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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Simple approach to tunneling using the method of finite elements

Carl M. Bender, Fred Cooper, Vincent P. Gutschick, Michael Martin Nieto

Published Sep 15, 1985• 18 citations
Publisher

Abstract

We use the method of finite elements to solve the operator field equations for a simple quantum-mechanical model of tunneling. For the case of just one finite element we can express the tunneling in terms of a Borel transform. We obtain excellent numerical results for short times.
DOI
10.1103/physrevd.32.1486

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Quantum tunneling is an important model in elementary quantum mechanics. However, since solution of its wave function involves transcendental equation, many elementary quantum mechanics textbooks often focus on the explanation of transmission coefficient, and rarely mention the evolution of wave function. Although this process is mentioned in some textbooks, it is often solved by graphic method, which is not conducive to beginners' understanding of the process. In this paper, a numerical calculation method for beginners is proposed to solve this problem. According to Dirac's vectorization idea, the state vectors and operators in the Schrodinger equation are discretized in the form of vectors and matrices, respectively, and the time-dependent evolution of wave function tunneling under several potential barriers is solved by Julia numerically.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
We present a numerical Fourier-transform method for solving the time-dependent Schrödinger equation in spherical coordinates. The time propagation is accomplished with a novel implementation of a Crank–Nicholson scheme combined with a Feit–Fleck split-operator method. Calculations on model problems give results that are in excellent agreement with those obtained by other methods. The procedure avoids the numerical instabilities associated with the division by the sine of the polar angle. The propagator is unitary, does not require the calculation of any matrix elements, and uses no matrix multiplications other than fast Fourier transforms.
Canonical discrete-time operator difference equations are introduced as an alternative approach to the numerical solution of a quantum field theory. We apply these techniques to the solution of the operator Heisenberg equations of motion describing the problem of quantum-mechanical tunneling. Our numerical solutions accurately depict the time evolution of 〈qn〉〈{q}^{n}〉〈qn〉 and of a local probability measure.