Critical comments on the quantization of the angular momentum: II. Analysis based on the requirement that the eigenfunction of the third component of the operator of the angular momentum must be a single valued periodic function

TL;DR

This paper reexamines the assumptions behind the quantization of angular momentum eigenvalues, showing that non-integer eigenvalues are theoretically possible and challenging traditional constraints used in quantum mechanics.

Contribution

It demonstrates that the common requirement for single valuedness and periodicity of eigenfunctions is not derived from first principles, allowing for non-integer angular momentum eigenvalues.

Findings

Eigenfunctions with non-integer eigenvalues are mathematically admissible.

The traditional integer eigenvalue constraint can be dropped without inconsistency.

A new definition of complex power functions preserves translational invariance.

Abstract

We discuss the requirement of single valuedness and periodicity of eigenfunction of the third component of the operator of angular momentum. This condition, imposed on a non observable, is often used to derive that the eigenvalues of angular momentum could be only integer. We reexamine the arguments based on this requirement and alternate condition imposed by Pauli and show that they do not follow from the first principles and therefore these constraints can dropped. Consequently, we arrive to the same conclusion as in [1]: there exist regular, normalizable eigenfunctions with the non-integer eigenvalues thus a non-integer angular momentum is perfectly admissible from the theoretical viewpoint. The issue of the nature of eigenvalues forming the spectrum of the angular momentum remains open. What can be derived from the first principles is that to a fixed value of the angular momentum L corresponds a discrete spectrum of eigenvalues of the third component of the angular momentum, m, defined by the relation |m|=L-k, k=0,1,...,[L], where [L] is an integer part of L. As a mathematical byproduct of our analysis of eigenfunctions, we present an alternate definition of a power of a complex number allowing to retain initial translational invariance of a base.