A detailed quantum mechanical treatment of the motion of a plane pendulum is given. Exact solution of the Schrödinger equation shows that the energy eigenfunctions and eigenvalues go over to those of a plane rotator for vanishing acceleration due to gravity g and to those of a harmonic oscillator in the limit of infinitely massive bob. In view of this, perturbation calculations have been done using rotator and oscillator basis, and conditions for convergence of the perturbation series in both cases have been discussed.
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Pradhan et al. (1973) studied this question.