  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 $\big \langle \bar V^3 \big \rangle$ and
$\big \langle (\bar V')^2 \big \rangle$ where $\bar V$ is the $D$-dimensional
Coulomb potential. We evaluate a number of finite $D$-dimensional expectation
values such as $\big \langle r^{-2+4\epsilon} \partial_r^2 \big \rangle$ and
$\big \langle r^{4\epsilon} p^4 \big \rangle$ that have $\epsilon \rightarrow
0$ limits that differ from their three-dimensional counterparts $\big \langle
r^{-2} \partial_r^2 \big \rangle$ and $\big \langle p^4 \big \rangle$. 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.

Quantum Physics;High Energy Physics - Phenomenology