Is the collapse of the particle wave function?

[Nick V]

Is the collapse of the wave function of the electron in the double slit experiment based purely on the act of observation? Or could it be that the way the instrument used to measure the electron caused it to collapse by how it physically interacted with the electron? Keep in mind the delayed choice experiment.

 

[Nugatory]

Is the collapse of the wave function of the electron in the double slit experiment based purely on the act of observation? Or could it be that the way the instrument used to measure the electron caused it to collapse by how it physically interacted with the electron? Keep in mind the delayed choice experiment.

QM says that the probability of the electron landing at a particular point on the screen is affected by any interactions (for example, with the slits, or with measuring devices behind the slits) it has on the way to the screen. There's nothing special about the subset of interactions that (for historical reasons) we call "observations"; if the interaction is such that we could in principle know which path the electron took, the effect of the interaction will be to eliminate the interference pattern.

That's pretty much a "no" answer to your first question and a "yes" to the second... but you should note that the word "collapse" does not appear in a precise statement of exactly what QM does and doesn't tell us.

 

[atyy]

but you should note that the word "collapse" does not appear in a precise statement of exactly what QM does and doesn't tell us.

Are you allowing a Bohmian or Many-Worlds interpretation in this statement? The standard Copenhagen-style interpretation does include a mathematically precise statement that corresponds to what is colloquially called collapse, or more formally called state reduction or the conditional state or the a posteriori state.

 

[Nugatory]

Are you allowing a Bohmian or Many-Worlds interpretation in this statement? The standard Copenhagen-style interpretation does include a mathematically precise statement that corresponds to what is colloquially called collapse, or more formally called state reduction or the conditional state or the a posteriori state.

That's a fair criticism, and I wouldn't even have made the comment if OP had said "state reduction" or similar.

 

[atyy]

That's a fair criticism, and I wouldn't even have made the comment if OP had said "state reduction" or similar.

Well it's good to agree (I think) on the physics. I have come to realize that "collapse" is a loaded term, where Copenhagenists like me routinely mean "state reduction", while others think of something far more colourful.

 

[afcsimoes]

From a text of QM at Wiki: "The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states (...)"
When we consider ALL the possible states we have a deterministic function, even if we don't know in advance what is the state of the system?
If we could know the state of the system and its evolution's equations, then we can write down deterministic equations with only those states? Without using probabilistic ones?

 

[atyy]

From a text of QM at Wiki: "The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states (...)"
When we consider ALL the possible states we have a deterministic function, even if we don't know in advance what is the state of the system?
If we could know the state of the system and its evolution's equations, then we can write down deterministic equations with only those states? Without using probabilistic ones?

In the orthodox Copenhagen interpretation, time evolution is deterministic between measurements, and random at the instant of a measurement. Although this approach is consistent with all data to date, there problem with this approach is that the theory does not say when a measurement is made, which is something that the user of the theory must put in from outside.

This problem is solved in the Bohmian interpretation. In Bohmian mechanics, time evolution is determinstic, while the randomness lies in the initial conditions. However, there is not a unique Bohmian dynamics, and there is also no consensus yet about whether a Bohmian view is possible for all aspects of the standard model of particle physics.

A third approach called Many-Worlds is possibly completely deterministic. In this approach, all measurement outcomes occur. However, it is not yet known if this approach is correct in all details.

 

[Nick V]

QM says that the probability of the electron landing at a particular point on the screen is affected by any interactions (for example, with the slits, or with measuring devices behind the slits) it has on the way to the screen. There's nothing special about the subset of interactions that (for historical reasons) we call "observations"; if the interaction is such that we could in principle know which path the electron took, the effect of the interaction will be to eliminate the interference pattern.

That's pretty much a "no" answer to your first question and a "yes" to the second... but you should note that the word "collapse" does not appear in a precise statement of exactly what QM does and doesn't tell us.

