Spin collapse in a magnetic field

[msumm21]

TL;DR Summary

Is magnetic field really exactly 0 in most places? Assuming it's not, isn't the spin of an electron constantly being "measured" and collapsing to a certain direction.

Basic descriptions of spin such as the beginning of Lindley's "Where does the weirdness go" state that an electron's spin doesn't exist or is "indeterminant" until measured (e.g. passed through a Stern-Gerlach field). However, isn't the magnetic field nonzero essentially everywhere (albeit small and "noisy")?

Are magnetic fields quantized so that they're 0 in most of space? If it's really nonzero in most of space, how is that we get the same spin result when passing through a 2nd SG (presumably the electron experiences another mag field in a "random" direction between the two SGs)?

 

[PeroK]

Summary:: Is magnetic field really exactly 0 in most places? Assuming it's not, isn't the spin of an electron constantly being "measured" and collapsing to a certain direction.

Basic descriptions of spin such as the beginning of Lindley's "Where does the weirdness go" state that an electron's spin doesn't exist or is "indeterminant" until measured (e.g. passed through a Stern-Gerlach field). However, isn't the magnetic field nonzero essentially everywhere (albeit small and "noisy")?

Technically, a SG magnet does not directly measure the spin of an electron, nor cause the spin component of the wave function to collapse. The wavefunction evolves during the time in the magnetic field and ends in a superposition of two spatial states, each associated with a distinct spin-state.

The only measurement is where the electron impacts a screen, from which the spin of the electron is inferred. At that point the superposition of two spatial states collapses. It's only when the result of the interaction with the magnetic field is recorded that the wavefunction collapses.

It's critical, in fact, to the theoretical sequential SG experiments that the electron state does not collapse when passing through the first magnet, but emerges as a superposition.

 

[msumm21]

Technically, a SG magnet does not directly measure the spin of an electron, nor cause the spin component of the wave function to collapse. The wavefunction evolves during the time in the magnetic field and ends in a superposition of two spatial states, each associated with a distinct spin-state.

Thanks, I think this answers most of my question.

I still don't understand the case of two aligned SGs. After passing through the 1st (before the 2nd), presumably there's a small but nonzero "random" magnetic field including everything else in the universe, outside this system we are focusing on. If so, presumably the field would generally not be aligned with the SGs and hence the state of the electron would change to a superposition two different spin states. If so, why would the 2nd SG deflect the electron the same way as the 1st.

 

[hutchphd]

If there are other fields that affect the beam the eigenstates of the system are no longer exactly simple.

 

[msumm21]

If there are other fields that affect the beam the eigenstates of the system are no longer exactly simple.

Thanks and wouldn't that always be the case? Is the field ever exactly 0 anywhere, if we really account for everything in the universe?

 

[PeterDonis]

Is the field ever exactly 0 anywhere

The field (aside from the one from our experimental apparatus) doesn't have to be exactly zero. It just has to be sufficiently small that the error it induces is small enough not to matter--meaning, much smaller than other sources of error.

For example, in the case of two successive SG magnets oriented in the same direction, if the extraneous fields in the lab are small enough that the error they induce is smaller than the error of your experimental apparatus in pointing the two SG magnets in exactly the same direction (which will most likely be the case), then the extraneous fields don't matter.

 

[PeterDonis]

if the extraneous fields in the lab are small enough that the error they induce is smaller than the error of your experimental apparatus in pointing the two SG magnets in exactly the same direction

Note, btw, that for an SG magnet, the error induced by an extraneous field will be due to the variation in the extraneous field on the distance scale of the experiment, not the field itself, since the SG magnet's action is due to the inhomogeneity of its field. So the Earth's magnetic field, for example, will induce negligible error, since it will be constant to a very good approximation on the distance scale of the lab.

 

[Nugatory]

Thanks and wouldn't that always be the case? Is the field ever exactly 0 anywhere, if we really account for everything in the universe?

It is not, and formally that means that when we write down and solve Schrodinger's equation using a Hamiltonian that includes only the electron and the magnetic field of the Stern-Gerlach device we're making an approximation. If we were to try for an exact Hamiltonian, we would include all those other effects in the Hamiltonian, and if we include everything no matter how small we'd end up with Schrodinger's equation for the wave function of the entire universe.

