Explain what is meant by the microcanonical ensemble for a quantum system. Sketch how to derive the probability distribution for the canonical ensemble from the microcanonical ensemble. Under what physical conditions should each type of ensemble be used?

A paramagnetic solid contains atoms with magnetic moment $\boldsymbol{\mu}=\mu_{B} \mathbf{J}$, where $\mu_{B}$ is a positive constant and $\mathbf{J}$ is the intrinsic angular momentum of the atom. In an applied magnetic field $\mathbf{B}$, the energy of an atom is $-\boldsymbol{\mu} \cdot \mathbf{B}$. Consider $\mathbf{B}=(0,0, B)$. Each atom has total angular momentum $J \in \mathbb{Z}$, so the possible values of $J_{z}=m \in \mathbb{Z}$ are $-J \leqslant m \leqslant J$.

Show that the partition function for a single atom is

$$Z_{1}(T, B)=\frac{\sinh \left(x\left(J+\frac{1}{2}\right)\right)}{\sinh (x / 2)}$$

where $x=\mu_{B} B / k T$.

Compute the average magnetic moment $\left\langle\mu_{z}\right\rangle$ of the atom. Sketch $\left\langle\mu_{z}\right\rangle / J$ for $J=1$, $J=2$ and $J=3$ on the same graph.

The total magnetization is $M_{z}=N\left\langle\mu_{z}\right\rangle$, where $N$ is the number of atoms. The magnetic susceptibility is defined by

$$\chi=\left(\frac{\partial M_{z}}{\partial B}\right)_{T}$$

Show that the solid obeys Curie's law at high temperatures. Compute the susceptibility at low temperatures and give a physical explanation for the result.