ANALYTICAL SOLUTIONS OF ARBITRARY ORDERS TO THE CLASSICAL AND QUANTUM OSCILLATORS WITH VELOCITY-DEPENDENT QUARTIC ANHARMONICITIES

Abstract

The classical oscillator with velocity-dependent anharmonicity (COVDA) arises when the velocity of the oscillator is reasonably high. By using an intuitive approach, we obtain an approximate analytical solution of arbitrary order to the problem of a COVDA. In addition to the third harmonic generation manifested by the nonlinear interaction, it is found that the solution contains the secular terms since the intuitive approach basically depends on the perturbation method. By assuming the small anharmonic constant, the secular terms are summed up for all orders and we obtain the renormalization of the frequency. The frequency of the oscillator decreases with the increase of the anharmonic constant. Interestingly, the magnitude of the shifts of the frequency of the oscillator with velocity-dependent quartic anharmonicity is identical with those of the oscillator with q-dependent quartic anharmonicity. However, the sign of the shifts for those two types of anharmonic oscillator is opposite in nature. These results indicate that the frequency shifts of the oscillator are actually the resultant effects (shifts) due to the q-dependent and the velocity-dependent anharmonicities. Finally, with the help of the correspondence principle, the solution of a quantum oscillator with velocity-dependent quartic anharmonicity (QOVDA) is obtained from the knowledge of the solution of its classical counterpart.