Classical and Quantum Ideal Gases

Abstract

Bose and Einstein’s prediction of Bose–Einstein condensation came out of their theory for how quantum particles in a gas behaved, and was built on the pioneering statistical approach of Boltzmann for classical particles. Here we follow Boltzmann, Bose and Einstein’s footsteps, leading to the derivation of Bose–Einstein condensation for an ideal gas and its key properties.

Problems

2.1

Consider a system with 6 classical particles, total energy of $6 \epsilon$, and 7 cells with energies $0, \epsilon , 2 \epsilon , 3 \epsilon , 4 \epsilon , 5\epsilon$ and $6 \epsilon$. Complete the table below by entering the cell populations for each macrostate, the statistical weighting for each macrostate W, and the average population per cell $\bar{N}(E)$ (averaged over macrostates). What is the most probable macrostate? Plot $\bar{N}(E)$ versus E. It should be evident that the average distribution approximates the Boltzmann distribution, despite the small number of particles.

Macrostates
Cell energy E	1	$\cdots$	11	$\bar{N}(E)$
6$\epsilon$	?	$\cdots$	?	?
5$\epsilon$	?	$\cdots$	?	?
$\vdots$	$\vdots$	$\cdots$	$\vdots$	$\vdots$
$\epsilon$	?	$\cdots$	?	?
0	?	$\cdots$	?	?
Statistical weighting W	?	$\ldots$	?

2.2

Consider a system with N classical particles distributed over 3 cells (labelled 1, 2,  and 3) of energy 0, $\epsilon$ and $2\epsilon$. The total energy is $E=0.5 N \epsilon$.

• (a) Obtain an expression for the number of microstates in terms of N and $N_3$, the population of cell 3.

• (b) Plot the number of microstates as a function of $N_2$ (which parameterises the macrostate) for $N=50$. Repeat for $N=100$ and 500. Note how the distribution changes with N. What form do you expect the distribution to tend towards as N is increased to much larger values?

2.3

Consider an ideal gas of bosons in two dimensions, confined within a two-dimensional box of volume $\mathcal {V}_\mathrm{2D}$.

• (a) Derive the density of states g(E) for this two-dimensional system.

• (b) Using this result show that the number of particles can be expressed as,

  $$\begin{aligned} N_\mathrm{ex}=\frac{2\pi m \mathcal {V}_{2D} k_\mathrm{B}T}{h^2}\int ^\infty _0 \frac{ze^{-x}}{1-ze^{-x}}dx, \nonumber \end{aligned}$$

  where $z=e^{\mu /k_\mathrm{B}T}$ and $x=E/k_\mathrm{B}T$. Solve this integral using the substitution $y=z e^{-x}$.

• (c) Obtain an expression for the chemical potential $\mu$ and thereby show that Bose–Einstein condensation is possible only at $T=0$.

2.4

Equation (2.25) summarizes how the internal energy of the boxed 3D ideal Bose gas scales with temperature. Derive the full expressions for the internal energy for the two regimes (a) $T<T_\mathrm{c}$ (for which $z=1$), and (b) $T\gg T_\mathrm{c}$ (for which $z\ll 1$). Extend your results to derive the expressions for the heat capacity given in Eq. (2.27).

2.5

Bose–Einstein condensates are typically confined in harmonic trapping potentials, as given by Eq. (2.28). Using the corresponding density of states provided in Sect. 2.5.10.1:

• (a) Derive the expression for the critical number of particles.

• (b) Derive the expression (2.29) for the critical temperature.

• (c) Determine the expression (2.30) for the variation of condensate fraction $N_0/N$ with $T/T_\mathrm{c}$.

• (d) In one of the first BEC experiments, a gas of 40, 000 Rubidium-87 atoms (atomic mass $1.45 \times 10^{-25}$ kg) underwent Bose–Einstein condensation at a temperature of 280 nK. The harmonic trap was spherically-symmetric with with $\omega _r=1130$ Hz. Calculate the critical temperature according to the ideal Bose gas prediction. How does this compare to the result for the boxed gas (you may assume the atomic density as $2.5 \times 10^{18}$ m$^{-3}$).

2.6

The compressibility $\beta$ of a gas, a measure of how much it shrinks in response to a compressional force, is defined as,

$$\begin{aligned} \beta =-\frac{1}{\mathcal {V}}\frac{\partial \mathcal {V}}{\partial P}. \nonumber \end{aligned}$$

Determine the compressibility of the ideal gas for $T<T_\mathrm{c}$.

Hint: Since $T_\mathrm{c}$ is a function of $\mathcal {V}$, you should ensure the full $\mathcal {V}$-dependence is present before differentiating.