Vibration-rotation Hamiltonians classical mechanics

Classical and Quantum-Mechanical Vibration—Inversion—Rotation Hamiltonian 

Vibrational, rotational, and vibrational/rotational energy levels are found by first transforming the classical Hamiltonians described in the previous section to the appropriate quantum mechanical operator H. The eigenvalue equation 

Thus the asymptotic quantum states are labeled by the vibrational and rotational quantum numbers, whereas the projection quantum number is treated classically. We have in the above Hamiltonian indicated that it depends upon time through the classical variables. Thus the quantum mechanical problem consists of propagating the solution to the TDSE 

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