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\sect{Electrons in effectively 1D box $V(x,y,z)=x^2+y^2$}

Consider ${N}$ electrons that are kept between 
the plates of a capacitor. 
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\[
V\left(x,y,z\right)=\left\{\begin{array}{ll}
\frac{1}{2}m\omega^{2}\left(x^{2}+y^{2}\right) &\textrm{ $0\leq z \leq L$}\\
\infty & \textrm{else}\\
\end{array}\right.
\]
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The system is in thermal equilibrium at zero temperature. 
Find the force that the gas exerts of the plates assuming 
that it can be treated as one-dimensional. 

Write the condition on ${N}$ for having this assumption valid.

Tip: Find first the one particle states, 
and illustrate them using a schematic drawing. 
Express your results using  ${N,L,m,\omega}$ only.