Lecture 1 – Introduction to the WKB Approximation

All material available online at www.tomostler.co.uk/teachingfollow links to WKB lecture 1

Introduction

• WKB = Wentzel, Kramers and Brillouin

Picture of Wentzel from this link.Picture of Brillouin from this link.Picture of Kramers from this link.Picture of Carlini from 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.

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:

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.

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.