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

In this question one must evaluate ${Z\left(\beta\right)}$ using the
next equation

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\[Z\left(\beta\right)=\sum_{n}e^{-\beta E_{n}}=\int
g\left(E\right)d\left(E\right) e^{-\beta E}\]
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\begin {itemize}
\item[(a)]
Particle in a three dimensional space
${H=\sum_{i=1}^{3}\frac{p_{i}^{\alpha}}{2m}}$

Calculate ${g\left(E\right)}$ and through that evaluate
${Z\left(\beta\right)}$

Guideline: for calculating ${\mathcal{N}\left(E\right)}$ one must
evaluate some points ${\left(n_{1} n_{2} n_{3}\right)}$- each point
represents a state - there's in ellipse ${E_{n_{1} n_{2} n_{3}}\leq
E}$

\item[(b)]

${N}$ particles with equal mass in a three dimensional space. assume
that it's possible to distinguish between those particles. Prove:
${\mathcal{N}\left(E\right)= const \cdot E^{\frac{3N}{2}}}$

Find the const. use Dirichlet's integral (private case) for
calculating the 'volume' of an ${N}$ dimensional Hyper-ball:
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\[\int...\int \Pi
dx_{i}=\frac{\pi^{\frac{N}{2}}}{\left(\frac{N}{2}\right)!}R^{N}\]
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${\sum x_{i}^{2}\leq R^{2}}$

Calculate ${g\left(E\right)}$ and from there evaluate
${Z\left(\beta\right)}$

\end {itemize}