Novel Outlook on the Eigenvalue Problem for the Orbital Angular Momentum Operator

TL;DR

This paper introduces a new approach to the eigenvalue problem of the orbital angular momentum operator, providing normalizable, single-valued eigenfunctions for any real eigenvalues, expanding the traditional spectrum.

Contribution

It presents a novel prescription for the complex power function, leading to generalized eigenfunctions and eigenvalues for the angular momentum operators, including non-integer spectra.

Findings

Eigenfunctions are normalizable and single-valued for any real eigenvalues.

The spectrum of the angular momentum operator is not limited to integers.

Eigenfunctions are expressed in terms of hypergeometric functions.

Abstract

Based on the novel prescription for the power of a complex number, a new expression for the eigenfunction of the operator of the third component of the angular momentum is presented. These functions are normalizable, single valued and are invariant under the rotations at 2\pi for any, not necessary integer m - the eigenvalue of the operator of the third component of the angular momentum. For any real m these functions form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunctions. The eigenfunctions and eigenvalues of the operator of the angular momentum operator squared, derived for the two different prescriptions for the square root are reported. The normalizable eigenfunctions of the operator of the angular momentum operator squared are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the operator of the angular momentum operator squared.