Ballentine gives an argument based solely on the operator formalism and which does not rely on the wave function being single-valued. The azimuthal angular momentum is defined as L z = Q x P y − − Q y P x Define new operators q 1 = Q x + P y 2 q 2 = Q x − − P y 2 p 1 = P x − − Q y 2 p 2 = P x + Q y 2 (Dimensional correctness may be maintained by inserting factors of mass and unit angular frequency numerically equal to one.) Then L z = 1 2 ( p 1 2 + q 1 2 ) − − 1 2 ( p 2 2 + q 2 2 ) But the two terms on the right are just the Hamiltonians for the quantum harmonic oscillator with unit mass and angular frequency H 1 = 1 2 ( p 1 2 + q 1 2 ) H 2 = 1 2 ( p 2 2 + q 2 2 ) and L 2 , L z , H 1 and H 2 all commute.