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 mainly 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 distributions hinges on the following result from quantum mechanics:

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

It so happens that fully antisymmetric states are called Fermions, while symmetric states are Bosons. 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 the distribution function (that is: the probability of finding a particle in a single-particle energy level (a microstate) with energy :math: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 limit where the chemical potential is large and negative (corresponding to a dilute system), 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 antisymmetric 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

At low energy, Fermions and Bosons follow the same statistical distribution.
1. This is sometimes true.
2. This is always true.
3. This is never true.
Consider the exchange operator $\hat{P}_{12}$ whose effect is to swap two particles. What is true among the assertions below?
1. The operator is unitary but not Hermitian.
2. The eigenvalues of the operator are positive.
3. The eigenvalues of the operator can only take two values for any wavefunction describing a pair of Bosons or a pair of Fermions.
4. None of the other claims are correct.
5. The eigenvalues of the operator are complex.
What can you say about the chemical potential ( $\mu$ )?
1. It is small when the density of matter ( $n$ ) is small.
2. It does not depend on the density of matter ( $n$ ).
3. It is small when the density of matter ( $n$ ) is large.
Consider a bosonic particle extracted from a large distribution of the same particles. A specific measurement shows that its energy is 1~eV below the chemical potential of the distribution. What can you conclude?
1. Researchers who performed the measurements should sign up for PHYS-2350 at RPI: their measurement is clearly wrong.
2. The entropy of the particle is very small.
3. The particle is very stable.
Among the following claims, which one is not true?
1. All the other claims are correct.
2. Multiple Fermions can occupy the same quantum state.
3. Bosons have an integer spin value and Fermions have an half integer spin value.
4. Bosons and Fermions are treated as indistinguishable particles.
5. Multiple Bosons can occupy the same quantum state.
At high energy (large $E-\mu$ values), we can use the Botzmann, Fermi-Dirac, or Bose-Einstein disributrions to study any gas of particles (e.g., photons, electrons,…)
1. This is false because the three distributions do not converge to one another at high energy.
2. This is a very good question. I wil make sure to read page 598 of the slides posted on SLACK to understand the answer.
3. This is true because this corresponds to the low density limit.

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!