Intensity of sequential spin measurments in QM

Question

The following is a question from Sakurai's book Modern Quantum Mechanics, 2nd edition, page 60:

A beam of spin $\frac12$ atoms goes through a series of Stern-Gerlach-type measurements as follows:

1. The first measurement accepts $s_z=\hbar/2$ atoms and rejects $s_z=-\hbar/2$ atoms.
2. The second measurement accepts $s_n=\hbar/2$ atoms and rejects $s_n=-\hbar/2$ atoms, where $s_n$ is the eigenvalue of the operator $\mathbf S\cdot\hat{\mathbf n}$, with $\hat{\mathbf n}$ making an angle $\beta$ in the $xz$-plane with respect to the $z$-axis.
3. The third measurement accepts $-s_z=\hbar/2$ atoms and rejects $s_z=\hbar/2$ atoms.

What is the intensity of the final $s_z=-\hbar/2$ beam when the $s_z=\hbar/2$ beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final $s_z=-\hbar/2$ beam?

Proposed Solution: I assume the intensity is defined as the probability amplitude squared of getting $s_z = - \frac{\hbar}{2}$ in the final measurement. So you start with the state $| \psi \rangle = \frac{1}{\sqrt{2}} | + \rangle + \frac{1}{\sqrt{2}} | - \rangle$ before any measurements are made, you then project onto the eigenstates which correspond to the measurements you accept, hence you first project onto $| + \rangle \langle + |$ and then $| +; \hat{n} \rangle \langle +; \hat{n}|$ (taking $\alpha = 0$) and finally $|-\rangle \langle - |$, this process hence yields the intensity given by $$\bigg(\frac{1}{\sqrt{2}} \langle + | + \frac{1}{\sqrt{2}} \langle - |\bigg)\bigg(|- \rangle \langle - |+; \hat{n} \rangle \langle +; \hat{n} | + \rangle \langle + |\bigg) \bigg(\frac{1}{\sqrt{2}} | + \rangle + \frac{1}{\sqrt{2}} | - \rangle\bigg).$$

Apparently this is not correct, does anyone know where I went wrong?

Answer 1

Using the notation you posted in the comments, $s_z |\pm\rangle = \pm\frac \hbar2 |\pm\rangle$ and $s_n |\pm,n\rangle = \pm \frac\hbar2|\pm,n\rangle$, then you have correctly summarized the effect of the experiment as the right-to-left application of the projectors $$ |- \rangle \langle - | \cdot |+; \hat{n} \rangle \langle +; \hat{n} | \cdot | + \rangle \langle + |, $$ where you want to maximize the norm of the wavefunction after it's been acted on by that operator.

On the other hand, I've got no idea you got the superposition $| \psi \rangle = \frac{1}{\sqrt{2}} | + \rangle + \frac{1}{\sqrt{2}} | - \rangle$ from, but it's wrong; the text does not make any reference to the state of the system before it reaches the first magnet. If your thinking there was "take a state that's equally likely to be in $|+\rangle$ as it is in $|-\rangle$, then that's not how you describe it, because the state you wrote down is coherent and it doesn't appear up by magic. Why not write down, say, $| \psi \rangle = \frac{1}{\sqrt{2}} |+\rangle - \frac{1}{\sqrt{2}} |-\rangle$ instead?

Thus, you don't know what the initial state is, but you do know that you want the norm after the first projector comes up to be one, or in other words, what you really want to maximize is $$ p=\frac{\bigg\||- \rangle \langle - |\cdot|+; \hat{n} \rangle \langle +; \hat{n} |\cdot| + \rangle \langle + | \cdot |\psi\rangle\bigg\|^2}{\bigg\|| + \rangle \langle + | \cdot |\psi\rangle\bigg\|^2}. $$ Now, this seems like it depends on what the initial state is, but it doesn't really (since the experiment doesn't, either) unless you happened to start with on $|\psi\rangle = |-\rangle$, in which case the experiment doesn't make sense. As such, you're free to take whatever arbitrary state you want, for convenience, so you can just start with $|\psi\rangle = |-+\rangle$ and be done with $|\psi\rangle$, in which case the calculation reduces to maximizing $$ p=\bigg\||- \rangle \langle - |+; \hat{n} \rangle \langle +; \hat{n} | + \rangle \bigg\|^2 =\bigg| \langle - |+; \hat{n} \rangle \langle +; \hat{n} | + \rangle \bigg|^2 . $$

Answer 2

"The s_z=|+> beam surviving the first measurement is normalized to unity". "What is the intensity of the final s_z=|-> beam"? You seem to start and end with superpositions from which I don't see where you get them from. But those won't matter anyway, since there is only one Stern-Gerlach device that really does something. The first one creates the particles and the last one annihilates them. What remains is an amplitudo

$\langle-|+;n\rangle\langle+;n|+\rangle$.

As for the intensity, you should take the squared norm of this amplitudo, as usual in QM