I have atoms in an unknown spin state, which we can write in complete generality as |ψ⟩ = a| ↑⟩ + b| ↓⟩. Suppose that when I send them through a Stern-Gerlach analyzer along z, I find that 36% of them go up and 64% of them go down.
a) What can you conclude about a and b (remembering that they are in general complex numbers)?
b) Suppose this same state ψ were sent through an x-analyzer. Write down the probability for them to be deflected in each direction in terms of a and b. What are the maximum and minimum possible values for P(±x), the probabilities of being deflected along ±x?
c) Suppose only the 36% that went up are sent through this x-analyzer what can you say about P(±x)?
d) Suppose only the 64% that went down are sent through the x-analyzer what can you say about P(±x)?
e) Redo the calculations of parts a and b above for a case where 50% of the atoms go up and 50% go down.