To see it, let us consider two spinless particles with masses m

_{1}and m

_{2}coupled with any central potential, say, with . The quasi-particle with the reduced mass describes the relative particle motion and depends on

**particles in a two-particle compound system. It is its orbital momentum which is quantized in terms of integer numbers, namely, ranges from to .**

*both*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.

I saw the discussion on Lubos' blog and of course agree with you. It's quite simple when one thinks in terms of relative coordinates and reduced masses. And your note makes it clear that the quantization only comes about due to what amounts to a sort of renormalization of the masses from particle masses to reduced masses.

By the way, I think that arguing with people, especially Lubos, is a great idea. It sharpens the understanding.