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(Solved): Two localized and non-interacting electrons are placed in a non-uniform external magnetic field whic ...
Two localized and non-interacting electrons are placed in a non-uniform external magnetic field which
points everywhere in the z-direction. Their spin dynamics is governed by the Hamiltonian
H=\epsi lon_(1)\sigma _(1)^(z)+\epsi lon_(2)\sigma _(2)^(z)
where \epsi lon_(i) are energy scales and \sigma _(i)^(x,y,z) are the Pauli matrices associated with the electrons i=1,2. The
electrons are initially entangled in a spin singlet state:
|\psi _(0):
(a) What is the probability that the two electrons will be detected in the spin singlet state at a later
time t ?
(b) Construct the state vectors of all spin-triplet eigenstates of the total spin along the field direc-
tion. What are the probabilities of detecting the two electrons in each triplet state at any time t ?
(c) What are the possible outcomes of measuring the total spin in a direction perpendicular to the
field, say S^(x)=S_(1)^(x)+S_(2)^(x) ? How do the probabilities of these outcomes depend on time?
(d) How do these results change if we reorient the magnetic field in a different direction, y for ex-
ample?