Boson-magnetism Consider a gas of non-interacting bosons with mass $m > 0$ and spin 1. The Hamiltonian of each boson is given by
$H_1 = \frac{p^2}{2m} - \mu_B B S_1$
(2.1)
Where $S_1 \in \{-1, 0, 1\}$ is the spin of the particle, B is the external magnetic field and $\mu_B$ is a constant.
1. [5] In a grand canonical ensemble with chemical potential $\mu$, what are the average occupation numbers \{$<n_{-}>$, $<n_0>$, $<n_{+}>$\} of the one-particle states?
2. [5] Show that the average total number of each spin $N_{-}$, $N_0$, $N_{+}$ are given by
$N_s = \frac{V}{\lambda_T^3} g_{3/2}(ze^{\beta s \mu_B B})$
(2.2)
Where $g_{3/2}$ is a Bose-Einstein function (as defined in the Script and in the notes), $s \in \{-1, 0, 1\}$, $z = e^{\beta \mu}$, and $\lambda_T$ the thermal wavelength.
3. [5] Using the expressions obtained in part 2) above, express the magnetization of the system: $M = \mu_B (N_+ - N_{-})$.
