QUANTUM RATE THEORY FOR SOLIDS .2. 1-DIMENSIONAL TUNNELING EFFECTS

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. © 1969 The American Physical Society.