Quantum description of the asymmetric top using average and relative coordinates in an operator-based formalism

The quantum asymmetric top is rarely covered in quantum classrooms or textbooks. Yet it is one of the most interesting nontrivial applications of quantum mechanics to spectroscopy. Part of the challenge is the relationship between the lab frame and the body-fixed frame, which is not explained well in currently available treatments. We show how one can solve the quantum dynamics of asymmetric tops in a natural way by working with average and relative degrees of freedom. This allows one to derive all of the operator relationships between the two frames starting from quantum operators rather than relying on results from the classical solution. The derivation also avoids the need to use Euler angles to describe the motion. The solution can then be employed to examine pure rotational spectroscopy, as we illustrate with an example of rotations of the water molecule, which is an asymmetric top.