You are confusing classical physics (the orientation of the atom) with quantum mechanics (the spin of the atom).
The spin of the atom can only be +1 or -1, regardless of the coordinate system (i.e. orientation) you choose to measure it in.
It is comparable to the polarization of light. You can filter it in a certain direction, but it, too, is a quantum property. While light cannot pass through two 90deg rotated polarization filters. Ir can pass if you put a 45deg polarization filter between them. That can not be explained classically.
But the two lines they get on the pin-head-sized region is physical and has a particular physical orientation? How can the spin not be a physical orientation and yet the experiment result is all about the physical location of the lines?
I'm sure the issues are with the details of how the experiment is explained, but I still don't understand.
Quantum angular momentum is a different thing from the ordinary spin of a spinning top. It's not about mass revolving around a physical center. It's just there, almost as if you painted "The angular momentum is 10^-34 Joule-seconds, pointing to the left wall of the laboratory" on it.
Despite that, a magnet acts on it exactly the same as if it were a spinning piece of charged metal. So it doesn't start as a spatial difference, but it becomes one once you pass it through the field.
And one of the ways you can tell it's not the same as a spinning piece of metal is that the amount of spin is always exactly the same, regardless of how you orient the field. It's always that number I gave you above, called h-bar. The only question is whether it's positive or negative; it's going to be exactly one or the other.
That's not what would happen to a regular object. For a regular object, you'd sometimes get 100% of h-bar, and sometimes 50%, and sometimes 0%, and sometimes -100%, depending on the angle between the spin and your apparatus. Just like if you were trying to measure the width of a piece of wood with a ruler: it depends on how you angle the ruler. Somehow, for quantum things, it's always exactly 100% or -100%.
100% things go one direction; -100% things go the other direction. You get exactly two lines, separated physically in space, even though there was no such separation in the original charged particle.
It turns out that you can only measure one component of the spin vector at a time. So you choose some direction in space, call that direction the “z”
direction and orient your Stern-Gerlach apparatus in that direction. Important point: SG only separates two beams along this one direction so is only sensitive to differences in the z component of spin vector. You find two final beams. Effectively the measurement has “snapped” the z component of spin onto one of two discrete values. The system is rotationally symmetric in the sense that you could have arbitrarily chosen any other direction in space to call z and the measurement results would be similar. Indeed it’s very interesting to consider the results of orienting various SG apparati in different directions and then chaining together by feeding output of one as input beam of another (as is imagined in some textbooks).
