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 important in 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 saw 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 bosons can occupy the same state.

Key Definitions

Note

Fugacity:: Fugacity $z= e ^{\beta \mu}$ 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 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:

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