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\sect{Ideal Fermi gas in 2D box}

${N}$ fermions with mass ${\mathsf{m}}$ and spin ${\frac{1}{2}}$ are
in a box , it's dimensions are ${L\times L \times \gamma,
\left(\gamma<<L\right)}$. The system is kept in low temperature
${T}$. Find the pressure on the box walls and the heat capacity of
the gas in conditions where it's possible to consider as ${2}$
dimensional.
The one particle states are ${| n p_{x} p_{y}\rangle}$
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\[E_{n p_{x}
p_{y}}=\frac{1}{2m}\left(\frac{\pi}{\gamma}R\right)^{2}+\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}
\]
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It's possible to consider the gas as a ${2}$ dimensional as long as
the occupation of the levels ${1<n}$ is negligible.
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\[g\left(E\right)=2\cdot A\frac{m}{2\pi},\,
\frac{1}{2m}\left(\frac{\pi}{\gamma}\right)^{2}<E<4\cdot
\frac{1}{2m}\left(\frac{\pi}{\gamma}\right)^{2}
\]
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It's convenient to take in to calculation the
${E=\frac{1}{2m}\left(\frac{\pi}{\gamma}\right)^{2}}$ like an
attribute level to the uniparticles state energy, but there's a need
to be careful when you calculate the pressure on the upper and lower
walls.



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