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\sect{The functions $N(E)$ and $Z(T)$ for particle in a box with gravitation}


Find the distribution function ${Z\left(\beta\right)}$ of a particle
in a three dimensional box with a gravitation field along axis
${-Z}$. Assume the box dimensions are ${L\times
L\times\left(Z_{b}-Z_{a}\right)}$

Guideline: write the hamiltonian and calculate
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\[Z\left(\beta\right)=\int\int\int
\frac{dxdp_{x}}{z\pi}\frac{dydp_{y}}{2\pi}\frac{dzdp_{z}}{2\pi}e^{-\beta
H\left(x,y,z,p_{x}p_{y}p_{z}\right)}\]



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