Is the concept of "wave function collapse" obsolete?

[A. Neumaier]

what do you think comes out of the high-precision experiment with single microscopic quantum systems other than the predicted statistics for measurement outcomes? If there where something else observed, QT were obsolete and we'd have to look for a better theory. Do you have single example (a real experiment of course not some philosophical pseudoproblem)?

The same single system viewed over an extended time, hence having a single trajectory, not a collection of identically prepared particles measured independently, as the statistical intepretation in their traditional form assumes. Single particles visibly jumps (change in very short time) between stationary states.

We discussed this in two other threads here and here, and your answer was unsatisfactory, as always in such cases, replacing a detailed analysis by lots of generalities.

 

[A. Neumaier]

what derivation do you need for the SG experiment? After a silver atom is deflected by the magnetic field to one of two clearly separated directions (that must be ensured of course by the appropriate choice of the magnetic field) you know its spin state being in one of the two pure states

That's precisely the question: How do you know that? You cannot know it from the unitary evolution, and Born's rule doesn't apply yet since the unitary transformation due to the magnetic field is not yet a measurement (as it is in principle reversible). The only way to know it is to invoke the collapse!

 

[vanhees71]

Yes, but what about the experiment where you take particles of definite $S_z$ and then perform an $S_x$ measurement? Afterward they've gone from a state with definite $S_z$ to one of definite $S_x$ in a way that cannot be described by unitary evolution.

Again there's no problem. You just use a magnetic field with its large homogeneous part in $x$ direction. Then a silver-atom prepared to have $S_z=1/2$ is randomly deflected (with probability 50%) in the one or the other direction. Now the position of the particle is entangled with $S_x$ rather than with $S_z$, and you have prepared a particle with $S_x=-1/2$ or $S_x=+1/2$, depending of the direction it went when running through the $\text{SG}_x$ magnet. You cannot decide beforehand, which value of $S_z$ you get, you only know that in about half the cases you get $S_x=+1/2$ and the other half of the cases you get $S_x=-1/2$, but after the particle has run through the magnet, it's clear which of the two possible $S_x$ values it has.

You can understand this intuitively nearly by thinking in classical terms (being a bit sloppy in thinking about spin, which is not a classical observable to begin with): If you have the particle prepared in a $S_z=1/2$ eigenstate, the $S_x$ and $S_y$ components are indetermined. This implies that the magnetic moment vector has a determined $\mu_z$ component but indetermined $\mu_x$ and $\mu_y$ components. If the particle now enters an inhomogeneous magnetic field with a large homogeneous part in $x$-direction, the magnetic moment starts to precess rapidly around the $x$ direction. In moving through the magnetic field, which must also have some inhomogeneous part, the particle is deflected according to the force $\vec{F}=-\vec{\nabla} (\vec{\mu} \cdot \vec{B})$. Now the $\mu_y$ and $\mu_z$ components are rapidly oscillating since $\vec{\mu}$ rapidly precesses around the $x$ direction (with the Larmor frequency $\omega=g e B/(2m) \simeq g e B_0/(2m)$ (for a silver atom $g=g_{\text{electron}} \simeq 2$). Thus it is a good approximation to assume that the deflection is given by $\vec{F} \simeq -\vec{\nabla} (\mu_x B_x)$. The other components $\mu_y B_y$ and $\mu_z B_z$ can be assumed tobe so rapidly oscillating that they average out to 0 over the typical much longer time scales the silver atom moves inside the magnet. Let the magnet be along the $y$ axis and the particle's momentum well peaked around the $y$ axis too. Then we can assume that in the time it's inside the magnet it's not too far refleced from the $y$ axis, and we can approximate the $B$ field as
$$\vec{B}=(B_0 + \beta x)\vec{e}_x -\beta y \vec{e}_y.$$
Note that the last term must be there, because we need to fulfill $\vec{\nabla} \cdot \vec{B}=0$. Nevertheless since we can approximate due to the above argument
$$\vec{F} \simeq -\vec{\nabla} (\mu_x B_x)=-\mu_x \beta \vec{e}_x.$$
The particle gets deflected in $x$ direction in two opposite directions depending on the two possible signs $\mu_x$ can take.

