EXERCISE 6.6. An ensemble of silver atoms (each with spin) is prepared so that 60% of the atoms are in the S =+ eigenstate of S, and 40% of the atoms are in the S, = -eigenstate of S, (S, and , are the x and z components of the spin angular momentum operator).

(a) Compute the density matrix at time t = 0 in the basis of eigenstates of S.

(b) Assume that the silver atoms sit in a magnetic field, B = Boy, and have a magnetic Hamiltonian, =S B, where is the magnetic moment of a silver atom. Compute the density matrix at time t (in the basis of eigenstates of ).

(c) Compute (S(t)) at time t = 0 and at time t.