Monday, February 6, 2012

A hint of quantization as a classical-physics consequence of intrinsic spin

Equation (1) is a requirement that must be satisfied in order for the total angular momentum of the quasiclassical positronium atom to be a constant of the motion, assuming that the electron and positron are initially oriented so that their components perpendicular to the orbital angular momentum are anti-parallel.  In Eq. (1), L is the vector total orbital angular momentum (the sum of the orbital angular momenta of the electron and positron), n is a unit vector in the direction from the positron to the electron, and the intrinsic spin vectors are s.  The subscripts on the spin quantities indicate whether the spin is that of the electron or positron, and further whether it is the vector component parallel or perpendicular to L.

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,