Canonical density matrix for free electrons moving on a spherical surface

The energy levels and eigenfunctions for electrons that are confined to moving on the surface of a sphere but are otherwise free are utilized to calculate the corresponding canonical density matrix. This matrix is then expanded in an asymptotic series for a large sphere radius R. The leading term naturally corresponds to free electrons moving on a plane and the first correction term to this, O(${\mathit{R}}^{\mathrm{-}2}$), is exhibited explicitly. This matrix is then utilized to construct both the Dirac density matrix and the Green function in the same approximation. This Green function, in relation, say, to K-doped ${\mathrm{C}}_{60}$, where added electrons are localized near the surface of a sphere, serves to emphasize first the quasi-two-dimensional character of the motion, and then to exhibit the corrections for a finite ball radius.