Path distributions for describing eigenstates of orbital angular momentum

TL;DR

This paper introduces a new path distribution framework for understanding eigenstates of orbital angular momentum, extending classical models to include nonclassical paths and s-states, with potential implications for quantum state analysis.

Contribution

It develops a generalized stationary-phase analysis to derive real-valued, non-negative path distributions that describe orbital angular momentum eigenstates, including the case of zero quantum number.

Findings

Distributions are real-valued and non-negative functions of a momentum variable.

The approach includes nonclassical paths described by elastica.

Provides a replacement for the traditional vector model, including s-states.

Abstract

The manner in which probability amplitudes of paths sum up to form wave functions of orbital angular momentum eigenstates is described. Using a generalization of stationary-phase analysis, distributions are derived that provide a measure of how paths contribute towards any given eigenstate. In the limit of long travel-time, these distributions turn out to be real-valued, non-negative functions of a momentum variable that describes classical travel between the endpoints of a path (with the paths explicitly including nonclassical ones, described in terms of elastica). The distributions are functions of both this characteristic momentum as well as a polar angle that provides a tilt, relative to the z-axis of the chosen coordinate system, of the geodesic that connects the endpoints. The resulting description provides a replacement for the well-known "vector model" for describing orbital angular momentum, and importantly, it includes treatment of the case when the quantum number $\ell$ is zero (i.e., s-states).