By Barry G. Adams

ISBN-10: 3540578013

ISBN-13: 9783540578017

ISBN-10: 3642579337

ISBN-13: 9783642579332

This booklet offers an creation to using algebraic tools and sym bolic computation for easy quantum structures with functions to massive order perturbation concept. it's the first e-book to combine Lie algebras, algebraic perturbation thought and symbolic computation in a kind appropriate for college kids and researchers in theoretical and computational chemistry and is with ease divided into elements. the 1st half, Chapters 1 to six, offers a pedagogical creation to the $64000 Lie algebras so(3), so(2,1), so(4) and so(4,2) wanted for the learn of easy quantum platforms comparable to the D-dimensional hydrogen atom and harmonic oscillator. This fabric is acceptable for complex undergraduate and starting graduate scholars. Of specific significance is using so(2,1) in bankruptcy four as a spectrum producing algebra for numerous very important structures akin to the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This method presents an enticing and critical substitute to the standard textbook technique utilizing sequence recommendations of differential equations.

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**Additional info for Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory**

**Example text**

21). , n2)' This shows that boson realizations of a Lie algebra can be used to find general representations ([SA86], [SC65]). 1 Introduction The Lie algebra so(2,1) plays a fundamental role in the algebraic reformulation and study of the radial Schrodinger equation for many of the important model problems in quantum mechanics such as the hydrogen atom, harmonic oscillator and one-electron diatomic ions. In this context so(2,1) appears as a spectrum-generating algebra in the sense that energy formulas such as the Bohr formula are all obtained in a unified and purely algebraic manner without resorting to the series solutions of second order radial differential equations.

16) Adding these results together gives m 2 ~ A which means that the eigenvalue spectrum of J3 is bounded above and below (case (4)) and the only unirreps are the finite dimensional ones. Let ml be the smallest value of m and let m2 be the largest value of m. Then J+tPAm2 = 0 and J_ tPAml = 0 which will be the case if AAm2 = 0 and B Aml = O. 16) ml and m2 satisfy A - ml(ml - 1) = 0 and A - m2(m2 + 1) = O. Eliminating A gives (ml +m2)(ml -m2 -1) = 0 so either ml = -m2 or ml = m2 + 1. The latter case is impossible since ml ~ m2.

26) and their generalizations will only be needed when we consider specific hamiltonians and their expressions in terms of the so(2,1) generators. We now consider operators of the form Rm PR in order to obtain three operators that close under commutation. Since hamiltonians are second order differential operators in the coordinate representation we can assume that n = 0,1,2 in the term RmpR. Therefore we choose the set {Ra, RbPR , RcPM and try to find a, band c such that this set of operators is closed under commutation.

### Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory by Barry G. Adams

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