The spin 1/2 h-bar of an electron is a remnant of the orbital angular momentum of positronium?

In the hydrogen atom, the electron orbital angular momentum in the z direction is an integer multiple of h-bar. Classically (that is, in Newtonian mechanics), almost the entire orbital angular momentum is in the movement of the electron, and only a tiny fraction in the movement of the proton.

In the positronium "atom", the orbital angular momentum is divided evenly between the electron and the positron. This may be the origin of the strange electron spin 1/2 h-bar. An electron and a positron are always created together. If they form some kind of a primitive positronium atom where the particles do not yet possess a spin, then after flying away, the electron and the positron will evenly share the 1 h-bar of orbital angular momentum of the positronium atom. In this model, the electron and the positron would have parallel spins. In the real world, they seem to have opposite spins.

Classically, the distance between the electron and the positron is 2r in the positronium atom, where r is the Bohr radius of the hydrogen atom. Both particles move around the center of mass in a circular orbit whose radius is r. Since the electric pull on the electron in positronium is only 1/4 of the pull in a hydrogen atom, the orbital velocity of the electron has to be 1/2 of that in a hydrogen atom, so that the centripetal acceleration agrees with the electric pull.

The de Broglie wavelength is defined as

       λ = h / p.

Since the electron momentum p in a positronium atom is just half of the hydrogen atom, the electron only completes half of a de Broglie wavelength in its circular orbit. In previous blog posts we have said that a "natural" periodic movement of a single particle must contain an integer number of wavelengths, to avoid destructive interference. But we noted that if two particles are moving "in unison", then we may consider them as a single system, and it is enough that the system completes a full number of wavelengths in one period. Now we realize that the positronium atom is just such a system.

In a previous blog post we developed the "pipe model" of an electron-positron pair. The particles rotate in unison at the opposite ends of the pipe. A problem is to find a way how the particles can preserve their tandem movement even when one of the particles is accelerated or its spin is rotated.

If we think of a positronium atom in a laboratory, both the electron and the positron have a spin 1/2 h-bar, and they may also have a multiple of h-bar of orbital angular momentum. How do we model the annihilation then?