Tunneling in one dimension: General theory, instabilities, rules of calculation, applications

Techniques used to analyze the semiclassical limit in quantum mechanics are illustrated by one dimensional tunneling problems. A set of rules obeyed by the logarithmic derivative of the ground state wave function which are valid in any situation are presented. Double well problems are discussed. Tunneling through sequences of two types of barriers in situations of increasing complexity is analyzed. Inductive methods to derive the behavior in a long sequence from the analysis of short subsequences are obtained. Results are sufficient to treat physically interesting problems. A hierarchical model where tunneling can be analyzed over all possible scales is treated.