At least one textbook [1] uses sequential Stern-Gerlach devices to introduce to students that the components of angular momentum are incompatible observables. Viz., the $z$-up beam from a SG device with magnetic field in the $\hat z$ direction (an SG$z$ device) is passed through an SG$x$ device, and is found to split into two beams. Passing say, the $x$-up beam through an SG$z$ device, it too splits.
Of course, knowing quantum mechanics this is exactly what we expect.
But to someone who does not know quantum mechanics, is this convincing that there is no $\mid+x,+z\rangle$ state? I am not so sure it is if we consider it as a real experiment, with finite precision.
We know that the beam entering the SG$x$ device has $S_z = \hbar/2$, we do not know anything about its $S_x$. We know that the beams leaving the SG$x$ device have $S_x = \pm \hbar/2$, respectively. By adding the second SG$z$ we wish to test if $S_x$ and $S_z$ can have definite values simultaneously, but there is then an assumption that the SG$x$ device does not disturb the value of $S_z$, or at least does so with a very small spread. But already in the classical picture the Stern-Gerlach device is not such a device.
In the $SG$z device the $\mathbf B$-field has a large homogeneous component $B_0\hat z$, such that the angular momentum around $\hat z$ is approximately conserved while the other components average to 0, and the force, on average, has only a $\hat z$ component [2]. But in the SG$x$ device the angular momentum precesses around $\hat x$, with a period that is quite short, $T = 10^{-9}$ s or less.
If the particle beam has a spread of velocities $v$ such that the spread in times-of-flight $t$ is not small compared to $T$, we should not expect the second beam to be $z$-polarized, even classically. The relation between the spreads is $\Delta t = t \Delta v /v$. In the original experiment [2] we can estimate $v$ and $t$ as being on the order of $10^2$ m/s and $10^{-4}$ s, requiring $\Delta v /v $ on the order $10^{-5}$. This seems entirely unreasonable for a thermal source, considering the finite width of the collimator and if nothing else the force component neglected initially seems liable to produce a spread of at least this order.
I tried to search the literature to see if the sequential experiment has actually been carried out, but could not find anything. I did find Ref. 3 that seems to talk about two-spinors, but I cannot access it.
References
- Townsend, J.S. (2000). A Modern Approach to Quantum Mechanics. University Science Books
- Stern, O. (1988). A way towards the experimental examination of spatial quantisation in a magnetic field. Zeitschrift für Physik D Atoms, Molecules and Clusters, 10(2), 114-116.
- Darwin, C. G. (1927). The electron as a vector wave. Proceedings of the Royal Society of London A, 227-253.
Best Answer
In your first paragraph you describe a Stern-Gerlach device as one with a magnetic field in the $\hat z$ direction. And then later you talk about having a large homogeneous component of the magnetic field. I'm not sure you have an accurate physical model of a Stern-Gerlach device.
The Hamiltonian for a Stern-Gerlach has magnetic fields components combined with the Pauli matrices like $B_x\hat\sigma_x+B_y\hat\sigma_y+B_z\hat\sigma_z$ well proportional to that. It is the quantum version of a magnetic moment in an external magnetic field and in this case the magnetic moment is proportional to the spin hence $\vec\mu\cdot\vec B$ is proportional to above.
The classical force comes from the gradient of this quantity. So it is inhomogeneous magnetic fields that you use to measure spin.
And while you want the field to only have $\hat z$ components to measure just the z component of spin you need the magnetic field to have a gradient (be inhomogeneous) to deflect the beam. And it is the direction that the field gets stronger that is as important as which direction it points. So it is in no way similar to just having a magnetic field that points some direction.
That said. It is rather straightforward to measure the spin z component twice in a row or three times in a row and you get the same answer each time as you got the first time. So it is in the nature of the outcome that it give those results again.
Same if you do two or three spin x component measurements. So it is in the nature of the outcome of the first experiment that the result be the kind of thing that gives those same results again and does so reliability.
These experiments are easy to do, so I don't really think that is what you are asking about.
Now if you measure z then x then z you do not always get the same result for the second z measurement as you got for the first z measurement. This has been done.
So we know for a certainty that the spin x "measurement" has changed the particle's state. Because it used to have a reliability under spin z measurements and then it no longer has that reliability.
I don't know what details you think need to be involved here, we definitely changed the particle when we measured a complementary (i.e. not equal) component.