I think my question before is a little unclear but let me edit it...
When the electrons go through the double slit they create a wave like pattern. However, when there is a tracker to measure the electrons, they create a double band particle pattern. Could it be that the instrument used to measure the electrons is physically interacting with them causing them to collapse the wave like pattern, or is it just the act of observing that causes them to collapse the wave like pattern into a particle like pattern?

 

[Nugatory]

I think my question before is a little unclear but let me edit it...
When the electrons go through the double slit they create a wave like pattern. However, when there is a tracker to measure the electrons, they create a double band particle pattern. Could it be that the instrument used to measure the electrons is physically interacting with them causing them to collapse the wave like pattern, or is it just the act of observing that causes them to collapse the wave like pattern into a particle like pattern?

It is the interaction. But do remember that it is impossible to observe without interacting: all observations are interactions, but not all interactions are observations.

 

[bhobba]

When we consider ALL the possible states we have a deterministic function

Can you expand on what you mean because I can't follow it.

Its simple really. The state encodes within it the probabilities of the outcomes of observations. It evolves deterministically, but what it represents doesn't change.

Thanks
Bill

 

[Nick V]

It is the interaction. But do remember that it is impossible to observe without interacting: all observations are interactions, but not all interactions are observations.

Ok, but does the delay choice experiment play any part against the belief that the wave like nature of the electron in double slit collapses do to the instrument physically interacting with the electron, and that it is really just the act of observing that causes the wave like behavior to collapse?
Sorry if this is a stupid question.

 

[vanhees71]

There's only one explanation for the delayed-choice experiments: quantum theory! It's not understandable with outdated concepts like wave-particle dualism. Delayed choice is just what it says: You prepare a lot of quantum systems in an appropriate state and make specific measurements on each of those. Then you can choose afterwards, just using the once and for all fixed measurement protocol, different subensembles showing either an interference pattern or imply which-way information as in the now pretty famous "quantum eraser" experiment by Walborn, et al.

See the explanation here:

https://www.physicsforums.com/threa...xperiment-affects-result.785711/#post-4935759

 

[atyy]

https://www.physicsforums.com/threa...xperiment-affects-result.785711/#post-4935759

I guess we have discussed this many times, and I do not accept that the minimal interpretation does not have collapse or state reduction.

State reduction is undoubtedly part of the minimal interpretation, and is also known as the conditional state or the a posterior state. This is found in almost all standard textbooks, from Dirac, Landau & Lifshitz, Nieslen and Chuang, Weinberg, Sakurai & Napolitano, Shankar etc, and there really should be no controversy here.

Where there could be some controversy is whether the state reduction can be derived from unitary evolution and the Born rule alone. As far as I understand, it cannot, and requires an additional assumption. The main reason is that the Born rule only gives probabilities for a single time, whereas the conditional probability requires a rule for the joint probability of observations at different times. For spacelike observations, the Lorentz invariance can be used in place of the state reduction postulate, but the Lorentz invariance and spacelike separation is still an additional postulate.

 

[vanhees71]

Again I disagree. State collapse is unnecessary and leads to conceptually to very problematic issues with relativistic causality. I'd ban the word "collapse" from any writings and talk about quantum theory (I hope I've not made myself guilty of this sin in my manuscripts yet, because I've also been exposed to this mantra of the otherwise brillant textbooks mentioned in the previous posting; I've to check :-)). All this we have discussed at length over the Christmas break :-).

 

[maline]

Vanhees, would you mind clarifying for me? The bottom line is we only observe part of the WF. How can this be explained without either MWI or a collapse?

 

[vanhees71]

We observe the wave function by doing experiments on ensembles of equally prepared systems, using the usual statistical technques to test a hypothesis, which in this case is provided by quantum theory. So what do you mean by the statement that we only observe part of the wave function? Whatever it is, why do you think, you need MWI or a collapse (which I consider as either esoterics that cannot be empirically tested or intrinsically inconsistent, respectively).

