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Fermi-Dirac Statistics

Let us, first of all, consider Fermi-Dirac statistics. According to Equation (8.19), the average number of particles in quantum state can be written

 (8.21)

Here, we have rearranged the order of summation, using the multiplicative properties of the exponential function. Note that the first sums in the numerator and denominator only involve , whereas the last sums omit the particular state from consideration (this is indicated by the superscript on the summation symbol). Of course, the sums in the previous expression range over all values of the numbers such that and 1 for each , subject to the overall constraint that

 (8.22)

Let us introduce the function

 (8.23)

which is defined as the partition function for particles distributed over all quantum states, excluding state , according to Fermi-Dirac statistics. By explicitly performing the sum over and 1, the expression (8.21) reduces to

 (8.24)

which yields

 (8.25)

In order to make further progress, we must somehow relate to . Suppose that . It follows that can be Taylor expanded to give

 (8.26)

where

 (8.27)

As always, we Taylor expand the slowly-varying function , rather than the rapidly-varying function , because the radius of convergence of the latter Taylor series is too small for the series to be of any practical use. Equation (8.26) can be rearranged to give

 (8.28)

Now, because is a sum over very many different quantum states, we would not expect the logarithm of this function to be sensitive to which particular state, , is excluded from consideration. Let us, therefore, introduce the approximation that is independent of , so that we can write

 (8.29)

for all . It follows that the derivative (8.27) can be expressed approximately in terms of the derivative of the full partition function (in which the particles are distributed over all quantum states). In fact,

 (8.30)

Making use of Equation (8.28), with , plus the approximation (8.29), the expression (8.25) reduces to

 (8.31)

This is called the Fermi-Dirac distribution. The parameter is determined by the constraint that : that is,

 (8.32)

Note that if becomes sufficiently large. On the other hand, because the denominator in Equation (8.31) can never become less than unity, no matter how small becomes, it follows that . Thus,

 (8.33)

in accordance with the Pauli exclusion principle.

Equations (8.20) and (8.30) can be integrated to give

 (8.34)

where use has been made of Equation (8.31).

Next: Photon Statistics Up: Quantum Statistics Previous: Formulation of Statistical Problem
Richard Fitzpatrick 2016-01-25