  This paper is the continuation of a previous one [L. {\v{S}}amaj and B.
Jancovici, 2007 {\it J. Stat. Mech.} P02002]; for a nearly classical quantum
fluid in a half-space bounded by a plain plane hard wall (no image forces), we
had generalized the Wigner-Kirkwood expansion of the equilibrium statistical
quantities in powers of Planck's constant $\hbar$. As a model system for a more
detailed study, we consider the quantum two-dimensional one-component plasma: a
system of charged particles of one species, interacting through the logarithmic
Coulomb potential in two dimensions, in a uniformly charged background of
opposite sign, such that the total charge vanishes. The corresponding classical
system is exactly solvable in a variety of geometries, including the present
one of a half-plane, when $\beta e^2=2$, where $\beta$ is the inverse
temperature and $e$ is the charge of a particle: all the classical $n$-body
densities are known. For the quantum one-component plasma, two sum rules
involving the truncated two-body density (and, for one of them, the density
profile) have been derived, a long time ago, by heuristic macroscopic
arguments: one sum rule is about the asymptotic form along the wall of the
truncated two-body density, the other one is about the dipole moment of the
structure factor. In the two-dimensional case at $\beta e^2=2$, we have now
explicit expressions up to order $\hbar^2$ of these two quantum densities, thus
we can microscopically check the sum rules at this order. The checks are
positive, reinforcing the idea that the sum rules are correct.