You can make this fully quantum by reading everywhere operators instead of usual c-numbers and use time-dependent perturbation theory with
$$\hat{H}_0=\frac{\vec{p}^2}{2m} + \hat{\mu}_x (B_0+\beta \hat{x}), \quad \hat{H}_1=-\hat{\mu}_y \beta \hat{y}.$$
Then you can calculate the time evolution of a Pauli wave function, initially given as an apprpriate Gaussian wave packet with only a component referring to $\sigma_z=+1/2$. The time evolution with $\hat{H}_0$ gives a two-bump wave packet, where particles in one bump have $\sigma_x=+1/2$ and $\sigma_x=-1/2$. Then you can use first-order time-dependent perturbation theory to show that the correction due to the perturbation $\hat{H}_1$ is indeed small due to the rapid Larmor oscillation of the $\mu_y$. Thus you get (nearly) perfect entanglement between $\sigma_x$ and the particle's position coordinate $x$, i.e., blocking one of the two partial beams leads to a (nearly perfect) beam of particles with $\sigma_x=+1/2$.

 

[A. Neumaier]

Then a silver-atom prepared to have S_z=1/2 is randomly deflected (with probability 50%) in the one or the other direction.

No, since the magnetic field defines a unitary dynamics, it evolves into a superposition of both directions, not into the classical mixture you claim!

 

[PeterDonis]

Afterward they've gone from a state with definite $S_z$ to one of definite $S_x$ in a way that cannot be described by unitary evolution.

You can describe the process of going through the S-G magnet by unitary evolution; it just can't be a unitary evolution of the spin state alone, because the spin degree of freedom alone doesn't have a definite state. It's entangled with the momentum degree of freedom; unitary evolution under the applicable Hamiltonian for the S-G device is what entangles them.

If your experiment also has a detector screen that detects each output beam from the S-G device, then (on a collapse interpretation) the wave function will collapse into a state of definite $S_x$. But the S-G device by itself (the magnetic field) doesn't collapse anything; it just entangles spin and momentum.

 

[Michael Price]

I never said overlaps require the Born Rule, they obviously don't.

post #60 where you said "It does.". I really can't be bothered to debate when someone claims black is white and then denies making the claim.

 

[DarMM]

But the S-G device by itself (the magnetic field) doesn't collapse anything; it just entangles spin and momentum.

That's why I mentioned doing the $S_x$ measurement.

 

[DarMM]

post #60 where you said "It does.". I really can't be bothered to debate when someone claims black is white and then denies making the claim.

"Ceasing to overlap" requires the Born rule, the overlaps themselves do not.

In #61 you said "Overlaps do not require the Born rule". I agree and never said the overlaps require the Born rule.

However I did say in #60 that removing the overlaps requires the Born rule and I stand by that claim as it is true in any treatment of decoherence you'll find in textbooks.

 

[Michael Price]

It's entangled with the momentum degree of freedom; unitary evolution under the applicable Hamiltonian for the S-G device is what entangles them.
...
But the S-G device by itself (the magnetic field) doesn't collapse anything; it just entangles spin and momentum.

This is true is you examine any measurement closely - is just two systems becoming correlated (or entangled, if you prefer), without any collapse.

 

[DarMM]

Overlaps do not require the Born rule. I am out of here.

Since I think this was a typo, i.e. you meant to say "removing overlaps does not require the Born rule" can you explain how you derive overlaps dying off without the Born rule?

 

[PeterDonis]

That's why I mentioned doing the $S_x$ measurement.

