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.

The particle in a two-dimensional well is a useful model for the motion of electrons around the indole ring (3), the conjugated cycle found in the side chain of tryptophan. we may regard indole as a rectangle with sides of length 280 pm and 450 pm, with 10 electrons in the conjugated p system. as in case study 9.1, we assume that in the ground state of the molecule each quantized level is occupied by two electrons. (a) calculate the energy of an electron in the highest occupied level. (b) calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels. 9.27 electrons around the porphine ring (4), the conjugated macrocycle that forms the structural basis of the heme group and the chlorophylls. we may treat the group as a circular ring of radius 440 pm, with 20 electrons in the conjugated system moving along the perimeter of the ring. as in exercise 9.26, assume that in the ground state of the molecule quantized each level is occupied by two electrons. (a) calculate the energy and angular momentum of an electron in the highest occupied level. (b) calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels.