Coulomb Expectation Values In D=3 And D=3-2 Epsilon Dimensions

We explore the quantum Coulomb problem for two-body bound states, in D = 3 and D = 3 - 2 epsilon dimensions, in detail, and give an extensive list of expectation values that arise in the evaluation of QED corrections to bound-state energies. We describe the techniques used to obtain these expectation values and give general formulas for the evaluation of integrals involving associated Laguerre polynomials. In addition, we give formulas for the evaluation of integrals involving subtracted associated Laguerre polynomials, those with low powers of the variable subtracted off, that arise when evaluating divergent expectation values. We present perturbative results (in the parameter epsilon) that show how bound-state energies and wave functions in D = 3 - 2 epsilon dimensions differ from their D = 3 dimensional counterparts and use these formulas to find regularized expressions for divergent expectation values such as <(V) over bar (3)> and <((V) over bar')(2)> where (V) over bar is the D-dimensional Coulomb potential. We evaluate a number of finite D-dimensional expectation values such as < r(-2+4 epsilon)partial derivative(2)(r)> and < r(4 epsilon) p(4)> that have epsilon -> 0 limits that differ from their three-dimensional counterparts < r(-2)partial derivative(2)(r)> and < p(4)>. We explore the use of recursion relations, the Feynman-Hellmann theorem, and momentum-space brackets combined with D-dimensional Fourier transformation for the evaluation of D-dimensional expectation values. The results of this paper are useful when using dimensional regularization in the calculation of properties of Coulomb bound systems.