%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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, the magnetic field $B$ is strong that is directed in the $Z$ direction. % 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. $\int_{0}^{\infty}x^{3}e^{-x^{2}}dx=\frac{1}{2},\int_{0}^{\infty}\frac{x^{3}}{e^{x^{2}}-1}dx=\frac{\pi^{2}}{12}$