A finite-dimensional representation of the quantum angular momentum operator

TL;DR

This paper introduces a finite-dimensional matrix representation of the quantum angular momentum operator using trigonometric interpolation, enabling numerical solutions that exactly reproduce spherical harmonics and eigenvalues.

Contribution

It presents a novel finite-dimensional matrix representation of the angular momentum operator that preserves key properties and eigenvalues, facilitating numerical analysis.

Findings

Eigenvalues match the exact form n(n+1).

Eigenvectors coincide with spherical harmonics at specific points.

The matrix representation inherits properties of the continuum operator.

Abstract

A useful finite-dimensional matrix representation of the derivative of periodic functions is obtained by using some elementary facts of trigonometric interpolation. This NxN matrix becomes a projection of the angular derivative into polynomial subspaces of finite dimension and it can be interpreted as a generator of discrete rotations associated to the z-component of the projection of the angular momentum operator in such subspaces, inheriting thus some properties of the continuum operator. The group associated to these discrete rotations is the cyclic group of order N. Since the square of the quantum angular momentum L^2 is associated to a partial differential boundary value problem in the angular variables $\theta$ and $\phi$ whose solution is given in terms of the spherical harmonics, we can project such a differential equation to obtain an eigenvalue matrix problem of finite dimension by extending to several variables a projection technique for solving numerically two point boundary value problems and using the matrix representation of the angular derivative found before. The eigenvalues of the matrix representing L^2 are found to have the exact form n(n+1), counting the degeneracy, and the eigenvectors are found to coincide exactly with the corresponding spherical harmonics evaluated at a certain set of points.