Rigid Rotor Hamiltonian

Hamiltonian

For a rigid rotor with moment of inertia $I$,

$$H = \frac{\hat L^2}{2I}.$$

The eigenfunctions are spherical harmonics, and the spectrum is

$$E_\ell = \frac{\hbar^2\ell(\ell+1)}{2I}, \qquad \ell=0,1,2,\ldots$$

Each $\ell$ level has degeneracy

$$2\ell+1.$$

Assumptions

• The rotor length is fixed.
• The configuration space is the sphere $S^2$.
• The moment of inertia is constant.
• Vibrations, centrifugal distortion, spin, and external fields are omitted.

Common Mistakes

• Treating the rotor as a one-dimensional oscillator.
• Expecting a zero-point energy of $\hbar\omega/2$.
• Forgetting the angular measure on the sphere.
• Confusing $\ell$ with a length or oscillator quantum number.

Canonical Links

• Rigid Rotor
• Particle on a Sphere
• Rotational Spectra
• Angular Momentum Operator

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.