|Equation (1): A condition for constancy of total angular momentum|
Equation (1) is fairly easily obtained based on Thomas's equation of motion of the spin (which can alternatively be obtained from the Bargmann-Telegdi-Michel covariant equation of spin motion) specialized to the positronium atom, as is done in my positronium paper separately for the electron and positron, and then taking their difference to obtain an equation of relative motion for the spins. Then putting in the initial condition that is the main result in my positronium paper, that antiparallel L-perpendicular spin components are a condition for total angular momentum constancy, Eq. (1) results. If Eq. (1) could be satisfied at all points on the orbit (that is, as the vector n traces out the relative particle positions around the orbit) then the needed relative orientation would be maintained and constant angular momentum would be maintained. (Total angular momentum constancy is in turn a necessary condition for nonradiativity.)
To understand how Eq. (1) comes close to obtaining a quantum condition from classical electrodynamics with intrinsic spin,
suppose that the left hand side was zero. Then, the total angular momentum would be a constant of the motion if the sum of the spin components parallel to the orbital angular momentum has a magnitude of three-halves of the orbital angular momentum. This would be like quantum theory if the 2 and 3 were instead both 1, and if we also had an ancillary dynamical condition that constrained the magnitude of the parallel components to be h-bar over two.
So, the right hand side of Eq. (1) is of the form needed as a quantum condition. With the additional conditions as already identified it would provide a dynamical justification for one of Arnold Sommerfeld's three quantum rules, the azimuthal quantum rule, in his generalization of the Bohr model. Manfred Bucher has recently shown how the Sommerfeld-Bohr model can be reinterpreted to agree with modern quantum theory surprisingly well.
I think it's plausible that the problems with Eq. (1) might be resolved when the electrodynamic laws are augmented, as I believe they must be, to include the anti-Euler force of the Thomas precession. This is a force that is recognized as necessary when it is recognized that the magnetic force is the anti-Coriolis force of the Thomas precession.
In the case that incorporation of the anti-Euler force allows for the left-hand side of Eq. (1) to generally vanish, then Eq. (1) will still apply to the circular orbit case, where the anti-Euler force vanishes due to constancy of the Thomas precession angular velocity. Thus, Eq. (1) will continue to hold for the circular orbit and rule out the circular orbit as an allowed condition. This is in agreement with the Bohr-Sommerfeld-Bucher model.