chapter  15
- Approximation Methods III: WKB and Asymptotic Methods
Pages 18

The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin [1-3]. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. This corresponds to states of the systems with large kinetic energy and hence high quantum number. In other words ~ is small. Therefore, the approximation is known as semiclassical approximation. The method is useful for estimating eigenfunctions and tunnelling probabilities for smooth potentials or for potentials with only a few discontinuities. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The results obtained by the WKB technique are also obtained employing two other different methods, namely, the semiclassical path integral and the instanton procedure [4,5]. In the WKB method, solutions on either side of a turning point are matched with a third solution valid near the turning point. This difficulty is avoided in an asymptotic method of Keller [6]. We discuss the WKB and the asymptotic method in this chapter.