Disentanglement of a Singlet Spin State in a Coincidence Stern-Gerlach Device

Abstract

We analyze the spin coincidence experiment considered by Bell in the derivation of Bells theorem. We solve the equation of motion for the spin system with a spin Hamiltonian, H z , where the magnetic field is only in the z -direction. For the specific case of the coincidence experiment where the two magnets have the same orientation the Hamiltonian H z commutes with the total spin I z , which thus emerges as a constant of the motion. Bells argument is then that an observation of spin up at one magnet A necessarily implies spin down at the other B. For an isolated spin system A-B with classical translational degrees of freedom and an initial spin singlet state there is no force on the spin particles A and B. The spins are fully entangled but none of the spin particles A or B are deflected by the Stern-Gerlach magnets. This result is not compatible with Bells assumption that spin 1/2 particles are deected in a Stern-Gerlach device. Assuming a more realistic Hamiltonian H z + H x including a gradient in x direction the total I z is not conserved and fully entanglement is not expected in this case. The conclusion is that Bells theorem is not applicable to spin coincidence measurement originally discussed by Bell.