# Amount of microstates - phase space volume

1. Jun 20, 2015

### a_i_m_p

1. The problem statement, all variables and given/known data
Dear all, I am desperately trying to solve the following exercise, but unfortunately can't find any resources how to properly calculate the phase space volume.

Given is a system of $N>>1$ classical particles that are allowed to move in a cylinder with a Radius of $R$. The Hamiltonian is:
$H=\sum_{i=1}^N \frac{{p_i}_x^2+{p_i}_y^2+{p_i}_z^2}{2m}\ + \frac{m\omega^2z_i^2}{2}\, +V(x_i,y_i)$

and

$V(x,y)= \begin{cases} 0 ~~ \text{if} ~~ x^2+y^2 \leq R^2 \\ \infty ~~ \text{if} ~~ x^2+y^2>R^2 \\ \end{cases}$

Calculate the amount of microstates $\Omega(E)$ within the energy band $E<H<E+\delta E$.

2. Relevant equations
I am allowed to use the following equations for solving the problem:

$d\Gamma =\prod_{i=1}^{f}dq_idp_i$where $f=3N$(degrees of freedom)
$\Gamma = \int_{H \leq E}d\Gamma$
$Let\ \frac{\delta E}{E}\ <<1 => \Delta \Gamma \approx (\frac{\partial \Gamma}{\partial E}) \delta E$
$\Omega (E)=\frac{1}{h_0^{3N}}\Delta \Gamma$

3. The attempt at a solution

I started solving the problem by calculating $\Delta \Gamma$:

First, we transform the hamiltonian so that its shape in phase space is converted from an Ellipsoid to a sphere:

$z_i=\frac{1}{m \omega}z_i'$

$H=\frac{1}{2m}\sum_{i=1}^N (p_i^2+z_i^2)=:\frac{1}{2m}r^2$

$H\leq E => r \leq \sqrt{2mE}$

Now, $\Gamma (E)=\frac{1}{(m\omega)^N} \int_{H \leq E}d^{3N}p~d^{N}x~d^Ny~d^Nz'$

$=\frac{1}{(m\omega)^N} \int_0^\sqrt{2mE}d^{4N}r (\int_0^R r' dr' \int_0^{2\pi} d\phi)^N$

I'm pretty sure that this step was wrong because the rest is basic calculus and the result has the wrong dimensions.

Does anyone have an idea how to correctly calculate the phase space volume? My thoughts were to split the volume element into a 4N-dimensional sphere(the first integral) and see the rest (dx,dy) as the front face of the cylinder where the particles are kept.