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).
