# Applications of the worldline Monte Carlo formalism in quantum mechanics

## TL;DR

This paper adapts the worldline Monte Carlo method from quantum field theory to non-relativistic quantum mechanics, enabling fast, potential-independent trajectory generation for efficient path integral approximation and ground state energy estimation.

## Contribution

It introduces a universal, non-recursive Monte Carlo approach for non-relativistic path integrals, applicable to singular potentials and useful for ground state and classical limit studies.

## Key findings

- Efficient trajectory generation for various potentials.
- Reliable estimates of Euclidean propagator and ground state energy.
- Applicability to singular potentials and classical limit analysis.

## Abstract

In recent years efficient algorithms have been developed for the numerical computation of relativistic single-particle path integrals in quantum field theory. Here, we adapt this "worldline Monte Carlo" approach to the standard problem of the numerical approximation of the non-relativistic path integral, resulting in a formalism whose characteristic feature is the fast, non-recursive generation of an ensemble of trajectories that is independent of the potential, and thus universally applicable. The numerical implementation discretises the trajectories with respect to their time parametrisation but maintains a continuous spatial domain. In the case of singular potentials, the discretised action gets adapted to the singularity through a "smoothing" procedure. We show for a variety of examples (the harmonic oscillator in various dimensions, the modified P\"oschl-Teller potential, delta-function potentials, the Coulomb and Yukawa potentials) that the method allows one to obtain fast and reliable estimates for the Euclidean propagator and use them in a certain time window suitable for extracting the ground state energy. As an aside, we apply it for studying the classical limit where nearly classical trajectories are expected to dominate in the path integral. We expect the advances made here to be useful also in the relativistic case.