But what you are calling "the $S_x$ measurement" is not actually measuring spin, it's measuring position: the position on the detector where the electron hits after going through the S-G magnet. The only way this tells you anything about spin at all is by a chain of inference: to hit that position on the detector, the electron must have had momentum in a particular direction (the direction from the magnet to that point on the detector), and since its momentum was entangled with its $x$ spin, its $x$ spin must have been up (or down).

In other words, the collapse occurs at the detector, not at the S-G magnet, and what exactly is collapsing will depend on your interpretation.

 

[PeterDonis]

This is true is you examine any measurement closely - is just two systems becoming correlated (or entangled, if you prefer), without any collapse.

Not if "measurement" means something that has macroscopic, irreversible effects. In my response to @DarMM just now, I pointed out that the "measurement" in what is often referred to as a "spin measurement" is actually a measurement of position. The entanglement process that justifies (at least according to an appropriate interpretation) calling this a "spin" measurement happens before the measurement--i.e., before a visible spot is made on the detector screen--not during it. The making of the spot on the detector screen is the macroscopic, irreversible process; the entanglement of the spin and momentum degrees of freedom of the electron in the S-G magnetic field is not.

 

[DarMM]

In other words, the collapse occurs at the detector, not at the S-G magnet, and what exactly is collapsing will depend on your interpretation.

Of course and I agree that often measurements occur through an ancilla and thus are often POVMs. How does this relate to the point with @vanhees71 , i.e. the relations between an $S_x$ and $S_z$ preparation constituting collapse. Does the fact that this often occurs via an ancilla affect anything substantial?

 

[PeterDonis]

How does this relate to the point with @vanhees71 , i.e. the relations between an $S_x$ and $S_z$ preparation constituting collapse.

I would say that, in the scenario @vanhees71 described in post #69, the "collapse" comes in when one of the output beams of the S-G magnet is blocked. It doesn't matter what you do with the other beam after that.

(Note that in the case you mention of successive S-G magnets oriented $z$ and then $x$, in order to say the output of the second is "a state of definite $x$ spin", you have to block one of the output beams there as well, or else use a detector screen to make the output beams make bright spots that are macroscopically observable. But that's not what makes the first $S_z$ preparation collapse; blocking one of the $z$ output beams does that.)

 

[DarMM]

I would say that, in the scenario @vanhees71 described in post #69, the "collapse" comes in when one of the output beams of the S-G magnet is blocked. It doesn't matter what you do with the other beam after that.

(Note that in the case you mention of successive S-G magnets oriented $z$ and then $x$, in order to say the output of the second is "a state of definite $x$ spin", you have to block one of the output beams there as well, or else use a detector screen to make the output beams make bright spots that are macroscopically observable. But that's not what makes the first $S_z$ preparation collapse; blocking one of the $z$ output beams does that.)

I agree with all this of course. I was not so much concerned with how exactly the $S_x$ and $S_z$ measurements are done but that the $S_x$ preparation cannot be considered a sub-ensemble (in the probability theory sense) of the $S_z$ preparation it originates from and hence the non-filtering nature of the experiment is what is essentially collapse.

 

[DarMM]

In other words, the collapse occurs at the detector, not at the S-G magnet, and what exactly is collapsing will depend on your interpretation.

I'm a bit confused after thinking about his if you don't mind, what is the exact relation to the discussion here? I get that really spin measurements occur via an ancilla and that "collapse" is invoked when one has some macroscopic fact, but I'm unsure of how it relates to our discussion. What's the importance of the ancilla part?

So a $S_x$ filtered state doesn't seem to be a subensemble of a previous $S_z$ filtering that it originated from. Is this incorrect due to the use of the ancilla in some way?

 

[PeterDonis]

a $S_x$ filtered state doesn't seem to be a subensemble of a previous $S_z$ filtering that it originated from

I'm not sure I see why not. The $S_x$ filtering just picks out a subset of the particles that come through the $S_z$ filter. How is that not a subensemble?

