Amount of microstates - phase space volume

[a_i_m_p]

Homework Statement

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 $E<H<E+\delta E$.

Homework 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)
$\Gamma = \int_{H \leq E}d\Gamma$
$Let\ \frac{\delta E}{E}\ <<1 => \Delta \Gamma \approx (\frac{\partial \Gamma}{\partial E}) \delta E$
$\Omega (E)=\frac{1}{h_0^{3N}}\Delta \Gamma$

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.

Thanks in advance!

 

[mark726]

Thank you for your question. I understand that you are trying to solve an exercise involving calculating the phase space volume of a system of classical particles in a cylinder with a given Hamiltonian. I can offer some guidance on how to approach this problem.

Firstly, you have correctly identified the equations that can be used to solve this problem. The key equation is $d\Gamma =\prod_{i=1}^{f}dq_idp_i$, which gives the volume element in phase space. However, in order to use this equation, we need to determine the limits of integration for the position and momentum variables.

To do this, we need to consider the constraints on the system. The Hamiltonian you have been given has two terms - the kinetic energy term and the potential energy term. The potential energy term is a hard wall potential, which means that the particles are confined to a cylinder with radius $R$. This means that the position variables, $x_i$ and $y_i$, are constrained to lie within $-R \leq x_i \leq R$ and $-R \leq y_i \leq R$. The momentum variables, $p_i$, are not constrained, as the particles can move freely in the z-direction.

Using these constraints, we can determine the limits of integration for the position and momentum variables. For the position variables, we have $-R \leq x_i \leq R$ and $-R \leq y_i \leq R$. For the momentum variables, we have $-\infty \leq p_i \leq \infty$ in the z-direction, and $-\sqrt{2m(E-V(x_i,y_i))} \leq p_i \leq \sqrt{2m(E-V(x_i,y_i))}$ in the x and y directions, where $V(x_i,y_i)$ is the potential energy term in the Hamiltonian.

Using these limits of integration, we can now calculate the phase space volume, $\Gamma$, for a given energy, $E$. This can then be used to calculate the number of microstates, $\Omega(E)$, within an energy band, $E<H<E+\delta E$.

I hope this helps. If you have any further questions, please let me know.