Show that \begin{align} f(\mu+\delta) &= 1 - f(\mu-\delta) \end{align} This means that the probability that an orbital above the Fermi level is occupied is equal to the probability an orbital the same distance below the Fermi level being empty. These unoccupied orbitals are called holes.

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.

1. 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 \(\mu\).

2. 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)?

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).