Rigid Rotor States and Energy Levels

We are now in a position to determine the rigid rotor energy levels for tops in order of decreasing symmetry. The Hamiltonian (5.8) for a spherical top is functionally indistinguishable from that of a diatomic rotor, 

Owing to the spherical symmetry of its inertia tensor (/ = /j, = IJ, the spherical top s rotational levels depend only on the single quantum number J. 

In the oblate symmetric top, the rotational Hamiltonian is given by Eq. 5.10. The c axis is denoted the figure axis. According to the commutation rules obtained in Section 5.3, one possible commuting set of observables is J, J, and J. It is then possible to formulate rotational states JKM which simultaneously obey the eigenvalue equations 

This implies that the eigenstates JKM must behave as 

A schematic energy level diagram is given for the oblate top in Fig. 5.5. 

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