Algebraic Approach to Simple Quantum Systems

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.

Inhaltsangabe1 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.