Lie groups and spherical harmonics

Abstract

In this thesis we introduce Lie groups and prove some important properties of them. Then we take a look at the general theory of representations of Lie groups. After that we take a look at irreducible representations and the decomposition of finite dimensional representations into irreducibles. This will allow us to prove an impor- tant theorem, the Peter-Weyl theorem, which states that the space L^2(G) of square integrable functions on a compact Lie group G decomposes as a Hilbert direct sum of the linear span of matrix coefficients of irreducible representations. Then we will apply our knowledge of representations to find all irreducible representations of SO(3) using another important Lie group namely SU(2). Then, using Peter-Weyl, we show that any square integrable function on the two-sphere S^2 can be writ- ten in terms of spherical harmonics. Finally we apply this knowledge to solve the Schrödinger equation for an electron in the hydrogen atom.