PH3203 - Lecture 8

In this lecture I discuss the JWKB approximation. Initially I motivate the basic idea by using the relation between momentum and wavelength of a free particle wave - by looking for a natural generalization to slowly varying potentials. I then make the idea more precise by first finding an equation for the "phase" of the wavefunction, showing that it reduces to the naive result in the limit hbar tending to zero, and then resorting to a solution in the form of a series in powers of hbar. The first two terms of this expansion (the hbar^0 and hbar^1 terms) give us the JWKB approximation. I then discuss the validity of this approximation and show that this will break down in the vicinity of classical turning points. I then explain how we can approximate the potential in the small range around the turning points by linear potentials and use the solutions of this problem in term of the Airy functions to derive connection formulas between the JWKB solutions on both sides of a turning point. I then use the connection formulas to derive a cornerstone of old quantum theory - the Bohr-Sommerfeld quantization conditions. I use this to explain the idea of an area in phase space per quantum state and also to explain why the number of nodes in a bound energy eigenstate increases by one for each successive energy eigenvalue.