By Barry G. Adams

This booklet offers an creation to using algebraic equipment 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 idea and symbolic computation in a sort compatible for college students and researchers in theoretical and computational chemistry and is very easily divided into elements. the 1st half, Chapters 1 to six, offers a pedagogical advent to the real Lie algebras so(3), so(2,1), so(4) and so(4,2) wanted for the examine of easy quantum structures equivalent to the D-dimensional hydrogen atom and harmonic oscillator. This fabric is acceptable for complex undergraduate and starting graduate scholars. Of specific value is using so(2,1) in bankruptcy four as a spectrum producing algebra for a number of very important structures similar to the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This method presents an enticing and demanding substitute to the standard textbook strategy utilizing sequence recommendations of differential equations.

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

**Example text**

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. 4. Realizations of so(2,1) 47 [RbpR,RCP~] =Rb[PR,RC]P~+RC[Rb,p~]PR' = RC[Ra, P~]. 31 ) l , k=O [Rn,p~] = PR[Rn,PR] + [Rn,PR]PR = PR(inR n- l ) + (inW-I)PR = in(Rn- 1PR - i(n - 1)Rn-2) = 2inRn-1 PR + n(n - 1)Rn-2. 30) to obtain [R a, Rb PRJ = iaR a+b- l , [R bPR' RCP~] = i(2b - C)Rb+C-I P~ + b(b - 1)Rb+ c-2 PR' [Ra,Rcp~] = 2iaR a+c- IPR + a(a _1)Ra+c-2.

70) ""('j'm' =L "'('i'm' = (_I)i'-m'( j', 1 j){Tlj'IIVlbj)b/jlm/) L( _1)i'-m- "'('i' Q( -m qm j' 1 j) -m - q q m (T'j'11V II ,jHr/j', m + q). Only terms with j' = j - 1, j, or j + 1 contribute. In the reduced matrix elements are diagonal in the additional by , and there is no summation over ,'. 59). 5The notation (ilh mlm2ljm) is also common. 10. Exercises 39 Useful tabulations of the CG coefficients are available and they are usually expressed in terms of the 3-j symbol because of its higher symmetry properties ([ED57], [BR68]).

16) 20 Chapter 2. Commutator Gymnastics There are also several useful commutators involving p2 = pop = PJPJ and rop=xJPJ: [r,p2] = 2ip, [f(r),p2] =2ir- 1J'(r)(r op-i)+J"(r), [r op,p2] =2ip2, [r, r ° p] = ir, [p, r ° p] = -ip, [f(r),rop] =irf'(r). 10). 11) with r-1f'(r) in place of fer) to obtain pJr-1f'(r) = r-1J'(r)pJ = r- 1 + [PJ,r-1J'(r)] d J'(r)pJ - ixJr- 1dr (r- J'(r)) . 1 Substitute this result in the first term to obtain [f(r),p2] ~ (r-1J'(r))) xJ + ir-1f'(r)x;pJ = i (r- 1J'(r)pJ - ixJr- 1 = ir- 1J'(r)pJx J + r (r- 1J"(r) - r- 2J'(r)) + ir- 1J'(r)r ° p.