 

[maline]

Thanks for replying. I hope you don't mind if I rephrase my question. After I observe a single particle in a particular position, can the situation still be described, in theory, by a linear propagation of the original wf? If yes, the MWI follows. If not, why shouldn't this change be called a collapse?

 

[bhobba]

Thanks for replying. I hope you don't mind if I rephrase my question. After I observe a single particle in a particular position, can the situation still be described, in theory, by a linear propagation of the original wf? If yes, the MWI follows. If not, why shouldn't this change be called a collapse?

The Ensemble interpretation is fully compatible with a number of different interpretations that say more about what going on so to speak eg Primary State Diffusion, Bohmian Mechanics, Nelson Stochastics, Many Worlds, and GRW collapse theory. The reason is its a minimalist interpretation making as few assumptions as possible - basically the assumption it makes is a frequentest interpretation of probability rather than a Baysian one which would lead to something like Copenhagen - it's what people mean when they say shut up and calculate - they mean an interpretation with very minimalist assumptions. Some have collapse and some do not. For definiteness I will pick Bohmian Mechanics. It does not have collapse - particles have well defined momentum, position etc at all times. What happens is when an observation is made decoherence occurs and you end up with the standard mixed state. However since it has definite properties the mixed state is a proper one and no collapse occurred. The ensemble interpretation is ambivalent to how the mixed state after decoherence becomes a proper mixed state - it simply assumes it is - somehow. That somehow may or may not involve collapse.

To understand the issue of proper and improper mixed states see the following:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

Thanks
Bill

 

[atyy]

The Ensemble interpretation is fully compatible with a number of different interpretations that say more about what going on so to speak eg Primary State Diffusion, Bohmian Mechanics, Nelson Stochastics, Many Worlds, and GRW collapse theory. The reason is its a minimalist interpretation making as few assumptions as possible - basically the assumption it makes is a frequentest interpretation of probability rather than a Baysian one which would lead to something like Copenhagen - it's what people mean when they say shut up and calculate - they mean an interpretation with very minimalist assumptions. Some have collapse and some do not. For definiteness I will pick Bohmian Mechanics. It does not have collapse - particles have well defined momentum, position etc at all times. What happens is when an observation is made decoherence occurs and you end up with the standard mixed state. However since it has definite properties the mixed state is a proper one and no collapse occurred. The ensemble interpretation is ambivalent to how the mixed state after decoherence becomes a proper mixed state - it simply assumes it is - somehow. That somehow may or may not involve collapse.

To understand the issue of proper and improper mixed states see the following:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

As far as I can tell, and as I have said many times, I don't disagree at all and I believe this is a correct Ensemble Interpretation. Although there are some difference in terminology, I cannot tell if it differs in any way in content from my default interpretation, which I usually call "Copenhagen" or "Copenhagen-style". When I say "collapse" I simply mean exactly what bhobba says by "The ensemble interpretation is ambivalent to how the mixed state after decoherence becomes a proper mixed state - it simply assumes it is - somehow."

So what is clear in bhobba's post is that there has to be something beyond unitary evolution and the Born rule in order to calculate the joint probability for sequential measurements. This clarity is lacking in Ballentine making his interpretation dubious at best, and as far as I can tell it is rejected by vanhees71.

 

[bhobba]

This clarity is lacking in Ballentine making his interpretation dubious at best, and as far as I can tell it is rejected by vanhees71.

I strongly disagree - but I will let Vanhees speak to his position.

Thanks
Bill

 

[atyy]

To understand the issue of proper and improper mixed states see the following:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

As I said above, I agree with bhobba on physics, and at worst we only disagree on the interpretation of Ballentine. The linked article by Hansen is excellent, and as far as I can tell, I agree with it. Let me present Hansen's argument in a more mathematical way to make it clear that we have equations to discuss, and that the difference between proper and improper mixtures is not some vague interpretive thing.

