Quantum Mechanics for Scientists and Engineers

Many interesting quantum mechanical problems can be reduced to one-dimensional mathematical problems. This reduction is often possible because the problem, though truly three-dimensional, can be mathematically separated. For example, the three-dimensional hydrogen atom also mathematically separates to leave a radial equation that looks like a onedimensional effective Schrödinger equation. Most problems associated with electrons and planar surfaces or layered structures can be handled with one-dimensional models. Examples include field emission of electrons from planar metallic surfaces and most problems associated with semiconductor quantum well structures.

One-dimensional problems can be solved by a number of techniques. Here, we discuss one of these, the transfer matrix technique, and we also derive one key result of the so-called WKB method. We concentrate on the use of such techniques for solving tunneling problems, so we start with a brief discussion of tunneling rates.

Tunneling probabilities

Suppose we have a barrier, shown in Fig. 11.1 as a simple rectangular barrier. Electrons are incident on the barrier from the left. Some are reflected and some are transmitted. We presume that the electron energy E is less than the barrier height V 0 so that we are discussing a tunneling problem. We already know how to solve this problem quantum mechanically for the simple case of a rectangular barrier, with notation as shown in the figure. We discuss now how to solve such problems for more complex forms of barrier.