Quantum Rate Theory for Solids. II. One-Dimensional Tunneling Effects

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Abstract
The effects of tunneling are included in a one-dimensional rate theory which utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets (coherent states). An approximate tunneling formula for such wave packets is developed for a piecewise quadratic potential representing a well followed by a barrier; it is based on an exact solution for a Gaussian wave packet moving on a parabolic barrier. A new method for the numerical solution of the time-dependent Schrödinger equation is presented and used to check this approximate tunneling formula. The addition of tunneling to the effect of quantum statistics gives a rate formula which approximates the classical result at high-temperature levels and approaches a constant rate determined by tunneling at very low-temperature levels.