A finite-dimensional representation of the quantum angular-momentum operator

A useful finite-dimensional matrix representation of the derivative of peri odic functions is obtained by using some elementary facts of trigonometric interpolation. This N x N matrix becomes a projection of the angular deriva tive into polynomial subspaces of finite dimension and it can be interprete d as a generator of discrete rotations associated to the z-component of the projection of the angular-momentum operator in such subspace, inheriting t hus some properties of the continuum operator. The group associated to thes e discrete rotations is the cyclic group of order N. Since the square of th e quantum angular momentum L-2 is associated to a partial differential boun dary-value problem in the angular variables: theta and phi whose solution i s given in terms of the spherical harmonics, we can project such a differen tial equation to obtain an eigenvalue matrix problem of finite dimension by extending to several variables a projection technique for solving numerica lly two point boundary value problems and using the matrix representation o f the angular derivative found before. The eigenvalues of the matrix repres enting L-2 are found to have the exact form n(n + 1), counting tile degener acy, and the eigenvectors are found to coincide exactly with the correspond ing spherical harmonics evaluated at a certain set of points.