Lecture 21: Bose-Einstein and Fermi-Dirac Distributions

Note

The [Fermi-Dirac] solution… is probably the correct one for gas molecules, since it is known to be the correct one for electrons in an atom, and one would expect molecules to resemble electrons more closely than light quanta. – P. A. M. Dirac, 1926

Warning

This lecture corresponds to Chapter 29 of the textbook.

Summary

Attention

So far, this course has largely treated particles as classical objects. Statistical mechanics consists largely in enumerating states and assessing their probability: this is the fundamental basis of the concept of partition functions. This is particularly important, as we have seen, for indistinguishable particles. We took this into account by introducing approximations such as the case where the number of states is much larger than the number of particles. However, when we enumerated those states, we did not take into account the fact that some states might not be allowed by quantum mechanics. The entire concept of Bose-Einstein and Fermi-Dirac distribution hinges on the following result from quantum mechanics:

A quantum mechanical system of indistringuisahle particles is described by a wave-function that is either totally symmetric or antisymmetric upon exchange of two particles.

It so appears that fully anti-symmetric states are called Fermion while symmetric states are Boson. Why does it matter when calculating partition functions? It matters because only allowed states will contribute to the partition function.

The mathematical analysis of the situation described in the screencast shows that the partition functions are, for Fermions (+) and Bosons (-):

$\ln Z =\pm \sum_{i} \ln \left(1 \pm e ^{\beta\left(\mu-E_{i}\right)}\right).$

Using this result, we find that distribution function (that is: the probability of finding a particle in a macrostate described by an energy $E$ at temperature $T$ ) for each case is given by:

$f(E)=\frac{1}{ e ^{\beta(E-\mu)}\pm1}$

where Fermions are described by the Fermi-Dirac distribution and Bosons are described by the Bose-Einstein distribution.

In the small chemical potential limit (which corresponds to the low density limit), the Fermi-Dirac and Bose-Einstein distributions coincide with the Boltzmann distribution: this makes sense since, in that limit, most single-particle states are singly occupied and the effect of symmetrization is not critical. The situation at high density is very different and this is where manifestations of quantum mechanics are prevalent.

Key Definitions

Note

Fermi-Dirac distribution:: Probability of finding a Fermion in energy $E$ at temperature $T$ . This corresponds to anti-symmetric wave functions.
Bose-Einstein distribution:: Probability of finding a Boson in energy $E$ at temperature $T$ . This corresponds to symmetric wave functions.

A full list of terms, including the ones provided here, can be found in the Index.

Learning Material

Copy of Slides

The slides for Lecture 21 are available in pdf format here: pdf

Screencast

Test your knowledge

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: