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

A_{j} = \prod_{i \neq j} W_{k, i},

(5.4)

the probability for occupying the state $j$ with an electron is proportional to

A_{j} \exp (- \frac{E_{j} - μ}{k T}) .

(5.5)

The electron in state $j$ adds the energy $(E_{j} - μ)$ to the complete energy.
The probability that the state $j$ is not occupied is proportional to

A_{j},

(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 $j$ :

W_{j} = \frac{A_{j} \exp (- \frac{E_{j} - μ}{k T})}{A_{j} + A_{j} \exp (- \frac{E_{j} - μ}{k T})} = \frac{1}{1 + \exp (\frac{E_{j} - μ}{k T})} .

(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 $j$ .