Let us imagine a new mechanics in which the allowed occupancies
of an orbital are 0, 1, and 2. The values of the energy associated
with these occupancies are assumed to be \(0\), \(\varepsilon\), and
\(2\varepsilon\), respectively.
Derive an expression for the ensemble average occupancy
\(\langle N\rangle\), when the system composed of this orbital is in
thermal and diffusive contact with a resevoir at temperature \(T\)
and chemical potential \(\mu\).
Return now to the usual quantum mechanics, and derive an expression
for the ensemble average occupancy of an energy level which is
doubly degenerate; that is, two orbitals have the identical energy
\(\varepsilon\). If both orbitals are occupied the toal energy is
\(2\varepsilon\). How does this differ from part (a)?