Undergrad Understanding Nuclear Rotation: Quantum Numbers and Wavefunctions

[kelly0303]

Hello! I am a bit confused by the quantum numbers used to describe the rotation of a nucleus. In Wong's book these are J, M and K, which represent the rotational quantum number, its projection along the lab z-axis and its projection along the body intrinsic symmetry axis, respectively. However, as far as I remember from quantum mechanics, in describing the rotor motion (on which the nuclear rotation is based, at least as a first approximation), we needed just J and M in order to fully specify one of the sates of the system. Why do we need one extra number (the body frame projection of J) in the case of the nucleus. Moreover, the wavefunctions of the rotor were spherical harmonics, while for the nucleus they seem to be the Weigner D functions. Why do we have different quantum numbers and different wavefunctions, if the description of the nucleus is based on the rotor motion (here I am only talking about cylindrically symmetric nuclei). Here is a link to the nucleus rotation description that I mentioned (almost identical to Wong's). Thank you!

 

[vanhees71]

This is the quantum theory of a rigid rotator, if I understand your explanation right. It's an effective model, describing the nucleus as a rigid body and then quantize this problem. Obviously you also assume it's like a socalled symmetric top, i.e., components of the tensor of inertia in body-fixed coordinates, choosing a principle-axis body-fixed frame of reference, are $\Theta=\mathrm{diag}(A,A,C)$. Then rotation around the body-fixed 3-axis is a symmetry and thus the corresponding angular-momentum component conserved.

For a complete treatment of the rigid rotator ("spinning top") see Landau and Lifshitz vol. 3 in the chapter on multiatomic molecules (in my German edition it's in paragraph 103).