 

[Michael Price]

Not if "measurement" means something that has macroscopic, irreversible effects. In my response to @DarMM just now, I pointed out that the "measurement" in what is often referred to as a "spin measurement" is actually a measurement of position. The entanglement process that justifies (at least according to an appropriate interpretation) calling this a "spin" measurement happens before the measurement--i.e., before a visible spot is made on the detector screen--not during it. The making of the spot on the detector screen is the macroscopic, irreversible process; the entanglement of the spin and momentum degrees of freedom of the electron in the S-G magnetic field is not.

Yes, I agree, it the production of the spot in the screen that makes the measurement irreversible and permanent. Reversible measurements are, in principle, possible, and it is easy to forget this.

 

[DarMM]

I'm not sure I see why not. The $S_x$ filtering just picks out a subset of the particles that come through the $S_z$ filter. How is that not a subensemble?

In the probability theoretic sense of an ensemble for the spin random variables. They are of course a subset of the particles you sent through.

 

[PeterDonis]

In the probability theoretic sense of an ensemble for the spin random variables.

I'm still not sure I understand. Is this just due to the fact that the $S_z$ and $S_x$ observables don't commute?

 

[DarMM]

Yes basically, so if you detect a particle to be in a spin state (however you do it) and then to be in a spin state associated with another direction it has "jumped states" in a way not described by unitary evolution.

In the classical probabilistic case you wouldn't have this because you could just assume the subsequent measurements reduce the support of the probability distribution. It's not a jump to another ensemble, it's just a subensemble.

In a Bayesian view collapse is sort of Bayesian updating + necessary information loss.

So my point to vanhees is that we seem to need collapse for sequences of measurements, because quantum measurements are not just filtrations as in the classical probabilistic case.

You introduced the point of the momentum ancilla into this, I'm just not sure of its purpose. It's something to do with the detector?

 

[PeterDonis]

You introduced the point of the momentum ancilla into this, I'm just not sure of its purpose. It's something to do with the detector?

It's more just to emphasize that describing the process as "detecting a particle to be in a spin state" requires interpretation. In Bohmian mechanics, for example, you aren't doing that; you're just detecting the particle's position, and which output beam of a S-G magnet the particle is in is purely due to its position, not its spin ("spin" doesn't really exist in Bohmian mechanics).

 

[DarMM]

It's more just to emphasize that describing the process as "detecting a particle to be in a spin state" requires interpretation. In Bohmian mechanics, for example, you aren't doing that; you're just detecting the particle's position, and which output beam of a S-G magnet the particle is in is purely due to its position, not its spin ("spin" doesn't really exist in Bohmian mechanics).

Ah sorry I see now. Indeed my language should have been more neutral. What would be the correct phrasing do you think?

Instead of saying "the particle has gone from one state to another" the most neutral statement would be "our probability assignments have gone from one form to another"

 

[PeterDonis]

Instead of saying "the particle has gone from one state to another" the most neutral statement would be "our probability assignments have gone from one form to another"

That phrasing makes "collapse" a non-problem, since there is no requirement that our probability assignments must obey unitary evolution.

 

[DarMM]

What's an interpretation neutral phrasing then?

Our assignments outside of measurements have to obey unitary evolution I thought due to that being an automorphism of the observable algebra. If you read nothing more into the formalism than probability assignments for macroscopic outcomes (and I thought all interps allow you to do this, their common core would be this effective use of the formalism) it seems to me the above is what you would say.

 

[PeterDonis]

What's an interpretation neutral phrasing then?

The only really neutral phrasing is to just describe the macroscopic observation ("a spot was observed at such-and-such point on the detector screen") and leave it at that.

 

[DarMM]

The only really neutral phrasing is to just describe the macroscopic observation ("a spot was observed at such-and-such point on the detector screen") and leave it at that.

That's not a neutral phrasing of the quantum formalism though. It's not just interpretation of QM neutral it's theory neutral.

