Basic Formulation of the Wentzel–Kramers–Brillouin (WKB) Theory

This chapter is our first encounter with the WKB method applied to the Schrödinger equation, whose solutions then become a good description of a semiclassical particle. We will review the most crucial results of the WKB method in conventional quantum mechanics, as well as its derivation beyond the traditionally considered expansion terms. We are also going to discuss how the WKB approximation is applied to the various types of differential equations in math and other fields. We are looking to learn to solve some traditional WKB types of differential equations, in which the highest-order derivative is multiplied by a small parameter. Apart from that, we will find out that the WKB approximation and the corresponding series expansions lead to some reasonably precise solutions for a much wider class of differential equations with several small parameters in different terms, which is out of the scope of the standard definition of the WKB method.