The WKB Approximation
Immediately after the birth of the quantum theory, Wentzel, Kramers and Brillouin introduced in 1926 one of the first methods to obtain approximate solutions to the Schrödinger equation. A method that might be used when rigorous analytical solutions are not possible. In most of the actual quantum systems we face this problem. It is known that only for exceptional systems like the harmonic oscillator, the hydrogen atom and the piecewise constant potentials, it is possible to solve the Schrödinger equation analytically. It is therefore important to study an approximate method to solve the Schrödinger equation. In this chapter we present the basic ideas behind the WKB approximation and we will learn how to deal with this approximation in the transfer matrix representation. For simplicity reasons, our discussion will be mostly devoted to one-dimensional systems. Another widely used approximation method, is the so-called perturbation theory. In Chap. 12 we will discuss basic aspects of this theory.