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\sect{Effusion from a box with Bose gas and magnetic field}

Bosons that have  mass ${m}$ and spin~1 with gyromagnetic ratio~${\gamma}$ 
are placed in a box. The temperature ${T}$ is below the condensation temperature. 
A strong magnetic field $B$ is applied in the $z$ direction. 
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A hole that has small area ${\delta A}$ is drilled in the box
so the particles can flow out. The flux is separated into 3~beams 
using a Stern-Gerlach aparatus. Each beam is directed into a different container.


\begin{itemize}

\item[(a)] Write the single particle Hamiltonian.

\item[(b)] Find the velocity distribution ${F_{S_{z}}(v)}$ for ${S_{z}=-1,0,1}$.

\item[(c)] Define what does it mean a strong magnetic field, and explain why and how
it helps for the solution of the next item.

\item[(d)] Find how many particles are accumulated in each container after
time $t$.

\item[(e)] Find what would be the velocity distribution for horizontal 
filtering ${S_{x}=-1,0,1}$ of the beam.

\end{itemize}

Express your answer using ${\mass, \gamma, B, \delta A, T, t}$. 
In the last item assume that $F_{S_{z}}(v)$ is known, 
irrespective of whether the second item has been solved.

\[
\int_{0}^{\infty}x^{3}e^{-x^{2}}dx=\frac{1}{2},\int_{0}^{\infty}\frac{x^{3}}{e^{x^{2}}-1}dx=\frac{\pi^{2}}{12}
\]

