A beam of identical neutral particles with spin 1/2 travels along the y-axis. The beam passes through a series of two Stern-Gerlach spin analyzing magnets, each of which is designed to analyze the spin projection along the z-axis. The first Stern-Gerlach analyzer only allows particles with spin up (along the z-axis) to pass through. The second SternGerlach analyzer only allows particles with spin down (along the z-axis) to pass through. The particles travel at speed v0 between the two analyzers, which are separated by a region of length d in which there is a uniform magnetic field B pointing in the x-direction. Determine the smallest value of d such that only 25% of the particles transmitted by the first analyzer are transmitted by the second analyzer.
Rabi's formula: In this instance I said w0 = 0, so the probability is given by P+→- = sin2(w1t/2)
The Attempt at a Solution
I know that I want 25% of the particles to come out of the second analyzer. Since both analyzers measure along the z-axis, and go from spin-up to spin-down, I have been trying to solve this using Rabi's formula for spin-flip. I set P+→- = sin2(w1t/2) = 1/4.
However, I am now feeling stuck and I think I have missed some things along the way:
1st: I forgot that the particles are said to be travelling along the y-axis, and I don't know if this matters in the problem. I currently have the input state before the particles enter the B-field as |ψ(0)> = |+>.
2nd: I need to solve for the distance, so I need to find the value of t from solving P+→- = 1/4. But I don't know what to use for my value of w1. For an electron I see that w = eB/me, but I don't know what to use for these neutral particles.