Given

(1)
\begin{align} \epsilon = \frac{h^2}{8ma^2} (s_x^2 + s_y^2) \end{align}

(1) Find the density of states and then use it to find the q_trans:

We can write:

(2)
\begin{align} R^2 = (s_x^2 + s_y^2) = \epsilon\frac{8ma^2}{h^2} \end{align}

We assume that the number of states ($\Phi(\epsilon)$) with energy $\epsilon$ can be approximated by 1/4 the area of a circle with radius R. Then we have

(3)
\begin{align} \Phi(\epsilon) = \frac{\pi R^2}{4} = \frac{2\pi\epsilon ma^2}{h^2} \end{align}

Then the number of states between $\epsilon$ and $\epsilon +d\epsilon$ is given by:

(4)
\begin{align} \Phi(\epsilon+d\epsilon)-\Phi(\epsilon) = \frac{2\pi ma^2}{h^2}d\epsilon = \omega(\epsilon)d\epsilon \end{align}

Now using:

(5)
\begin{align} q_{trans} = \int_0^\infty \omega(\epsilon)exp(-\beta\epsilon) d\epsilon \end{align}

gives:

(6)
\begin{align} q_{trans} = \frac{2\pi mkT}{h^2}A \end{align}

(2) Find $q_{trans}$ using another method:

This time we will sum over states instead of summing over levels, we write:

(7)
\begin{align} q_{trans} = \int_0^\infty \int_0^\infty exp(\frac{-\beta h^2}{8ma^2}s_x^2)exp(\frac{-\beta h^2}{8ma^2}s_y^2) ds_x ds_y = (\int_0^\infty exp(\frac{-\beta h^2}{8ma^2}s^2)ds)^2 \end{align}

Evaluating the integral gives:

(8)
\begin{align} q_{trans} = \frac{2\pi mkT}{h^2}A \end{align}

Which is fortunately the same we obtained by summing over levels.

(3) Find heat capacity, U, S, and EOS

To find the EOS we use:

(9)
\begin{align} P = kT (\frac{\partial ln(q)}{\partial A})_T = \frac{kT}{A} \end{align}

To find E:

(10)
\begin{align} E = kT^2 (\frac{\partial ln(q)}{\partial T})_A = kT \end{align}

Finally to find $C_A$:

(11)
\begin{align} C_A = (\frac{\partial E}{\partial T})_A = k \end{align}