First let's just assume unitary evolution and the Born rule, and no (the following are equivalent) collapse or state reduction or change from improper to proper mixtures. We have a quantum system and quantum ancilla. They start to interact at time=0, and finish interacting at time=t via unitary evolution. At time t, an observable A is measured on the ancilla and an observable B is measured on the system. Decoherence does not alter the picture for this argument, as it only restricts which observables A we can measure in a practical way. Since A and B are commuting observables that are measured at the same time, the Born rule gives P(A,B), and we can define the reduced density operator of the system by tracing over the ancilla, and we can define the conditional state or a posteriori state of the system using Bayes's rule. As long as we only calculate things derivable from P(A,B), the conditional state can be derived from unitary evolution and the Born rule alone, since it just amounts to writing Bayes's rule applied to P(A,B) in a different way. (See eg. http://arxiv.org/abs/0810.3536, Eq 6.7, 6.8 and Section 6.2.3.)

But if we use the conditional state or a posteriori state to calculate P(B,C), where C is a measurement on the system at a later time t+dt, then we are using the conditional state in a way that goes beyond applying Bayes's rule to P(A,B), and amounts to a postulate beyond unitary evolution and the Born rule, and this is collapse or state reduction or a change from an improper to proper mixture.

There is also very interesting discussion of the issue by Ozawa in http://arxiv.org/abs/quant-ph/9706027 and http://arxiv.org/abs/quant-ph/9711006. But I reference those papers only to note that as late as 1997, he writes that the consensus is that "partial trace does not derive state reduction", which is the same issue that bhobba brings up about improper and proper mixtures. Other texts that explicitly consider the issue and agree with bhobba are Haag https://www.amazon.com/dp/3540610499/?tag=pfamazon01-20 (1996, p301) and Haroche and Raimond https://www.amazon.com/dp/0198509146/?tag=pfamazon01-20 (2006, p82).

 

[vanhees71]

There is no general rule concerning what happens after the measurement of an observable. It depends on the experimental setup. If there is a measurement and you filter somehow the systems within the ensemble according to this measurement's result it's a form of state preparation, and you can calculate the probability of finding a value of an observable measured in a consecutive measurement. Then the first measurement is called a von Neumann filter measurement. Of course, usually you don't describe the filter process in full glory by using quantum theory. This is also an overkill, because measurement apparati are macroscopic systems and can be well described by classical or semiclassical means.

A good example is quantum optics. The quantum opticians on the one hand deal with single photons (or other many-photon Fock states like the entangled biphotons from parametric down conversion) and on the other with macroscopic optical devices as known from classical optics. The latter are usually described classically, e.g., birefringent crystals used for parametric down conversion via an index of refraction (linear response) and some non-linear "susceptibilities". The same holds for phase shifters, lenses, polarizers, beam splitters,... I don't think that a fully microscopic description exists and is possible in practice. Of course, it's fully sufficient to describe the measured effects, which are among the most fascinating and important findings about fundamental quantum physics made in recent decades!

 

[maline]

Okay, vanhees, I think I get it- please correct me if I misunderstood. You are saying that what we usually call a reduction can be thought of as simply a selection (and normalization) of those elements of the original wf that are relevant for predicting future observation probabilities. For the present observation, the Born rule can be defined without any reduction, as atyy explained. As far as what "actually happens" to the rest of the wf, you are taking the position that we have no reason to believe the wf "exists"; all we know is how it predicts probabilities, and we have no hope of ever interpreting this in terms of some underlying process because models that agree with QM, by definition, cannot make testable predictions of their own.

 

[atyy]

Okay, vanhees, I think I get it- please correct me if I misunderstood. You are saying that what we usually call a reduction can be thought of as simply a selection (and normalization) of those elements of the original wf that are relevant for predicting future observation probabilities. For the present observation, the Born rule can be defined without any reduction, as atyy explained. As far as what "actually happens" to the rest of the wf, you are taking the position that we have no reason to believe the wf "exists"; all we know is how it predicts probabilities, and we have no hope of ever interpreting this in terms of some underlying process because models that agree with QM, by definition, cannot make testable predictions of their own.

