Understanding how Stern–Gerlach apparatuses work I'm studying about the  Stern–Gerlach experiment. I'm having some hard time to figure out how the Stern–Gerlach apparatuses work. I have the following question:
Transfer a beam of electrons through the next series of experimental machines (Stern - Gerlach):

Given that in all Stern apparatuses - your electrons with the positive spin projection on the axis indicated on the machine are in the upper beam coming out of the machine and the electrons with the negative spin projection on the axis indicated on the machine are in the lower beam coming out of the machine.
Given that $3/4$ of the electrons in the original beam have a spin state $S_z=\frac{1}{2}\hbar$ and $1/4$
of the electrons in the original beam have a spin state $S_z=-\frac{1}{2}\hbar$, find the percentage of electrons from the original beam that are in the top beam coming out of the Stern machine - your last (rightmost) machine in the series.
What I did: In the upper part of the first SG tool, $\frac{3}{4}$ of the electrons are moving through, in the lower part of the second SG tool $\frac{1}{2}$ of the electrons are moving through, in the upper part of the third SG tool $\frac{1}{2}$ of the electrons are moving through and  in the upper part of the fourth SG tool $\frac{1}{2}$ of the electrons are moving through. This leads to:
$$
\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{3}{32}=9.375%
$$
I know that this is the right solution but I'm not sure I fully understand it. So I have the following question to sum up:

*

*In each one of the {2,3,4} steps I reduces half of the electrons because there are two outputs in the SG tool or is it something has to do with the spin state $S_z=\frac{1}{2}\hbar$. If it's because there are two outputs, then what $S_z=\frac{1}{2}\hbar$ is telling us in this question? Also, when then we don't reduce by half in the first step?

*What if I had $SG_z\to SG_z$ at first or at some point? How should I happened same axis?

*Is there a general function that represents the SG apparatuses that gives the output based on the input information?

 A: A SG apparatus measures the spin projection of electrons in a given axis (the one where the magnetic field is applied). For example, a SG in the z direction measures the probability of finding electons with spins in the up and down states in the z axis.
Once your electrons pass through a SG device, their wave function collapse because of the measurement, obtaining two different outcomes where the intensities of the beams depend on the initial state (before passing through the machine). Each outcome can be characterized as a different state in the projected basis.
In your example, the initial state gives $3/4$ of probabilities to the state up and $1/4$ to the state down in the z axis, so the wave function of these electrons is
$$
|\Psi\rangle=\frac{\sqrt{3}}{2}|\uparrow\rangle_z+\frac{1}{2}|\downarrow\rangle_z
$$
After passing through the SG number 1 (in the z direction), electrons of the upper beam will have a state like $|\Psi\rangle=|\uparrow\rangle_z$ because they are projected in this direction. In the same way, the electrons of the lower beam have the state $|\Psi\rangle=|\downarrow\rangle_z$. Now, considering that there are no perturbations that alter the states of the electrons, we can make the beams pass through more SG machines knowing their states.
The important point comes now: if you are going to pass a beam through a SG with a different direction than the previous one, you can write the wave function in the new basis in order to see the probabilities of projection. In the example, the second SG machine is in the y direction, so we rewrite the states of the electrons in the new basis:
$$
|\uparrow\rangle_z=\frac{1}{\sqrt{2}}|\uparrow\rangle_y+\frac{1}{\sqrt{2}}|\downarrow\rangle_y
$$
$$
|\downarrow\rangle_z=-\frac{i}{\sqrt{2}}|\uparrow\rangle_y+\frac{i}{\sqrt{2}}|\downarrow\rangle_y
$$
where, for both beams, probabilities of projection in the new direction are 1/2. This happens always that we change the direction of the of the SG machine, since the electrons "forget" their previous state .
Answering to the questions:

*

*The fact that you have two possible outcomes is responsible of the factor 1/2, but it is independent of introducing the upper beam. As I showed right before, the lower beam has the same probabilities. It happens because of the measurement and the change of direction in the next apparatus.


*If you use the same direction twice, your second machine would show only the projection of the beam that enters, because the electrons are already projected in that direction. So changing the SG number two in the example by a SG in the z direction, there would not be lower beam because you introduce the upper beam projected in the z direction.


*The function you are asking for is just the wave function of the electrons, and of course, you need to know the direction of the SG apparatus.
Maybe my explanation is not good enough because I'm skipping many details, but a good detailed reference could be the first chapter of "Modern Quantum Mechanics" by Sakurai. I hope this can help you!
