By Barry G. Adams

This ebook presents an creation to using algebraic equipment and sym bolic computation for easy quantum structures with purposes to massive order perturbation idea. it's the first booklet to combine Lie algebras, algebraic perturbation concept and symbolic computation in a kind compatible for college students and researchers in theoretical and computational chemistry and is very easily divided into components. the 1st half, Chapters 1 to six, presents a pedagogical creation to the real Lie algebras so(3), so(2,1), so(4) and so(4,2) wanted for the learn of easy quantum structures corresponding to the D-dimensional hydrogen atom and harmonic oscillator. This fabric is acceptable for complicated undergraduate and starting graduate scholars. Of specific significance is using so(2,1) in bankruptcy four as a spectrum producing algebra for a number of vital structures reminiscent of the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This technique presents a fascinating and critical substitute to the standard textbook procedure utilizing sequence strategies of differential equations.

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**Example text**

1), the different choices give rise to different classes of irreps. If {T2, Td or {T2, T2} is chosen we obtain unirreps which would be suitable for a description of continuum or scattering states whereas the choice of {T2, T3} is suitable for a description of bound states. Therefore we make the latter choice and introduce an abstract vector space of states such that T2 and Ta are diagonal: T 21/JQq = Q1/JQq, T31/JQq = q1/JQq. 9) that T+ and L are raising and lowering operators for the Ta eigenvalue q: T3(T+1/JQq) = (q + l)T+1/JQq, Ta(T-1/JQq) = (q - l)T-1/JQq.

25) can be expressed in the equivalent form [J+, V_] [J3, V±] [J+, V+] = 2V3, = ±V±, [L, V+] [J±, V3] = -2V3' = =fV±, = [L, V_] = [h V3 ] = O. 34) Using these commutation relations and the identities derived in the preceding section we can evaluate matrix elements of the form (j'm'1V3Ijm) and 32 Chapter 3. Angular Momentum Theory and so(3) (j'm'IV±lim). The relationship between m' and m which corresponds to nonvanishing matrix elements is called the m-selection rule and similarly the relationship between j' and j is called the j-selection rule.

The following two rules are often useful for moving functions of an operator inside or outside a commutator: f(A)[g(A), BJ = [g(A), f(A)B], [A, g(B)]J(B) = [Af(B),g(B)]. 4) Chapter 2. Commutator Gymnastics 18 Here we assume that a function of an operator is a polynomial or a formal power series in the operator without regard to the convergence of the power series. Thus if f(x) = Ej aixi is a formal power series in the real variable x and A denotes an operator then by f(A) we mean the series obtained by formally replacing x by A.