Everybody knows that the orbital momentum is "quantized" and its z-projection has integer eigenvalues in units of . Too few, however, know that it is, in fact, a quasi-particle angular momentum which is integer-valued, not the particle one!
To see it, let us consider two spinless particles with masses m1 and m2 coupled with any central potential, say, with . The quasi-particle with the reduced mass describes the relative particle motion and depends on both particles in a two-particle compound system. It is its orbital momentum which is quantized in terms of integer numbers, namely, ranges from to .
The particle-1 and particle-2 orbital (angular) momenta and , calculated relative to the system center of inertia, are just certain, generally non-integer fractions of .
In particular, and are not obliged to have integer values! Why? Because they are simply non-integer functions of operator and are determined with its eigenvalues. For example, in the non-relativistic Positronium (where the particle masses are equal), if , the electron orbital momentum has the following values of projections: -1/2, 0, +1/2.
Surprise! And the same statement is valid for the positron projections in Positronium. "Half-integer" projections of a particle orbital momentum are thus possible and there is no kidding in it.
The particle orbital momenta in a bound system with respect to the system center of inertia are not momenta of independent, non-correlated subsystems. On the contrary, they are related and the rule of addition of independent angular momenta, usually taught in courses of Quantum Mechanics, is not applicable to them.
Unknown Physics Of Particle Orbital Momentum