Path distributions for describing eigenstates of orbital angular momentum

• URL: http://arxiv.org/abs/2308.02884v6
• Date: Tue, 01 Oct 2024 09:50:01 GMT
• Title: Path distributions for describing eigenstates of orbital angular momentum
• Authors: Randall M. Feenstra,
• Abstract summary: distributions are derived that provide a measure of how paths contribute towards any given eigenstate. The resulting description provides a replacement for the well-known "vector model" for describing orbital angular momentum.
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• 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).

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