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Bell's Stern-Gerlach experiment, consisting of 2 sets of detectors each with 3 randomly chosen detection directions, is one of the most important experiments cited as proving that entanglement has strange properties - most often apparent super-luminal communication.

The conclusion that this experiment disproves non-super-luminal hidden variables theories rests on the assumption that if the particles contained hidden variables that determine whether they are measured with spin-up or spin-down, the measurements of spin would be the same (both spin-up or both spin-down) only 4/9ths of the time. Alternatively, entanglement theory expects the measurements to match 50% of the time. This experiment is well explained by this video: https://www.youtube.com/watch?v=ZuvK-od647c .

In the following I'll say a spin is in a particular direction, if I mean more specifically that the average direction of the superposition of a particles spin is in a particular direction.

The math he uses in that video seems incredibly simplistic, and I noticed that if you chose a direction of spin perpendicular to one of the detectors, the measurement of that one detector is essentially useless - there is a 50% chance of it measuring spin-up and 50% of spin-down. See the following picture:

enter image description here

Here, the green arrow represents the particle's direction of spin. Detector 1 would still give either spin-up or spin-down, but for such a particle it essentially gives us no real information - its 50/50. Extending this observation to other directions, the chance of a detector measuring a particle who's spin is pointing straight down (not spin-down, in the diagram, the direction between detector 3 and 2) of being detected as spin-up by detectors 2 and 3 would be only 75%, and the probability of detector 1 measuring spin-down would be 100%.

These numbers calculate that even with hidden variables, the expected probability of the spin measurements being different is still 50% - just as actually seen in the real experimental results. Here's my work:

http://www.btetrud.com/files/BellSternGerlach.ods

My question is, under the assumption that many physicists have done a lot more thinking about this than I have and have determined I'm wrong, why am I wrong?

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    $\begingroup$ I think you're misinterpreting the hidden variables theory. The idea is not that there's a hidden "real" spin direction; it's that there's an infinity of hidden up or down bits, one for each possible spin direction. $\endgroup$ – Warren Dew Sep 19 '15 at 0:13
  • $\begingroup$ The conjecture that there is a hidden "real" spin direction certainly could be called a hidden variable theory. Is there something experimental that proves that impossible? $\endgroup$ – B T Sep 19 '15 at 0:16
  • $\begingroup$ That's a different hidden variable theory. Let's flesh it out a bit: let's suppose the particle now encounters a detector at 60 degrees to its spin. How is it determined whether the detector detects "spin up" or "spin down"? If it's randomly, independent of its pair particle, then there's a possibility of the 60 degree detectors having matching results for the two particles in the pair, when empirically that does not happen. (cont) $\endgroup$ – Warren Dew Sep 19 '15 at 0:22
  • $\begingroup$ If it's not independent of the pair particle, then we still need communication between the particles, or else some other hidden variable to ensure that the particles still have opposite measurements no matter what direction they are measured in. $\endgroup$ – Warren Dew Sep 19 '15 at 0:23
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    $\begingroup$ @WarrenDew Why would you need communication? The wavefunction could evolve according to the Schrödinger equation, it's a PDE. $\endgroup$ – Timaeus Sep 19 '15 at 2:07
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The experiment rests on the assumption that if the particles contained hidden variables that determine whether they are measured with spin-up or spin-down, the measurements of spin would be the same (both spin-up or both spin-down) only 4/9ths of the time.

The experiment doesn't rest on assumptions, the experiment is an experimental setup that produces certain experimentally observed results. And there are hidden variable theories that predict the same results as Quantum Mechanics predicts for Stern-Gerlach measurements. Of any type, and for any spin and any entanglement. But when someone says that hidden variables predict 8/18 and quantum mechanics predicts 9/18 then they aren't discussing hidden variable theories that are designed to make correct predictions, they are discussing hidden variable theories designed to be wrong. And then you can experimentally prove that wrong hidden variable theories are wrong.

But there will still be other hidden variable theories that agree with the experiments because there are hidden variable theories that agree with quantum mechanics.

One example is to have a hidden variable for the configuration and another for a spin state. That's, right have a whole spin state as a hidden variable in addition to position. Then to correctly get a result for a Stern-Gerlach device you ... write down the Schrödinger equation for a Stern-Gerlach device. A beam goes in one side, it widens, it splits, and then meanwhile the spin state changes so that in the beam deflected one way the spin changes to become projected onto one eigenspace and the spin changes to become projected onto another eigenspace for another branch of the original incoming spin and so on. And this isn't the Born rule, this is literally just what the Schrödinger equation predicts happens when you write down the actual experimental conditions of the actual combination of object plus Stern-Gerlach device.

And the Schrödinger equation predicts the exact size of each branch of the split beam, and predicts the size according to the L2 norm of the projections. Again, that's all just the Schrödinger equation, no born or probability yet.

So now we have a wave where the beam is split, and for each branch the beam splits into, the spin for that particle is an eigenstate of the right operator. And the sizes of the beams are sized right so amongst other things repeated measurement of the same type don't split the beam and just deflect it.

And to get the statistical prediction you can either say that one of the split beams is real by having the hidden variable of position be in that branch of the split beam. Or instead, you can have, in addition to the Stern-Gerlach device, you can have and also write down the Schrödinger equation for the whole ensemble and the aggregator and get a wave that is super tiny in regions where the state of the aggregator is not super close to the average predicted by the Born rule, and since devices that observe have to be tolerant to small enough perturbations the other parts actually fail to affect the designed dynamics of the aggregator so you have a clear prediction of having averages close to the Born average for large ensembles whose results are aggregated by a physical device.

