Semi-classical Determination of the Energy Levels of a $x^{4/3}$ Potential

Given a classical solution to a 1-dim in space system, for which all the solutions are periodic, the application of the modified Bohr-Sommerfeld quantization condition [1] allows a determination of semi- classical estimates for the energy levels of the associated quantum system. We consider a $x^{4/3}$ potential and use the methods of harmonic balance and iteration to calculate accurate approximations to the classical periodic solutions [2]. With these results, a general semi-classical energy spectrum can be determined. To judge the accuracy/validity of these calculations, we use a simple functional form for the ground state wave function in a variational calculation of the associated energy and compare this value with our semi-classical result. \\[3pt] [1] A. B. Migdal and V. P. Krainov, ``Approximation Methods in Quantum Mechanics (W. A. Benjamin, New York, 1969). \\[0pt] [2] R. E. Mickens, Journal of Sound and Vibration 292 (2006), 964-968.