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.
This can happen for example when rigid molecules with an electric dipole moment are placed in an eletromagnetic field. Check this out: http://en.wikipedia.org/wiki/Diatomic_molecule#Rotational_energies http://chemwiki.ucdavis.edu/Physica...Rotational_Spectroscopy_of_Diatomic_Molecules
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.
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.