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\sect{Boltzmann gas confined in capacitor}

An ideal gas is formed of $N$ spinless particles of mass $\mass$ 
that are inserted between two parallel plates (Z direction). 
The horizonatl confinement is due to a two dimensional harmonic 
potential (XY direction). Accordingly, 

\[ V(x,y,z)  = \left\{ 
  \begin{array}{l l}
    \frac{1}{2}\mass\omega^2(x^2+y^2) & \quad \text{$z_1$} < \text{$z$} < \text{$z_2$}\\
    \infty & \quad \text{else}
  \end{array} \right.\]

The diatance between the plates is ${L=z_2-z_1}$.
In the first set of questions (a) note that the partition function $Z$ can be factorized. 
In the second set of questions (b) an electric field $\mathcal{E}$ is added in the Z direction. 
Assume that the particles have charge~$e$.
Express your answers using $N,\mass,L,\omega,e,\mathcal{E},T$.

\Dn
\begin{itemize}

\item [(a1)]
Calculate the classical partition function  $Z_1(\beta; L)$ via a phase space integral.
Find the heat capacity $C(T)$ of the gas.

\item [(a2)]
Calculate the quantum partition function for large~$L$.
Define what is large $L$ such that the Z motion can be regarded as classical.

\item [(a3)]
Find the heat capacity $C(T)$ of the gas using the partition function of item (a2). 
Define what temperature is required to get the classical limit.

\item [(a4)]
Calculate the forces $F_1$ and $F_2$ that the particles apply on the upper and lower plates.

\item[(b1)]
Write the one-particle Hamiltonian and calculate the classical partition function $Z_1(\beta ; z_1, z_2, \mathcal{E})$

\item[(b2)]
Calculate the forces $F_1$ and $F_2$ that are acting on the upper and lower plates. 
What is the total force on the system? What is the prefactor in ${(F_1-F_2) = \alpha N T / L }$. 

\item[(b3)]
Find the polarization $\tilde{\mathcal{P}}$ of the electron gas as a function of the electric field. 
Recall that the polarization is defined via the formula $\bar{d}W=\tilde{\mathcal{P}}dE$.

\item[(b4)]
Find the susceptibility by expanding $\mathcal{P}(\mathcal{E})=(1/L)\tilde{\mathcal{P}} = \chi\mathcal{E}+O(\mathcal{E}^2)$. 
Determined what is a weak field $\mathcal{E}$ such that the linear approximation is justified. 

\end{itemize}


%\includegraphics[scale=1]{Capture.png}



