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

 Aj=ijWk,i,(5.4)

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

 Ajexp(EjμkT).(5.5)

The electron in state j adds the energy (Ejμ) to the complete energy.
The probability that the state j is not occupied is proportional to

 Aj,(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:

 Wj=Ajexp(EjμkT)Aj+Ajexp(EjμkT)=11+exp(EjμkT).(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 Aj is independent of the occupation of state j.


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© J. Carstensen (Quantum Mech.)