Algebraic Approach to Simple Quantum Systems: With by Barry G. Adams

By Barry G. Adams

This ebook presents an creation to using algebraic equipment and sym­ bolic computation for easy quantum platforms with functions to massive order perturbation concept. it's the first publication to combine Lie algebras, algebraic perturbation idea and symbolic computation in a kind appropriate for college students and researchers in theoretical and computational chemistry and is comfortably divided into components. the 1st half, Chapters 1 to six, presents 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 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 corresponding to the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This strategy offers an enticing and demanding substitute to the standard textbook procedure utilizing sequence ideas of differential equations.

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There are several important ideas related to the above derivations. 69) would lead to positive energies corresponding to scattering states and a TI eigenvalue equation. 74) justifies our earlier choice of {T2,T3} as the set of commuting operators to diagonalize. 70) of T, do form a realization of so(2,1). This follows since the components L j of the orbital angular momentum (and hence L2) commute with the so(2,1) generators: [Lj, T k ] = 0, j, k = 1,2,3. 81 ) by identifying the abstract basis function Iqk} with CPnl(R) and that they form a basis for the direct sum algebra so(2,1) EEl so(3), corresponding to the direct product 80(2,1) 0 80(3) of the associated Lie groups, in the sense that the set of 6 operators Lj,Tj, j = 1,2,3 generate the Lie algebra so(2,1) EEl so(3).

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, ... ,j}, 2j = 0,1,2,... 18) J(j + m)(j - m + 1) Ij, m - 1). 20) 30 Chapter 3. Angular Momentum Theory and so(3) Strictly speaking the unirreps for integer values of j (corresponding to the orbital angular momentum) are unirreps of so(3) and those for non-integer (half-integer) values are the so-called spinor representations of su(2).

In this chapter we first consider the so(2,1) representation theory following the same approach used in the previous chapter for so(3). These two algebras have a similar structure and the general representation theories for both are closely related. However we are only interested in the unitary irreducible representations (unirreps) which are obtained by requiring that the generators be hermitian with respect to a suitable scalar product. In this case the unirreps of so(2,1) are infinite dimensional in contrast to the unirreps of so(3) which are finite dimensional.

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