# Pathria 2.4 -- show that the number of microstates available to a rigid rotor is (M/ħ)^2

1. Feb 27, 2015

### halley00

1. The problem statement, all variables and given/known data
By evaluating the "volume" of the relevant region of its phase space, show that the number of microstates available to a rigid rotor with angular momentum less or equal to M is (M/ħ)2.

2. Relevant equations
Consider the motion in the variables θ and φ, with M^{2}=(p_{θ})2+{p_{φ}/{sin(θ)}2.

3. The attempt at a solution
I just integrated the "volume": (1/h²)∫0πdθ∫0dφ∫0Mdpθ0Msinθdpφ,

For some reason I'm missing a π. Why? I'm getting a "volume" 4π(M/h)2
Maybe I'm making a mistake in the choice of the limits of integration.

Sorry, I don't know how to write the equations in clean way.

2. Feb 27, 2015

### TSny

Hello. Welcome to PF!

Note that while integrating over $p_\phi$, the variable $p_{\theta}$ is fixed at some value. Using the relation $M^2 = p_{\theta}^2 +\frac{ p_{\phi}^2}{\sin^2 \theta}$, what are the limits for the integration over $p_{\phi}$?

(Can the angular momenta components $p_\theta$ and $p_\phi$ be negative?)

3. Jul 18, 2017

### Met119

No angular momentum can't be negative so those components can't be negative either. But I don't know what the limits for p_phi are supposed to be. (I'm also having trouble with this problem.)

4. Jul 19, 2017

### TSny

The components of angular momentum can be negative as well as positive.

For the limits of integration, it might help to consider a simpler, but similar, problem. Suppose you have a single particle moving in two dimensions. In Cartesian coordinates the magnitude of linear momentum, p, is related to the components of momentum, px and py, by

p2 = px2 + py2.

Suppose you want to integrate over the momentum portion of phase space to get the "volume" corresponding to all momentum values 0 < p < P0. How would you choose the limits of integration in ∫dpy ∫dpx?