# Statistical Mechanics -- partition function, change to polar coords

1. Apr 14, 2017

### binbagsss

1. The problem statement, all variables and given/known data

Hi I have the following definition for the partition function of $N$ particles in $s$ dimensions:

I am looking at computing the partition function for this Hamiltonian:

The solution is here:

2. Relevant equations
above

3. The attempt at a solution

I don't understand the top line of this solution? The definition is to integrate over $\Pi_{i=1}^{i=n} \Pi_{\alpha=1}^{\alpha=2}dP_{i,a}$ not $\Pi_{i=1}^{i=2N} d\vec P_i$

I can change to polar coordinates to get the integration over $d\vec P_i$ (just looking at the integration over $P$ and I have the same result, however in the first line of the solution there are no factors of $2\pi$ from the change of variables to polar coordinates, so how has the definition of the partition function been used? (or is it just by chance I've got the same answer)i.e. $dP_x dP_y \neq d^2\vec P$ does it?:

2. Apr 19, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.