Physical meaning of $L_z$ perturbation of rigid rotator The rigid rotor is a classical exercise in introductory quantum mechanics classes. The Hamiltonian is
$$H = \frac{L^2}{2m} \, .$$
Often you are also asked to consider the effect of adding a perturbation such that the new Hamiltonian has the form
$$H = \frac{L^2}{2m} + aL_{z} \,,$$
where the perturbing term expresses a coupling of the energies to the projection of the angular momentum along the $z$-axis. Is there a simple physical explanation behind this term? What would it represent in a realistic system?
 A: Actually the example of your $H$ is spherical rotor, which is not so interesting.  I suppose the example you have could describe a spherical rotor in a magnetic field. 
If you're willing to go beyond this, the most common rotors are axially symmetric, with Hamiltonians of the type
$$
H= \alpha L^2+\beta L_z^2\left(\frac{1}{I_3}-\frac{1}{I_1}\right)
$$
where $I_3$ and $I_1$ are the principal moments of inertia.  Those are very popular starting points in the study of nuclear deformations and their spectrum typically contains rotational bands (i.e. sequences of states connected by quadrupole transitions and with spectrum proportional to $L(L+1)$ within a band.
There is a very famous problem of a single particle with constant angular momentum $j$ moving in the field of an axially deformed nucleus (particle plus rotor nuclear model, or Nilsson Hamiltonian).  A simple version of this Hamiltonian would be
$$
H=Q L_z^2-\omega L_x\, .
$$
The $L^2$ is constant and has been omitted.  There is a nice paper by Aage Bohr and Ben Mottelson on the semiclassical analysis of this Hamiltonian (Phys.Scr. vol. 22 (1980) 461-467).
More generally a class of models known as Lipkin-Meshkov-Glick, with Hamiltonians of the type
$$
H=\varepsilon_0+2\omega L_z+\lambda (L_x^2-L_y^2)+\gamma(L^2-J_z^2)
$$
have been extensively studied.  See this time E. Romera et al, Phys. Scr. vol. 89 (2014) #095103 for again a semiclassical analysis.
There are all sorts of additional variations.

Sorry for the clear oversight of some molecules with symmetries, but I am more familiar with the nuclear side of this type of Hamiltonian.
