Continued fractions and the harmonic oscillator using Feynman’s path integrals

Abstract

The simple harmonic oscillator plays a prominent role in most undergraduate quantum mechanics courses. The study of this system using path integrals can serve to introduce a formulation of quantum mechanics which is usually considered beyond the scope of most undergraduate courses. However, given the current interest in the interpretation and foundations of quantum mechanics, nonstandard approaches such as Feynman’s path integral formalism can be helpful in developing insights into the structure of quantum mechanics. In this paper we evaluate the path integration appearing in Feynman’s treatment in a natural and direct manner utilizing a symbolic computational program. This approach makes the use of the path integral formulation of quantum mechanics accessible to most undergraduate physics majors. As a by-product of our approach, we find a representation of the reciprocal of the sinc function, sinc (x)≡sin(x)/(x), in terms of an infinite product of partial approximates of a continued fraction. We have not found this representation in the literature.