I am a bit confused about the rotational motion in molecules. Assuming the bond length is constant, the motion can be described as a rigid rotor. In the center of mass frame the energies are given by BJ(J+1) and the wavefunctions are spherical harmonics. However when we measure the energies or the angular momenta, we do it in lab frame. So I am a bit confused. Is the formula for the energy the same both in lab and CM frame? And if not, what is the formula in the lab frame? Also, is the wavefunction the same in both frames or, in other words, is the angular moment of the molecule the same in both frames. Actually I am a bit confused about how is the angular momentum defined in the CM frame. Isn't the molecule stationary in that frame? Yet the wavefunctions in the CM frame (spherical harmonics) do show a clear angular momentum dependence. Can someone help me clarify these things? Thank you!
The molecule is not stationary in the center-of-mass frame. Any rigid body dynamics can be separated into motion of the center of mass, and motion about center of mass. The second includes rotational motion, which is what you are considering. Only translational motion is absent in the CM frame. The question of whether angular momentum, or energy is the same in the lab frame depends on what other kinds of motion the molecule has in the lab frame. Does it have translational degrees of freedom, for example? If not, the energy is the same in both frames. The angular momentum here is intrinsic angular momentum, due to rotations within the molecule, and intrinsic spins of the constituents. This too should be the same in both frames, unless you are considering some odd situation like the molecule revolving around some axis in the lab frame.
Like NewUser mentioned, the motion can always be broken down into motion of centre of mass and motion about centre of mass. In our case, we interact with the system using light. And this causes transitions in the energy levels.
The system factors into one continuous energy spectrum corresponding to the kinetic energy of the free centre of mass and one discrete energy spectrum corresponding to the rigid rotor. The former causes light to scatter and the latter causes absorption. So if you look at absorption spectra, you'll find the signature of the discrete rotational energy levels.