Integral transforms of the quantum mechanical path integral: hit function and path averaged potential

TL;DR

This paper introduces two novel integral transforms of the quantum mechanical path integral, providing probabilistic insights into particle trajectories and potential interactions within quantum systems.

Contribution

The paper presents the hit function and path averaged potential as new integral transforms that encode physical information about quantum paths.

Findings

The hit function quantifies the contribution of paths crossing a point.

The path averaged potential measures the likelihood of potential values along paths.

These transforms offer new tools for analyzing quantum path integrals.

Abstract

We introduce two new integral transforms of the quantum mechanical transition kernel that represent physical information about the path integral. These transforms can be interpreted as probability distributions on particle trajectories measuring respectively the relative contribution to the path integral from paths crossing a given spatial point (the hit function) and the likelihood of values of the line integral of the potential along a path in the ensemble (the path averaged potential).