Describing Actual Quantum Systems Using the WKB Theory

This chapter is a brief demonstration of how the WKB theory—a method to obtain the approximate solutions of the Schrödinger equation—is employed and appears to be extremely helpful in analyzing the properties and behavior of various realistic quantum mechanical systems. It is logical to start it by formulating the applicability conditions of the semiclassical model: what is exactly the object that we can investigate by using the WKB approximation? How can we estimate the precision of the results obtained from this method? After that, we will derive the connection formulas between the wave function in the classically allowable and classically forbidden regions. Using a few specific examples, we show that the WKB momentum quantization relations are extremely helpful in finding the allowable quantized values of the observable quantities for the bound states. Special attention will be directed to calculating the semiclassical tunneling probabilities, which is likely the most important, comprehensive, and widely used application of the WKB theory.