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Torque-free precession

  1. Mar 6, 2012 #1
    In classical mechanics, an asymmetric rotating object will generally precess. Expressed in the body-fixed normal system of the object, we have [itex]I_i \dot{\omega_i}=(\vec{L}\times \vec{\omega})_i[/itex] where [itex]L_i=I_i\omega_i[/itex].

    Choosing a simple example where [itex]I_1=I_2[/itex], we obtain [itex]\dot{\omega_3}=0[/itex] and, for [itex]\Omega=\frac{I_1-I_3}{I_1}\omega_3[/itex],
    [itex]\dot{\omega_1}=\Omega \omega_2[/itex]
    [itex]\dot{\omega_2}=-\Omega \omega_1[/itex]
    describing the precession. Thus, [itex]\vec{\omega}(t)=(A\cos(\Omega t) , A\sin(\Omega t), \omega_3)[/itex].

    My question is; can this motion be described quantum mechanically?

    My first guess was to write the Hamiltionian as [itex]\hat{H}=\frac12 \hat{\vec{\omega}}I\hat{\vec{\omega}}[/itex] with [itex]I[/itex] being the inertia tensor. The difficulty is then to describe [itex]\hat{\vec{\omega}}[/itex] in terms of [itex]\hat{x},\hat{p_x}[/itex] etc.

    Am I going about this the wrong way?
    Is there any treatment of this problem available? I tried searching, but all the treatments of precession I found were related to magnetic moment precession.

    Any help is greatly appreciated.
  2. jcsd
  3. Mar 6, 2012 #2
    Last edited by a moderator: May 5, 2017
  4. Mar 6, 2012 #3
    That would still be concidered precession by an external torque, which is not what I am interested in here. Diatomic molecules don't experience free precession. I am sorry if I worded the problem poorly.

    What I am interested in is the kind of precession the rotational axis of the Earth experiences, but at the quantum level. For example, a free spinning molecule of white phosphorus (tetrahedral molecule) would experience precession.
  5. Mar 7, 2012 #4
    I see what you mean.

    Try the following Hamiltonian:

    [itex]\hat{H} = \frac{1}{2} \sum\limits_{ij} \hat{L}_i I^{-1}_{ij} \hat{L}_j[/itex]

    where [itex]I^{-1}_{ij}[/itex] is the invserse of the inertia tensor. In the normal system [itex] I^{-1}_{ij} = \delta_{ij} \frac{1}{I_i}[/itex]

    The angular momentum operator L is well defined, and the moment of inertia can be taken as constant.

    If I am not mistaken, then L does not commute with the Hamiltonian, so that you get precession.
  6. Mar 7, 2012 #5
    Great! I'll try it.
    Thank you very much! :smile:
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