Of course in practice we do no such thing, not if we want useful answers. We approximate by ignoring any effect that is small compared with accuracy that we need, just as in classical physics we derive and use Kepler's laws while ignoring the gravitational effect of the distant stars.

 

[vanhees71]

Technically, a SG magnet does not directly measure the spin of an electron, nor cause the spin component of the wave function to collapse. The wavefunction evolves during the time in the magnetic field and ends in a superposition of two spatial states, each associated with a distinct spin-state.

The only measurement is where the electron impacts a screen, from which the spin of the electron is inferred. At that point the superposition of two spatial states collapses. It's only when the result of the interaction with the magnetic field is recorded that the wavefunction collapses.

It's critical, in fact, to the theoretical sequential SG experiments that the electron state does not collapse when passing through the first magnet, but emerges as a superposition.

I would say that a filter preparation a la von Neumann (in the literature usually called a von Neumann filter measurement) is completed when you absorb all electrons in one of the two beams after the magnet, because then you have only electrons in a specific spin state (up or down wrt. the direction of the magnetic field) left.

Of course you can do another Stern Gerlach experiment with this "collapsed" electrons (I don't like the talk about "collapse" at all, but that's unfortunately also also common slang in the textbook literature).

What you can't do what you can do in principle with the superposition (spin-position entangled state) is to bring the two partial beams together again and precisely reverse the split in the previous SG experiment with another SG experiment with the magnetic field precisely pointing in the reflected direction. This is, because the unitary time evolution is reversible while this is not the case if you filter out one of the partial beams, because here you have an interaction with the absorbing material, i.e., an open quantum system, leading to dissipation and irreversibility.

 

[msumm21]

The field (aside from the one from our experimental apparatus) doesn't have to be exactly zero. It just has to be sufficiently small that the error it induces is small enough not to matter--meaning, much smaller than other sources of error.

I may not understand this, but I'll state how I interpreted this as an answer to my question. Between the SG magnets, the mag field is indeed non-zero but very small and "randomly" oriented (?). However, it's so small that the state of electron spin does not change significantly from the superposition acquired by SG#1.

If so, does this mean a "weak" vertical SG would not transition the state to a superposition of up/down? To get the hard up/down superposition a "strong enough" SG is required?

If we were to try for an exact Hamiltonian, we would include all those other effects in the Hamiltonian, and if we include everything no matter how small we'd end up with Schrodinger's equation for the wave function of the entire universe.

Yes I understand that may be impractical when doing the math. I guess what I'm questioning is how it's OK to simplify the problem this way -- why the background magnetic fields can be neglected and still give the same result. It at first seemed to me that the "random" field between SG1 and SG2 would be like placing an SG3 between 1 and 2 with a random orientation and hence destroy the fact that an electron would be deflected "up" via SG2 just because it went up via SG1.

 

[PeroK]

I may not understand this, but I'll state how I interpreted this as an answer to my question. Between the SG magnets, the mag field is indeed non-zero but very small and "randomly" oriented (?). However, it's so small that the state of electron spin does not change significantly from the superposition acquired by SG#1.

Yes, essentially. In general, a magnetic field will have a scrambling effect on the spin state.

If so, does this mean a "weak" vertical SG would not transition the state to a superposition of up/down? To get the hard up/down superposition a "strong enough" SG is required?

What you need is a magnetic field that varies significantly in the z-direction. It's the variation in the field across the uncertainty in the electon's z-position that creates the superposition of wave-functions.

Yes I understand that may be impractical when doing the math. I guess what I'm questioning is how it's OK to simplify the problem this way -- why the background magnetic fields can be neglected and still give the same result. It at first seemed to me that the "random" field between SG1 and SG2 would be like placing an SG3 between 1 and 2 with a random orientation and hence destroy the fact that an electron would be deflected "up" via SG2 just because it went up via SG1.

The trick of any experimental set-up is to isolate the experiment from external factors. If you can't do that, then you don't have an experiment. This is one reason that many of these experiments take extreme skill to execute. For example, it took decades before a test of Bell's theorem was undertaken. If you have rogue magnetic fields in the lab that are as significant as the one you are studying, then no experiment is going to be satisfactory.

