I need some help with this quantum mechanics question.

(a) (b) ZA 1.0 (S(0)) S(t)) 0.8 0.6 B 0.4 0.2 0 4 TT 6 TT X Wo ~ Wo Wo FIGURE 3.9 (a) Spin-flip probability for a uniform magnetic field with x- and z-components and (b) the corresponding precession of the expectation value of the spin.Two Physics 47] students want to have some fun. They set up an experiment similar to the one in the previous problem: as before. the rst Stem-Gerlach analyzer allows only particles with spin up along the z-axis to pass through. but this time the second Stem-Gerlach analyzer allows only particles with spin up along the x-axis to pass through. The analyzers are separated by a region of space where there is a uniform magnetic eld Bu. The students are free to choose the direction of 139, but they want to arrange it so that 100% of the silver atoms that were transmitted by the rst analyzer are also transmitted by the second analyzer. Student A nished the two previous problems and says that they should orient the magnetic eld 130 along the +53 direction so she can use her results from those problems. Student B looked at Figure 3.9(b) in McIntyre and thought it would be cool to orient the eld Bu in the x-z plane at an angle of45 with respect to the z-axis. (In spherical coordinates. Bu would be pointing in the direction given by 9 = KM and I1] = 0.) a] If they follow student A's suggestion, what is the minimum time the particles must spend in the magnetic field so that 100% of the particles transmitted by the initial z-analyzer are EEO transmitted by the nal iii-analyzer? Dene m0 = : and express your answer in terms of [.00. Hint: Although the eld is oriented along +y axis rather than the +x axis as in problems 1 and 2, the physics is the same, so the results should be the same. b} If they follow student B's suggestion, what is the minimum time the particles must spend in the magnetic eld so that 100% of the particles transmitted by the initial z-analyzer are transmitted by the nal iii-analyzer? Hint: You may be tempted to use Equation (3.48], but that's not necessary. Take seriously the claim that \"expectation values in quantum mechanics obey the laws of classical mechanics\" and think about how the direction of the spin vector varies in time. McIntyre's Figure 3.9(b) should be helpful. c] If they follow student A's suggestion and do the calculations correctly. but they accidentally orient the B-eld along the 1? direction instead of the +5? direction. what fraction of the particles transmitted by the initial z-analyzer will be transmitted by the nal iii-analyzer? Could they get that fraction back to [00% by changing the time that the particles spend in the magnetic eld? d] IF they Follow student B's suggestion but they accidentally orient the B-eld in the opposite direction, what fraction of the particles transmitted by the initial z-analyzer will be transmitted by the nal iii-analyzer