Lecture 22: Quantum Gases and Condensates

Note

Bose-Einstein condensation is one of the most intriguing phenomena predicted by quantum statistical mechanics. – Wolfgang Ketterle

Warning

This lecture corresponds to Chapter 30 of the textbook.

Summary

Attention

This lecture uses results from Lecture 21: Bose-Einstein and Fermi-Dirac Distributions to study the thermal properties of gases of quantum particles obeying either the Fermi-Dirac or Bose-Einstein distributions.

First, we study the Fermi gas: In a Fermi gas (i.e., a gas of fermions), fermions fill states up to $E_F$ at absolute zero. The Pauli exclusion principle ensures that fermions only singly occupy states. The results for a Fermi gas can be applied to the electrons in a metal. At non-zero temperature, electrons with energies within $k_B T$ of $E_F$ are primarily responsible for determining the properties. We find that the heat capacity in such a system scales linearly with temperature, in strong contrast with the physics of phonons we saw in Lecture 16: Phonons, where we found that heat capacity due to phonons scales as the third power of temperature. We conclude that heat capacity at room temperature is dominated by phonons.

Second, we study a Bosonic gas and realize that the treatment of the partition function using an integral rather than a sum leads to difficulties in understanding the behavior of the system at very low temperature. It so happens that at high density and low temperature, a new phase of matter develops: it is called the Bose-Einstein condensate where there is a diverging occupation for the lowest energy state. This is allowed by quantum mechanics since more than one boson can occupy the same state.

Key Definitions

Note

Fugacity:: Fugacity $z= e ^{\beta \mu}$ is a measure of the effective pressure or activity of particles. It is especially useful in describing non-ideal and quantum gases.is At low density, we have: $z= \ll 1$ . For bosons treated in a non-interacting scheme, we have that $0<z<1$ . The larger $z$ , the more evident are the quantum effects.
Fermi level:: Fermi level is the highest occupied energy level of a Fermionic state at $T=0$ .
Fermi surface:: Set of points in $k$ -space whose energy is equal to the chemical potential
Degeneracy:: Degeneracy describes the fact that two different states (i.e., different wave-functions) correspond to the same eigenvalue, e.g. energy.
Polylogarithm function:: The polylogarithm function is defined as:

$Li _{n}(z)=\sum_{k=1}^{\infty} \frac{z^{k}}{k^{n}}.$
Zeta function:: The Riemann Zeta function is defined as:

$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}.$
Gamma function:: The Gamma function is defined as:

$\Gamma(n)=\int_{0}^{\infty} x^{n-1} e ^{-x} d x.$

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 22 are available in pdf format here: pdf

Screencast

Test your knowledge

A Bose Einstein condensate has been demonstrated for laser-cooled alkali atoms. It is just a matter of time, and will probably lead to a Nobel Prize, before cryogenic technology reaches a sufficiently low temperature to demonstrate a Fermi Dirac condensate.
1. In fact, this has already been demonstrated in experiments performed in quantum cavities
2. False, because of the symmetry of the wave functions of Fermions, such a condensate is not possible.
3. There are some indications that this phase of matter may have been present at the early universe.
4. True, and the most likely systems to reach such a phase are positrons
The heat capacity of a metal has two contributions: phonons and electrons. Which one dominates at low $T$ ?
1. The contribution of phonons goes as $T^3$ and that of electrons goes as $T$ , therefore, at low $T$ , electrons contribute more.
2. They contribute almost equally to the heat capacity at all temperatures.
3. This is an excellent question. I’m glad you asked it.
4. The contribution of phonons goes as $T^3$ and that of electrons goes as $T$ , therefore, at low $T$ , phonons contribute more.
A Bose-Einstein condensate:
1. forms when the temperature is large and the density is low
2. forms the same way as a phase transition: it is due to increased interactions between bosons
3. can be formed with paired fermions so long as the pairs have integer spin values
4. forms when the temperature is large and the density is large
5. has never been observed, though it would be really cool if one could
What probability distribution corresponds to a collection of identical spin- $\frac{1}{2}$ particles?
1. The Bose-Einstein distribution.
2. The Fermi-Dirac distribution.
3. The answer to this question depends on the charge of the particles.
4. The Boltzmann distribution.
What is the Fermi energy?
1. The highest energy occupied by a Fermion at any temperature.
2. The highest energy occupied by a Fermion at $T=0$ . In this case (when $T=0$ ), it is the same as the chemical potential.
3. The highest energy occupied by a Fermion at $T=0$ . In this case (when $T=0$ ), it is not the same as the chemical potential.
4. The energy students must apply to succeed, firmly.
5. This is equivalent to Debye’s energy introduced when studying phonons.
What is the fugacity?
1. It is the name of term $k_BT$ , often written $\beta$ .
2. I really have no idea what this question has to do with the course.
3. It is a way to concisely represent the ratio between the chemical potential and the temperature.
4. It is a measure of my desire to end the semester and flee.
What probability distribution corresponds to a collection of classical particles?
1. The Fermi-Dirac distribution.
2. The Bose-Einstein distribution.
3. The Boltzmann distribution.
4. None of the other replies is correct.
In general when can you get away with treating the statistics of particles using classical theory and Boltzmann distribution?
1. At high temperature.
2. At low temperature.
3. At high density of particles.
4. At low density of particles.
5. Answers A and D are correct.

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!