# Stern Gerlach and Position Measurements

1. Feb 24, 2014

### bglasber

1. The problem statement, all variables and given/known data

Given the following stern-gerlach setup, find the probabilities for particles to arrive in detectors D1 and D2.
SGx = Stern Gerlach in x direction,
SGx = Stern Gerlach in z direction
(sorry for terrible ascii drawing)

|ψ> = a|+> + b|->, |ψ> is normalized.
................./D1
........./SGz.\.......D2
-- SGx .........SGx/
..........\SGz../....\ D3
...................\D4

Suppose that you take a measurement of the particle's position after doing the z-measurement so you can tell which of the four paths it took, what is the probability of detection in D1/D2?

2. Relevant equations
Chaining projectors, P$_{|+>}$ = |+><+|
Probability: |<ø|ψ>|$^{2}$

3. The attempt at a solution
Doing the initial probability calculations isn't too bad, we just chain projectors together and calculate the probability for the paths it can take to reach the detectors.
What confuses me though, is the second part, where we place a detector, because I am unsure of how this changes the question.

I thought that we would have already collapsed the wave function after passing through a detector, because the particle would be forced into the measured spin state |+>, |-x>, etc. ( Wouldn't this be the effect of the remaining ket in the projector chain? e.g.
|+> <+|+x> <+x|ψ> for D1?)

I'm guess I don't quite understand how knowing the path that the particle took changes the probability of it getting there? Is my answer supposed to change?