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

Abstract

Based on the novel prescription for the power function, (x + iy) m , a new expression for Ψ(x, y|m), the eigenfunction of the operator of the third component of the angular momentum, Mz, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x + iy) m = ρ m e imφ , are invariant under the rotations at 2π for any, not necessarily integer, m -the eigenvalue of Mz. For any real m the functions Ψ(x, y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of Mz. The eigenfunctions and eigenvalues of the angular momentum operator squared, M 2 , derived for the two different prescriptions for the square root, (m 2 ) 1/2 , (m 2 ) 1/2 = |m| and (m 2 ) 1/2 = ±m are reported. The normalizable eigenfunctions of M 2 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 M 2 .