Compute the correct answer of the given question
Question- A particle beam of spin-1/2 particles is sent through a Stern-Gerlach apparatus (SGA) -- that is, a magnetic -- in this case with the magnetic field oriented in the z-direction. The incident beam is "unpolarized" (that is, equal probability of up or down spins). Two beam exit this SGA. One of the exit beams is blocked and a second SGA with magnetic field in the y-direction is placed in front of the unblocked beam.
Part 1- What is the quantum mechanical state of a particle entering the first SGA in the up/down basis. Also express this as a spin vector (that is, a spinor).
Part 2- What is the fraction (relative flux - number of particles per second) of the original beam remaining in the unblocked exit beam from the second SGA? Why?
Part 3- What is the quantum mechanical state of a particle exciting the first SGA in the unblocked beam in the up/down basis? What is this state in the in/our (or y) basis? Also express this as a spin vector.
Part 4- We now place a third SGA (with z-direction magnetic field) in front of one of the exit beams from the second SGA. What fraction of the original beam remains in one of the exit beams from the third SGA?
I need expert's assistance to find the fraction of the original beam.