The Fermi statistics

5.3 The Fermi statistics

The Fermi statistics is a direct consequence of Eq. (5.3) assuming the independence of the occupation of individual states. Using the definition

\begin{equation*} A_j = \prod\limits_{i\neq j} W_{k,i} \qquad , \end{equation*}

(5.4)

the probability for occupying the state \(j\) with an electron is proportional to

\begin{equation*} A_j \exp \left(- \frac{E_j - \mu}{kT} \right) \qquad . \end{equation*}

(5.5)

The electron in state \(j\) adds the energy \((E_j - \mu)\) to the complete energy.
The probability that the state \(j\) is not occupied is proportional to

\begin{equation*} A_j \qquad , \end{equation*}

(5.6)

since in this case the energy zero is added to the complete energy.
This are all possibilities for a Fermion to occupy a state; consequently we find the probability for occupying the state \(j\):

\begin{equation*} W_j = \frac{ A_j \exp \left(- \frac{E_j - \mu}{kT} \right) }{A_j + A_j \exp \left(- \frac{E_j - \mu}{kT} \right)} = \frac{ 1 }{1 + \exp \left( \frac{E_j - \mu}{kT} \right)}\qquad . \label{fermi} \end{equation*}

(5.7)

The Fermi statistics describes the probability to occupy a one electron state in an ensemble. Essential for this calculation are independent electrons since for the above calculation we need that \(A_j\) is independent of the occupation of state \(j\).