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.