On Time Development of a Quasi-Quantum Particle in Quartic Potential (x2-a2)2/2g

In this article, I have precisely considered the time development of a quantum particle (of excited states) in the quartic potential (x2-a2)2/2g by means of the semiclassical path integral method. Using the elliptic functions, I have evaluated the tunneling phenomena and the quasi-quantum fluctuation around the quasi-classical paths. I found that the quasi-quantum fluctuation is expressed by the Lamé equation and was exactly solved. Then I have shown that the obtained kernel function is in agreement with exact solutions of the linear potential and the quadratic potential under certain limits as no time-development kernel function of the quartic potential has ever been found which contains the exact solution of the linear and the quadratic potential. It is natural because the classical motion in the quartic potential becomes those of the linear and the quadratic potential under the limits. Thus the obtained time-development kernel function also consists of the energy representation of the Green function of the quartic potential in the semiclassical path integral method given by Carlitz and Nicole (Ann. Phys.164 (1985) 411), which agrees with that of the WKB method in the operator formalism.