Path integral quantization of the symmetric poschl-teller potential

Abstract

Feynman's path integral approach to quantum mechanics [I] is expected to be an alternative to Schrodinger's. Nevertheless, a limited number of problems have been solved exactly by path integration. It is a surprizing fact that such a typical example as the hydrogen atom has been left unsolved until very recently [2-41. Because of its gaussian (fresnellian) nature, the functional integral if represented in cartesian variables can be evaluated only for quadratic potentials [5]. Use of polar coordinates has helped to relax this limitation [6], placing the Infeld-Hull potential V= ar2 + br-* in the list of path-integable examples. However, the change of variables in path integrals is not trikial and the result is seldom beneficial to calculations. In recent years, some useful techniques have been devised, so that coordinate transformations are more effectively utilized to carry out path integration for the Aharonov-Bohm effect [7], the hydrogen atom [2,3], the Morse oscillator [8], the Dirac-Coulomb problem [4] and the charge-monopole system [9]. Now we are generally able to evaluate a path integral if it is intrinsically reducible in the local limit to a confluent hypergeometric equation of the Infeld-Hull type. It is interesting that this situation is parallel to that of the Sq2.1) dynarnical group approach [lo,) 11.