 

[PeterDonis]

Between the SG magnets, the mag field is indeed non-zero but very small and "randomly" oriented (?).

No, it's very small and, to a very good approximation, its strength is constant. Which means it doesn't affect the superposition created by the SG magnet at all. Only a magnetic field of varying strength will do that.

Also, you need to consider the probable error in the relative direction of the second SG magnet with respect to the first. You want them to be exactly the same, but "exactly the same" is not achievable with 100 percent accuracy; there will be some error. The probable error due to that source is (I believe) much larger than the probable error due to extraneous magnetic fields inside the lab.

does this mean a "weak" vertical SG would not transition the state to a superposition of up/down?

No, it just means that the separation in the two output beams would be smaller, meaning that you would need to place whatever the next device in the experiment is (a detector or a second SG magnet) further away from the first SG magnet in order to make sure you were only capturing one output beam (assuming that the size, or more precisely the cross-sectional area, of the second device is held constant). Capturing both output beams instead of one, if you intend the second device to only capture one, is another source of possible error.

 

[PeterDonis]

what I'm questioning is how it's OK to simplify the problem this way -- why the background magnetic fields can be neglected and still give the same result

The basic reason, as I've said, is that the probable error due to stray magnetic fields inside the lab is much smaller than the probable error due to other sources (error in alignment of the second SG magnet relative to the first, and error in not completely isolating one output beam of the first SG magnet).

For a typical presentation in a textbook, the real reason is that trying to include any of those sources of possible error in the analysis would greatly complicate the analysis, and for pedagogical reasons that's not a good idea--one needs to understand the basic physics, as embodied in the ideal experiment, first, before trying to understand how various possible sources of error might affect the result. But in a real experiment, of course, you do have to analyze the possible sources of error and estimate their magnitude, in order to determine which errors can reasonably be ignored and which ones can't.

For estimating the error due to background magnetic fields, as has been noted, the key thing is not the strength of the background field but its variation on the distance scale of the lab, since it's variation in field strength along a specific direction that the SG device is using to separate the two spin eigenstates for that direction into two separate output beams.

 

[msumm21]

OK thanks all. So sounds like there can be a small field with very small variation between SG1 and SG2, but small enough to not impact the superposition created by SG1. So there's a limit to the field gradient, below which it doesn't impact the spin state? To simplify I'm thinking of a SG3 again, say horizontally aligned, while SG1 was vertical. If we make SG3 weak enough it won't affect the state of the particles?

I understand the part about uncertainty in the SG1 / SG2 alignment.

 

[PeterDonis]

So there's a limit to the field gradient, below which it doesn't impact the spin state?

No. But there is a limit below which the error due to a gradient in a stray field in the lab is too small to be measurable given all of the other sources of error which are present.

 

[msumm21]

No. But there is a limit below which the error due to a gradient in a stray field in the lab is too small to be measurable given all of the other sources of error which are present.

OK, so a weak, horizontally aligned SG3 modifies the state, but does not align it with SG3. The spin "up" beam exiting SG1 will still be substantially deflected up after passing through SG2, after passing through the background (SG3).

 

[PeterDonis]

so a weak, horizontally aligned SG3 modifies the state, but does not align it with SG3.

No. A weak, horizontally aligned, constant background field does not do anything significant in this scenario. A weak, horizontally aligned SG magnet does what I described in the last paragraph of post #12. A weak, varying background field is not going to vary only along one particular direction--see below.

the background (SG3).

The background is not an SG magnet. You can't treat them the same. The background field, even if its field strength does vary, is not going to vary only along one particular direction like an SG magnet's does.

 

[PeroK]

OK, so a weak, horizontally aligned SG3 modifies the state, but does not align it with SG3. The spin "up" beam exiting SG1 will still be substantially deflected up after passing through SG2, after passing through the background (SG3).

If you want a more in-depth analysis of what's happening in the SG experiment, try this:

https://physics.mq.edu.au/~jcresser/Phys304/Handouts/QuantumPhysicsNotes.pdf

Chapter 6 covers spin and SG. These notes generally give an accessible and insightful introduction to QM at a level above popular science.