# In Stern-Gerlach experiment, where does wavefunction collapse?

I was reading Sakurai's Modern QM and it talks about Stern-Gerlach experiment in chapter 1. As silver atom passes through non-uniform magnetic field and enters detector downstream, a measurement is made and the atom collapses into either Sz+ or Sz- eigenstate.

Now I am a bit confused, does silver atom's wavefunction collapses as it enters B-field or the detector? The former doesn't make sense because there are magnetic field everywhere on earth so does it mean wavefunction is always collapsed (in the direction of local magnetic field gradient)? Sure the non-homogeneity of earth's magnetic field is orders of magnitude smaller than the apparatus in Stern-Gerlach experiment, but if magnitude of B-field gradient is the answer does it imply that that there exist a threshold magnetic gradient below which wavefunction won't collapse? That doesn't sound right either.

I am more inclined to think that the particle detector collapses the wavefunction. If that's the case, what's so special about the detector that causes wavefunction to collapse? I mean, particle detector like scintillators ultimately relies on EM interaction between silver atom and detector material to generate electric signal. If magnetic interaction with the magnet downstream doesn't collapse the wavefunction, how could EM interaction inside the detector collapse the wavefunction?

• Maybe there is no wave function and the atoms are pushed and pulled, up or down by the magnetic field. Commented Nov 25, 2021 at 7:01
• These problems all go away after realizing that the silver atoms and our eyes do not have separate wavefunctions. Commented Nov 25, 2021 at 16:36

Wave function collapse is a change of wave function that we do at certain time of experimenting "by hand", "because we get new facts", as opposed to a change of wave function determined by past data and Schroedinger's equation. It is a fix for our inability to get the new fact purely from calculations. One such fact is detection of the atom at one of few possible beams or landing spots.

When the atom passes through magnetic field without interacting with position-revealing devices, we do not invoke collapse, because we have no reason to - it is fine evolving the wave function just using the Schroedinger's equation there.

When the atom is detected at a screen and we get new information about its position (and its spin state), this is more than calculated wave function implies, and this allows us to update the wave function we got from Schroedinger's equation using the new fact.

Aside: "Collapse of the wavefunction" unfortunately is a wide spread terminology that appeals to popularization of science articles. In the mathematics of quantum mechanics it means that "the boundary conditions have changed and a new wavefunction must describe the system after the so called collapse" . It is a word that describe an instant in time where the wavefunction changes.

In the experiment the silver atoms travel outside the magnetic field ,( the earth's field is very weak to enter in the calculations for the experiment, if you read the link below) and the beam splits according to the three values of the magnetic moment for the silver atom. So it is the magnetic field that picks the probable path of the individual atom.

In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path. The screen reveals discrete points of accumulation, rather than a continuous distribution,1 owing to their quantized spin.

For the experiment with silver atoms it is its intrinsic magnetic moment that shows up in the experiment. The history is given here.

Figure 12. Sketch of the Stern-Gerlach experimental apparatus. The result expected for atoms in an L=1 state (three components) is shown. From Weinert (1995).

The beam of silver atoms has variously oriented magnetic moments. Passing through the strong magnetic field , classical physics expects a uniform distribution, but the experiment shows the quantized (3 states for l=1) nature of the magnetic moment.

The wave function of the individual silver atom had the magnetic moment of the atom in a distribution according to the wavefunction describing it. The interaction of the magnetic moment of the individual atom with the magnetic field ( through the exchange of virtual photons) changes(collapses) the wave function to a new one that gives a specific angle for the trajectory according to the probability of the interaction with the magnetic moment. On the screen is a second interaction , of the individual atoms hitting the screen and interacting with its atoms.

• I am not sure that the wavefunction collapses when passing through the magnetic field, what happens is that the position and spin parts become correlated such that the upward trajectory gives spin up and similar for the downward trajectory with spin down. Collapse only occurs at the detector. Commented Nov 25, 2021 at 7:39
• @user7896 you must have not read my Aside in the answer. Collapse is a bad synonymous for change of wavefunction due to boundary conditions. The boundary conditions in the magnetic field are different than outside it, and a different wavefunction describes the system atom+magnetic field, by construction of the theory of quantum mechanics. The new wavefunctions at the detector are a second "collapse", new boundary conditions. Anyway the change in the beam is seen before the detector. Commented Nov 25, 2021 at 8:04
• Ah it seems that you are assuming the magnetic field to abruptly change leading to a consideration of boundary conditions. I was considering a smooth magnetic field leading to the interaction being treated as a scattering problem where there is no collapse. Commented Nov 25, 2021 at 8:20
• @user7896 look at the drawing of the experiment Commented Nov 25, 2021 at 8:32
• Even so the magnetic field will not be a step function, you are going to have a smooth tapering as you move away from the magnets. That was what I meant by saying that the interaction can be treated as a scattering with a smooth potential. Commented Nov 25, 2021 at 9:11