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:
- The first measurement accepts $s_z=\hbar/2$ atoms and rejects $s_z=-\hbar/2$ atoms.
- 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.
- 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?