Sequential Spin Measurements.
(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? Hint: Use Equation 4.155 to show that the probability of getting spin up (along b) in the first measurement is P+=cos2(θab/2), and (by extension) the probability of getting spin up in both measurements is P++=cos2(θab/2)cos2(θbc/2). 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 θbc. Answer: [1+tan2(θab/2)tan2(θbc/2)]−1.