# Determine the number of particles transmitted by S-G analyzer

## Homework Statement

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.

## Homework Equations

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.

blue_leaf77
Homework Helper
It will be useful to write the states involved in the problem such that spin states along a certain direction are distinguished from the other directions. For example, I propose to use ##|z;\pm\rangle## to denote the states along ##z## direction.
So, in the beginning (after the first SG) the state is ##|z;+\rangle##. Then it propagates through a region of uniform magnetic field oriented along ##x## direction. Which means, this intermediate region is associated with the time evolution operator ##U = \exp\left(-i\omega\frac{S_x}{\hbar}t\right)##. Now apply this operator to the initial state, namely ##U|z;+\rangle##. Hint: to do this calculation, make use of the completeness relation for ##|x;\pm\rangle##.
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)> = |+>.
It does not matter because the direction of the travel does not affect the Hamiltonian.
but I don't know what to use for these neutral particles.
Since, the particle is unknown, I think you can leave the answer in term of ##\omega##.

Last edited:
acdurbin953
Thank for the insight that the direction does not affect the Hamiltonian - I hadn't realized that.

Starting from the beginning, I let my prepared state be |ψ(0)> = |+>.
This enters the B-field and since the probability is dependent on time, I need to apply U = eiωt/2 to my time evolved state |ψ(t)>.
If I catch you drift, I should write |ψ(0)> in the x basis at this point? So |ψ(0)> = 1/√2 (|+> + |->).
Then the time evolved state passing through the B-field before it enters the second analyzer is |ψ(t)> = 1/√2 * (e-iωt/2 |+> + eiωt/2 |->)
The probability I am looking for with this state coming from the second analyzer is then |<-|ψ(t)>|2.
Next I applied Euler's, and the probability becomes P- = 1/2 |cos(ωt/2) + isin(ωt/2)|2 = 1/4

Is that looking correct? I am not sure what to do about all of the cos and sin terms I end up with when I square what is inside the abs value. It doesn't seem there is a clean way to extract t. Should I not use Euler's?

blue_leaf77
$$P_-(t) = \sin^2(\omega t/2) = \frac{1}{4}$$