# Another Stern-Gerlach Question

1. Dec 26, 2008

### msumm21

As described in the Wikipedia article about the Stern-Gerlach experiment, if an electron passes through the (Z-axis oriented) device it will hit a screen on the other side in one of 2 possible locations, call them the upper and lower locations. Sometimes the electron is "spin up" about the z-axis and it hits the screen in the upper location, other times it is "spin down" and hits the screen in the lower location.

This seems to imply that the electron is in the spin up or spin down state about the z-axis, but why can't it be in the spin up state about the x-axis or in any of the other infinite number of states? I know that a measurement of the spin about the z-axis will cause the spin state to be up or down about the z-axis, but I don't see that measurement taking place. I agree that the measurement of the location of the electron when it hits the screen gives away information about the spin (and therefore constitutes a spin measurement), but I don't see why that measurement is about the Z-axis. Why can't the electron hit the screen in other locations that are consistent with other spin orientations?

Looks like the total magnitude of the spin of an electron is sqrt(3)*hbar/2 and, when measured, its spin about a particular axis is either +hbar/2 or -hbar/2 (so there is another component of magnitude hbar/sqrt(2) in the plane normal to the axis of measurement). So, for example, why can't the electron hit the screen with no z-axis deflection corresponding to maybe a state with hbar/2 about the x-axis and hbar/sqrt(2) about the y-axis?

Thanks

2. Dec 27, 2008

Staff Emeritus
Because this is quantum mechanics. A spin-1/2 particle that is 100% polarized in the x-direction has a 50% chance of having it's spin measured in the +z direction and 50% chance of having its spin measured in the -z direction.

3. Dec 27, 2008

### msumm21

Of course I've read that, if it's in the state |up_x> (or |down_x>) then a subsequent measurement of the spin about the z axis it will put it in either the |up_z> or |down_z> state with a 50% chance of each. My question is when (in the Stern-Gerlach experiment) did the spin about the z-axis get measured? The only measurement I see is the measurement of the position at which the electron hit the screen, which does yield information about the spin, but not necessarily about the z-axis ... please see the original message where I explain my question.

4. Dec 27, 2008

### Staff: Mentor

Since the field of the Stern-Gerlach magnet is along the z-direction it forces an interaction (and measurement) of the z-component of the spin.

5. Dec 27, 2008

### msumm21

OK. I think I was mistaken in my last message when I said "which does yield information about the spin, but not necessarily about the z axis." Even if you were to imagine the particle as passing through the device in the |up_x> state, and therefore with a spin ranging from -hbar/sqrt(2) to +hbar/sqrt(2) about the z-axis, you would still be able to determine the amount of spin about the z-axis by the z position of the particle on the screen, so you are measuring the spin about the z axis when the particle hits the screen, so it must one of the two values hbar/2 or -hbar/2. Thanks.