At time t=0, an electron and a positron are formed in a state with total spin angular momentum equal to zero, perhaps from the decay of a spinless particle. The particles are situated in a uniform magnetic field B0 in the z direction.
(a) If interaction between the electron and the positron may be neglected, show that the spin Hamiltonian of the system may be written as
H^=ω0(S^1z−S^2z)
where S^1 is the spin operator of the electron, S^2 is the spin operator of the positron, and ω0 is a constant.
(b) What is the spin state of the system at time t? Show that the state of the system oscillates between a spin-0 and a spin-1 state. Determine the period of oscillation.
(c) At time t, measurements are made of S1x and S2x. Calculate the probability that both of these measurements yield the value ℏ/2.