In the following I'll say a spin is in a particular direction, if I mean more specifically that the average direction of the superposition of a particles spin is in a particular direction.

You can do better. For a single spin 1/2 particle the spin state literally is an eigenstate of $n_x\hat\sigma_x+n_y\hat\sigma_y+n_z\hat\sigma_z$ for some real unit vector $(n_x,n_y,n_z).$ So it basically the spin state literally is a unit vector (it is a bit more accurate to represent it as a spin plane when you consider nonrelativistic physics as a limit of relativistic physics, but in 3d they are pretty similar).

For a joint multiple particle spin state it isn't just a single vector but the same thing happens, as the beam for that one particle splits. The whole wave in configuration space splits and the joint spin state evolves continuously so that after the beam splits into different branches the joint spin state for each branch is the corresponding projection you'd use for the Born Rule, and again this is simply what the Schrödinger equation predicts as PDE for the continuous evolution.

The math he uses in that video seems incredibly simplistic, and I noticed that if you chose a direction of spin perpendicular to one of the detectors, the measurement of that one detector is essentially useless - there is a 50% chance of it measuring spin-up and 50% of spin-down.

I didn't watch the video, but the math for how the Schrödinger equation predicts that unit vector evolves is pretty simple. And yes, if the spin points in one direction and you setup your Stern-Gerlach device orthogonal to that you beam just splits straight in half so half is deflected one way and becomes spin up and the other half of the beam deflects the opposite way and becomes spin down.

So the hidden variable for position ends up dictating exactly which spin result you get. If the unknown hidden position was a little bit closer to one side you get spin up and if it was a little bit closer to the other side you get spin down.

Here, the green arrow represents the particle's direction of spin. Detector 1 would still give either spin-up or spin-down, but for such a particle it essentially gives us no real information - its 50/50.

Again, didn't watch the video. But I know what the Schrödinger equation predicts for the actual experimental setup from start to finish and it does give information. It gives the information about what the spin turned into.

And you couldn't hope for better, if you had an eigenstate of $\hat\sigma_z$ and you put it into a $\hat z$ oriented device you don't split the beam no matter how many times you do it, so the particle objectively was in that spin state. If you then stick it into into a $\hat x$ oriented device you, firstly, do split the beam. But worse, now when you stick it into a $\hat z$ oriented device you do split the beam. And that didn't used to happen, so the $\hat x$ oriented device objectively changed the state.

So Stern-Gerlach devices objectively change the spin states of things that aren't eigen to them, and they do give this real information about what they changed it into. But sure, this information was created out of garbage.

For instance, you can make Stern-Gerlach devices that send spin up in one direction and spin down in the other. But a different manufacturer can make a Stern-Gerlach device that is just as good at splitting the beams in reliable ways but sends those spin states consistently in the opposite directions.

It's like if you ask someone to sort your bills into two piles one for paid and one for unpaid. And what if the person on Mondays sorts one to the left and the other on the right. But the different person that comes on Fridays sorts them to the other sides of the desk. They are just as good at their jobs.

But for the Stern-Gerlach device whether an eigenstate of $\hat\sigma_z$ gets sorted into spin up by a $\hat x$ oriented device or spin down by a $\hat x$ oriented device will depend both on the position (which you don't know, and can't know) and on a silly choice of which brand Stern-Gerlach device you used. The frequency of the two results will be 50-50 for either brand and you will have no idea which result you will get. But the result is determined by things irrelevant to the spin of the particle it depends instead on the unknown, unknowable position and the brand of Stern-Gerlach.

And this is important. When people make up hidden variable theories, they do it in one of two ways. Carefully, and do it so it agrees with observations. Or try to make it have weird pointless properties such as making it behave in some random way that is inconsistent with getting correct results.

And here is the issue. There are correlations between position and spin that are actually different if you measure them partway through the spin measurement process and use these different brand Stern-Gerlach devices (or even afterwards since they literally deflect beams in different directions). So if you assume your theory doesn't care about the brand, then you are just going to be wrong because nature cares.

My question is, under the assumption that many physicists have done a lot more thinking about this than I have and have determined I'm wrong, why am I wrong?

There are hidden variable theories designed to agree with observations. Use them if you want to, if you think it is worth your time (you won't get any different predictions). And there are hidden variables that people make for other reasons. Don't use the wrong ones unless your goal is to be wrong.

If you goal is to criticize certain limited types of hidden variable theories, that's a reason. Still not sure why. And it would be wrong to think or claim you have said anything about the other theories.

Also keep in mind that the repeated Stern-Gerlach device technique shows that Stern-Gerlach devices change the state and so spin can't possibly be just some property that gets passively revealed by sending it through a Stern-Gerlach device otherwise you couldn't explain that measuring first in the $\hat z$ direction, then in the $\hat z$ direction, and finally then the $\hat x $ direction always gives the same result for the first and second yet measuring in the $\hat z$ direction then the $\hat x$ direction then the $\hat z$ direction doesn't always gives the same result for the first and third. Clearly the device changes the object. There is no debate about this by anyone that wants to agree with observations as simple as using one device on the same particle three times and rotating the device between runs sometimes.

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