This book provides an introduction to the use of algebraic methods and sym bolic computation for simple quantum systems with applications to large order perturbation theory. It is the first book to integrate Lie algebras, algebraic perturbation theory and symbolic computation in a form suitable for students and researchers in theoretical and computational chemistry and is conveniently divided into two parts. The first part, Chapters 1 to 6, provides a pedagogical introduction to the important Lie algebras so(3), so(2,1), so(4) and so(4,2) needed for the study of simple quantum systems such as the D-dimensional hydrogen atom and harmonic oscillator. This material is suitable for advanced undergraduate and beginning graduate students. Of particular importance is the use of so(2,1) in Chapter 4 as a spectrum generating algebra for several important systems such as the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This approach provides an interesting and important alternative to the usual textbook approach using series solutions of differential equations.

1 General Discussion of Lie Algebras

2 Commutator Gymnastics
3 Angular Momentum Theory and so(3)
4 Representations and Realizations of so(2,1)
5 Representations and Realizations of so(4)
6 Scaled Hydrogenic Realization of so(4,2)
7 Lie Algebraic Perturbation Theory
8 Symbolic Calculation of the Stark Effect
9 Symbolic Calculation of the Zeeman Effect
10 Spherically Symmetric Systems
A The Levi-Civita Symbol
B Lie Groups and Lie Algebras
C The Tilting Transformation
D Perturbation Matrix Elements
E Tables of Stark Effect Energy Corrections
F Tables of Zeeman Effect Energy Corrections
G Tables of Charmonium Energy Corrections
H Tables of Harmonium Energy Corrections
I Tables of Screened Coulomb Energy Corrections
J Solutions to Exercises
Index of Symbols Used.