Quasi-classical path integrals

Shigeki Matsutani

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.

Osvaldo Civitarese, Manuel Gadella

Lukong Cornelius Fai

It is interesting to know when to expect the wave nature of a particle propagation to be well approximated by the properties of the trajectory of a classical particle. Usually, whenever there is an integral expression for a given quantity, it is often easier to employ approximate methods compared to a differential equation. It is simply a matter of convenience representing a path by a classical path and fluctuations around it. For the quasi-classical approximation, however, it is critical to expand around the path leading to the dominant contribution, i.e. the classical path. The knowledge of the stationary phase approximation is very appropriate to apply to path integration. This heading examines some specific systems of particular interest in people investigation. The harmonic oscillator has practical applications in a variety of domains of modern physics, such as molecular spectroscopy, solid state physics, nuclear structure, quantum field theory, quantum statistical mechanics and so on.

L. E. Reichl

Eugene P. Gross

Eduardo Piña, N. Aquino

The path integral formulation of quantum mechanics in the \VKB approximation (quasi-classical) gives to us a useful tool to relate the classical systems (with or without chaos) with its quantum analogs. \Ve use this point of view to compute the eigenfunctions and energy eigenvalues for the quantum pendulum comparing with those obtained with the standard \VKB approximation. The method that we employed in this work was developed mainly by Gutzwiller, and one of its main features is that this method can be used in integrable and no integrable problems.