(a) At time t = 0 a large ensemble of spin-1/2 particles is prepared, all of them in the spin-up state (with respect to the z axis). They are not subject to any forces or torques. At time t₁ > 0 each spin is measured— some along the z direction and others along the x direction (but we aren’t told the results). At time t₂ > t₁ their spin is measured again, this time along the x direction, and those with spin up (along x) are saved as a subensemble (those with spin down are discarded). Question: Of those remaining (the subensemble), what fraction had spin up (along z or x, depending on which was measured) in the first measurement?
(b) Part (a) was easy—trivial, really, once you see it. Here’s a more pithy generalization: At time t = 0 an ensemble of spin-1/2 particles is prepared, all in the spin-up state along direction a. At time t₁ > 0 their spins are measured along direction b (but we are not told the results), and at time t₂ > t₁ their spins are measured along direction c. Those with spin up (along c) are saved as a subensemble. Of the particles in this subensemble, what fraction had spin up (along b) in the first measurement? Use Equation 4.155 to show that the probability of getting spin up (along b) in the first measurement is P₊ = Cos² (θ_ab/2), and (by extension) the probability of getting spin up in both measurements is Find the other three probabilities (P+→, P-+, and P--). Beware: If the outcome of the first measurement was spin down, the relevant angle is now the supplement of θ_ab.

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P++ cos² (ab (2) cos² (0bc/2).