 

[PeterDonis]

That's not a neutral phrasing of the quantum formalism though.

I would say a neutral phrasing of the quantum formalism is to just write down the equations and leave it at that.

 

[DarMM]

I would say a neutral phrasing of the quantum formalism is to just write down the equations and leave it at that.

But you have to apply them to an experiment. You can't just write them down, that seems to be the opposite extreme. That's why I think a fairly neutral statement is to say that the probabilities for future macroscopic effects are updated after a seeing a specific macroscopic effect in a way described by state collapse. All the interpretations would agree on that pragmatic use, they'd disagree on what else might be going on and what the meanings of terms are beyond their pure pragmatics.

 

[PeterDonis]

the probabilities for future macroscopic effects are updated after a seeing a specific macroscopic effect in a way described by state collapse

Yes, but as I said, this makes "collapse" a non-problem because updating probabilities does not require anything to have "actually happened" to the system. (Perhaps instead of "non-problem" it could be termed an "interpretation-dependent problem", and one interpretation is simply that nothing "actually happens" during collapse, it's just that we update the probabilities we'll use for future predictions.)

 

[DarMM]

Collapse and whether it needs to be explained is an interpretation dependent issue.

The interpretation neutral thing is how it is applied, i.e. see macroscopic effect then update the state to give probabilities to future macroscopic effects. That's what is actually done. I don't think this necessarily makes it a non-issue, it's just how it is used.

 

[vanhees71]

I agree with all this of course. I was not so much concerned with how exactly the $S_x$ and $S_z$ measurements are done but that the $S_x$ preparation cannot be considered a sub-ensemble (in the probability theory sense) of the $S_z$ preparation it originates from and hence the non-filtering nature of the experiment is what is essentially collapse.

The problem with all these philosophical (in my opinion irrelevant) discussions on interpretation is precisely the refusal to discuss real-world experiments. The SG experiment is the most simple example, where everything can be (nearly) analytically calculated within standard QM, and I don't see any mystery left.

Why you want to call blocking a partial beam of silver atoms by putting something in its way "collapse" I don't know. For sure looking at the experimental setup you immediately see that this "collapse" works with the usual local interactions of the silver atoms with the matter making up the "blocker". Nothing acts instantaneously at a distance! I call simply blocking a partial beam of silver atoms blocking this partial beam, and the partial beam defines indeed a subensemble of silver atoms with definite $S_x$ which have been prepared to have a definite $S_z=+1/2$. Of course, QT tells you that then $S_z$ doesn't take a definite value anymore, and you even understand it from the dynamics why this must be so!

 

[DarMM]

The problem with all these philosophical (in my opinion irrelevant) discussions on interpretation is precisely the refusal to discuss real-world experiments. The SG experiment is the most simple example, where everything can be (nearly) analytically calculated within standard QM, and I don't see any mystery left

This isn't philosophical or related to interpretations. I'm not even saying there is a mystery. You're the one saying there is no collapse, part of the textbook formalism of the theory. Yes I can easily calculate the whole set up in standard QM and as per textbook QM (Weinberg, Peres, Shankar) it uses the collapse postulate.

Why you want to call blocking a partial beam of silver atoms by putting something in its way "collapse" I don't know. For sure looking at the experimental setup you immediately see that this "collapse" works with the usual local interactions of the silver atoms with the matter making up the "blocker". Nothing acts instantaneously at a distance!

Nobody is saying there is instantaneous action at a distance. Where is this even coming from?

 

[Michael Price]

Nobody is saying there is instantaneous action at a distance. Where is this even coming from?

The collapse is a non-local, instaneous, action-at-a-distance effect. That's why a lot of us don't believe in it. That's also in the textbooks.

 

[DarMM]

defines indeed a subensemble of silver atoms with definite $S_x$ which have been prepared to have a definite $S_z=+1/2$.

It's not a subensemble in the the sense of probability theory that's literally a fact coming from the differences between quantum and classical probability.