Novel Outlook on the Eigenvalue Problem for the Orbital Angular Momentum Operator - George Japaridze; Anzor Khelashvili; Koba Turashvili (Physics, 2022)
Based on the novel prescription for the power function, <semantics>(x+iy)m</semantics>, the new expression for <semantics>Ψ(x,y|m)</semantics>, the eigenfunction of the operator of the third component of the angular momentum, <semantics><mover accent="true">M^</mover>z</semantics>, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, <semantics>(x+iy)m=ρmeimϕ</semantics>, are invariant under the rotations at <semantics>2π</semantics> for any, not necessarily integer, m—the eigenvalue of <semantics><mover accent="true">M^</mover>z</semantics>. For any real m the functions <semantics>Ψ(x,y|m)</semantics> form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of <semantics><mover accent="true">M^</mover>z</semantics>. The eigenfunctions and eigenvalues of the angular momentum operator squared, <semantics><mover accent="true">M^</mover>2</semantics>, derived for the two different prescriptions for the square root, <semantics>(m2)1/2</semantics>, <semantics>(m2)1/2=|m|</semantics> and <semantics>(m2)1/2=±m</semantics>, are reported. The normalizable eigenfunctions of <semantics><mover accent="true">M^</mover>2</semantics> 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 <semantics><mover accent="true">M^</mover>2</semantics>.