1. Lecture 1 – Introduction to the WKB Approximation All material available online at www.tomostler.co.uk/teaching www.tomostler.co.uk/teaching follow links to WKB lecture 1

2. Introduction WKB = Wentzel, Kramers and Brillouin Picture of Wentzel from this link.this link Picture of Brillouin from this link.this link Picture of Kramers from this link.this link Picture of Carlini from this link.this link BUT it was already discovered. In 1837 Liouville and Green published works using the method. Carlini also used a version to study elliptical orbits of plants.

3. Introduction The WKB approximation is one in a number of asymptotic expansions. Has been applied to a wide number of situations quantum mechanical tunneling. scattering. spinwave propagation through domain walls. Planetary motion. ….. The WKB method finds approximate solutions to the second order differential equation:

4. Aim of this lecture For a general potential V(x) this equation does not have an exact solution. In some cases the solution is exact but the form is often very difficult to work with. We will see an example of this where the exact solution is in the form of Bessel functions. The aim of this lecture is to show that we can derive an approximate solution for this equation in the limit of small ε.

5. Further reading M. H. Holmes – Introduction to Perturbation Methods, Springer-Verlag. Pages 161-165 (for version ISBN 0-387-94204-3). Next lecture Errors in the WKB. Practical: Maple based exercise. Application to the time independent Schrödinger equation.