajectory on the inverted potential and its associated action is pure imaginary. In the latter case, no tunneling path can be defined and the Huygens-type wave propagation should be carried out. In this case, tunneling is always accompanied by classical motion in the transversal direction and the associated action is complex. A general procedure is presented for the evaluation of energy splitting ΔE in the double well. Moreover, under the locally separable linear approximation, a simple and convenient formula for ΔE is derived and is confirmed to work well by comparison with the exact numerical calculations.
Three-dimensional Schrödinger's equation is analyzed with the help of the correspondence principle between classical and quantum-mechanical quantities. Separation is performed after reduction of the original equation to the form of the classical Hamilton–Jacobi equation. Each one-dimensional equation obtained after separation is solved by the conventional WKB method. Quasiclassical solution of the angular equation results in the integral of motion [Formula: see text] and the existence of nontrivial solution for the angular quantum number l = 0. Generalization of the WKB method for multi-turning-point problems is given. Exact eigenvalues for solvable and some "insoluble" spherically symmetric potentials are obtained. Quasiclassical eigenfunctions are written in terms of elementary functions in the form of a standing wave.