Undergrad Rotating Molecules: Energies, Angular Momentum & Wavefunctions

[kelly0303]

Hello! 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!

 

[DrClaude]

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?

Should the energy be frame dependent?

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.

The wave function is the wave function and is independent of any representation. However, its mathematical expression will differ dependent on the choice of representation (just like for the comparison between its representation in momentum space compare to position space).

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?

No, because what you separate is the translation of the center of mass. You then get a Hamiltonian corresponding to the relative displacement of the atoms (vibrations) and the overall orientation of the molecular frame (rotations).

Note that there are three rotational quantum numbers: the total angular momentum $J$, the projection of the total angular momentum on the z axis is the lab frame, $M_J$, and the projection of the total angular momentum on the z axis is the molecule frame, $K$. (The latter is not used for linear molecules, such as diatomic molecules, where only $K=0$ is possible.)