The propagator for the step potential and delta function potential using the path decomposition expansion

TL;DR

This paper derives the propagator for particles in step and delta function potentials using a direct path integral approach based on path decomposition and properties of Catalan numbers, providing a novel derivation method.

Contribution

It introduces a new derivation of propagators for step and delta potentials using path decomposition expansion and combinatorial properties of Catalan numbers.

Findings

Derivation of propagators using path decomposition expansion.

Application of Catalan numbers to path integral derivation.

Provides a direct, combinatorial approach to known propagators.

Abstract

We present a derivation of the propagator for a particle in the presence of the step and delta function potentials. These propagators are known, but we present a direct path integral derivation, based on the path decomposition expansion and the Brownian motion definition of the path integral. The derivation exploits properties of the Catalan numbers, which enumerate certain classes of lattice paths.