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lw>=a l+> + b l-> where a^2+b^2=1

If there is no magnetic field, the measurement of the spin is random and we get 50-50 chance to be spin up and spin down along that direction. The system state collapses into the one with eigen value after the measurement (interaction with SG-device). The system afterword can not be a superposition of two base states.

If there is a magnetic field and the particle enters it, a time evolution operator will act on it. The time evolution unitary operator is a function of the Hamiltonian which is a function of B and S. So, logically because of S, the state should also collapse into the one with the eigen value after that interaction. However, the time dependent state is still a superposition of the two base vectors lw(t)>=a exp(iyBt/2) l+> + b exp(-iyBt/2) l->

How a state is still pure after being acted upon by S part of the unitary operator?