# Volume of N dimensional phase space

1. Nov 6, 2006

### romeo6

Hi guys,

I have a volume integral in 3D phase space that looks like:

$$\int \frac{4\pi p^2 dp}{h^3}$$

Now, I want to generalize to N dimensions. How does this look:

$$\int \frac{\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}p^N dp}{N!h^{3N}}$$

Essentially, I've changed the 4 pi (which I think is the volume of a 2 sphere) into a generalized volume for an N sphere, and made some changes in the powers.

how does this look?

2. Nov 7, 2006

### dextercioby

Multiplied by p^{N-1} (with N the dimension of the momentum space) you need to have the area of the N-1 sphere.

What is with the N factorial ?

Daniel.

3. Nov 10, 2006

### romeo6

Do you mean I need to multiply my answer by p^{N-1}?

4. Nov 13, 2006

### dextercioby

It should be something like

$$\int_{\Omega} dV =\int_{0}^{\infty} p^{n-1} dp\int_{\partial \Omega} dS_{\Omega}$$

The second integral is the integral giving the area of the "n-1 sphere embedded in R^{n}.

Daniel.