Help with an ideal gas canonical ensemble partition function integral

[AndreasC]

TL;DR

I'm reading Mehran Kardar's book on statistical mechanics, and it's great but super terse. I am trying very hard to understand a particular integral that is used over and over again for ideal gases, with no luck so far. Any help? I've posted a relevant picture below.

Where does the volume even come from? Any help would be appreciated!

 

[Infrared]

Where does the volume even come from? Any help would be appreciated!

I'm not a physicist, but it looks just like $\int \prod_{i=1}^N\left(d^3\vec{q}_i\right)=V^N$ since the $q_i$ are presumably position coordinates.

 

[AndreasC]

I'm not a physicist, but it looks just like $\int \prod_{i=1}^N\left(d^3\vec{q}_i\right)=V^N$ since the $q_i$ are presumably position coordinates.

Oh, dammit, yeah, that where it comes from haha! Somehow I missed it. And then I guess the rest is just a 3N dimensional Gaussian integral over the momenta, right? Nice, that makes sense.