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

  1. Feb 27, 2015 #1
    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. jcsd
  3. Feb 27, 2015 #2

    TSny

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    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?)
     
  4. Jul 18, 2017 #3
    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.)
     
  5. Jul 19, 2017 #4

    TSny

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    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?
     
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