bose-einstein gasstatistical mechanicsThermal and Statistical Physics 2020
Consider one particle
confined to a cube of side \(L\); the concentration in effect is
\(n=L^{-3}\). Find the kinetic energy of the particle when in the
ground state. There will be a value of the concentration for which
this zero-point quantum kinetic energy is equal to the temperature
\(kT\). (At this concentration the occupancy of the lowest orbital is
of the order of unity; the lowest orbital always has a higher
occupancy than any other orbital.) Show that the concentration \(n_0\)
thus defined is equal to the quantum concentration \(n_Q\) defined by
(63): \begin{equation}
n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}
\end{equation} within a factor of the order of unity.

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.