Determine the number of particles transmitted by S-G analyzer

[acdurbin953]

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 traveling 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/m_e, but I don't know what to use for these neutral particles.

 

[blue_leaf77]

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 traveling 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##.

 

[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 = e^iωt/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 |+> + e^iωt/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]

Alright, actually after some check ups, the equation you already have from the final probability ##P_-(t) = \sin^2(\omega t/2)## is correct. If you want to follow step-by-step calculation, you will end up with that formula. So it's up to you whether you want to directly use the given formula or to firstly derive it. Either way, the equation you have is
$$
P_-(t) = \sin^2(\omega t/2) = \frac{1}{4}
$$
and you are asked to find the minimum value of ##t## such that ##t>0##. It should be easy.

 

[acdurbin953]

Right - yeah if the original equation I had used is correct I'll stick with that. Thanks for your help!

 

[Yutao]

is the answer w.r.t. v0??