By Barry G. Adams

This e-book offers an advent to using algebraic equipment and sym bolic computation for easy quantum platforms with purposes to massive order perturbation thought. it's the first publication to combine Lie algebras, algebraic perturbation concept and symbolic computation in a kind compatible for college kids and researchers in theoretical and computational chemistry and is with ease divided into components. the 1st half, Chapters 1 to six, offers a pedagogical creation to the real Lie algebras so(3), so(2,1), so(4) and so(4,2) wanted for the research of straightforward quantum structures resembling the D-dimensional hydrogen atom and harmonic oscillator. This fabric is appropriate 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 platforms equivalent to the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This method offers a fascinating and critical substitute to the standard textbook technique utilizing sequence strategies of differential equations.

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**Extra resources for Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory**

**Sample text**

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. If we let j = m2 then ml = -j and A = j(j + 1). Since m2 - ml = 2j must be a non-negative integer (J+ and L raise or lower m in unit steps) it follows that the J3 eigenvalue spectra are Sj = {m = -j,-j + 1, ...

Realizations of so(2,1) These realizations are sufficiently general for our purposes. In fact we shall use a = 1 for the hydrogen atom and a = 2 for the harmonic oscillator. 27) was used. 56). 18) 1 - a2 Solving for k gives k T+~=k(k+1). 23) have the form q = qo +1', with lowest eigenvalue qo I' = k + 1, = 0,1,2, ... k>-1. 61) 50 Chapter 4. Representations and Realizations of so(2,1) Before considering the general case of so(2,1) as a spectrum generating algebra for simple quantum systems and the generalization to D-dimensional cases we show how so(2,1) can be used to obtain the energy levels for the two most important cases: the 3-dimensional hydrogen atom and the 3-dimensional isotropic harmonic oscillator.

23) The "double bar" matrix elements are called reduced matrix elements. They cannot be evaluated unless further algebraic or physical properties of S are known. 18) ]2 is a scalar operator with reduced matrix elements (jIIJ2 1Ij) = j(j + 1). 25) where the summation over repeated indices is implied. 7. Selection Rules for a Vector Operator 31 operator. 24) also show that rand p are vector operators. We now obtain several identities relating J and V which will be used to determine matrix elements of components of V.