10. Analysis of the pairing of two electrons

Both in the Coulomb and Poincaré cases, when the spins are parallel, the two particles are very close to each other with a separation of order of Compton’s wavelength or shorter, and the motions of the CC’s take place in the same plane we obtain a stable bound motion of the two particles which remains stable under external electric fields. Under a transversal magnetic field with respect to the spin direction, the spin precesses and they are no longer parallel, so the pairing is destroyed and the two Dirac particles separate.The justification of this bound motion is because a repulsive force between the CC’s can become an attractive force between the CM’s as is shown in the figure 3-(a). Let us analyze the example depicted in the figure where the phases of the two particles are opposite
ψ
1
−
ψ
2
= 18
◦
0
, and the separation between the CM’s is 3/4 in natural units. Remember that Compton’s wavelength is 2
R
0
=
ℏ
mc
, which is 1 in natural units. In part (a) of the figure the CC’s are separated 1/4 while in part (b), after a half turn of the charges, they are at a distance 7/4.We analyze the interaction between the charges and compute the force acting on particle 1 (blue). The force on particle 2 is opposite and is not depicted in the figure.