Description of the associated Legendre functions in analogy to the state space of the free electromagnetic field

TL;DR

This paper proposes an analogy between associated Legendre functions and the state space of the free electromagnetic field, offering new insights into physical systems and connecting classical multipole expansions with quantum discreteness.

Contribution

It introduces a novel analogy that links associated Legendre functions to electromagnetic field state space, enhancing understanding of physical systems.

Findings

Associated Legendre functions can be understood via electromagnetic field analogy

Classical multipole expansion relates to quantum discreteness

New properties of Legendre functions are identified through this analogy

Abstract

Currently, some approaches to the associated Legendre functions based on different factorization methods are known. However, they have not allowed identifying new properties that permit to improve our knowledge of any physical system. In this letter, we show that the set of all the associated Legendre functions can be understood in analogy to the state space of the free electromagnetic field. Thanks to this correspondence we hope that any system, classical or quantum, described by such set of functions can be physically understood by using this analogy. We illustrate our results showing that the classical multipole expansion of the scalar and vector potentials is connected to the quantum mechanics discreteness property.