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As the occupation number only takes the values 0 and 1, the first bracket in expression (17) is equal to [e^{–β(ei – μ)}]; as for the other modes (k ≠ i) contribution, in the second bracket, it has already been computed when we determined the partition function. We therefore obtain:

(18)

Multiplying both the numerator and denominator by 1 + e^{–β (ei – μ)} allows reconstructing the function Z in the numerator, and, after simplification by Z, we get:

(19)

We find again the Fermi-Dirac distribution function (§ 1-b of Complement C_XIV):

(20)

This distribution function gives the average population of each individual state |ui〉 with energy e; its value is always less than 1, as expected for fermions.

The average value at thermal equilibrium of any one-particle operator is now readily computed by using (19) in relation (15).