Just to be clear, I don't agree that in what is usually called a minimal interpretation that one can do without state reduction. Bhobba's "minimal interpretation" (I agree with bhobba, I put "minimal interpretation" in quotes just to clarify that his terminology is different from mine) explicitly mentions approaches like hidden variables or many-worlds as a possibility so that one can do without postulating state reduction, so that state reduction can be derived. However, there is no consensus that any of these can reproduce the standard quantum mechanical predictions for all known quantum mechanical predictions with state reduction or a change from an improper to proper mixture simply postulated as a phenomenological rule. What vanhees71 seems to be saying is we don't have to postulate hidden variables, many worlds, or state reduction or a change from an improper to proper mixture - that is strange. Certainly from unitary evolution and the Born rule alone, we can at least partly derive state reduction as a consistency requirement for measurements at spacelike separation - so state reduction is needed and standard in what is usually called a minimal interpretation.

 

[vanhees71]

Why should I need something in addition to standard quantum theory? It's successful as it is! It predicts probabilities to an astonishing precision. There's no hint so far that there's something else than these probabilities. The wave function is the description of these probabilities. QT in this sense is just a special kind of description of a random process, no more no less.

Further I don't understand, what you mean by "part of a wave function" (I prefer to call what we discuss about the quantum state, because there are interesting things, where there is no wave-function description as for relativistic QT and photons particularly)? If I have an electron, described by a wave function (which makes sense in the non-relativistic approximation), I either detect it or not. In the latter case, I cannot say anything (not even, whether there was an electron at all). If I detect it and measure some of its observables, I just determine a fact that was either determined before through the state preparation or it was not, and then it's the outcome of a random experiment. The probabilities for the outcomes are predicted by the wave function describing the state, my electron was prepared in at the time of the measurement, and I can check this prediction in the usual way by repeating the preparation and measurement procedures independently for a many times and evaluate the findings with the usual mathematical tools of statistics. That's all. Nothing collapses. Or what collapses, if they draw the numbers of the weekly lottery?

 

[Feeble Wonk]

I've been reading through the thread with great interest. Please correct me if I'm wrong, but it seems to me that primary dispute here is regarding a fundamental conceptual difference as to what is being discussed. It's almost a semantic argument. Atyy seems to be taking the position that there is a "thing-in-itself" out there that is defined by the wave function (IS the wave function?), and therefore, in the absence of a Bohmian or Multiverse interpretation, the state reduction (collapse) concept is a necessary because it describes the "thing-in-itself" pre and post observation/measurement. In contrast, vanhees71 seems to be taking the position that QT says nothing about the "thing-in-itself" whatsoever, but is only a theory to describe the probability of an outcome to a specific type of measurement or observation. Is this even close to being accurate?

 

[vanhees71]

That seems to be very accurate!

 

[atyy]

I've been reading through the thread with great interest. Please correct me if I'm wrong, but it seems to me that primary dispute here is regarding a fundamental conceptual difference as to what is being discussed. It's almost a semantic argument. Atyy seems to be taking the position that there is a "thing-in-itself" out there that is defined by the wave function (IS the wave function?), and therefore, in the absence of a Bohmian or Multiverse interpretation, the state reduction (collapse) concept is a necessary because it describes the "thing-in-itself" pre and post observation/measurement. In contrast, vanhees71 seems to be taking the position that QT says nothing about the "thing-in-itself" whatsoever, but is only a theory to describe the probability of an outcome to a specific type of measurement or observation. Is this even close to being accurate?

That seems to be very accurate!

That is not accurate at all. The question is a mathematical statement. Collapse is the statement that after a measurement, we can use a classical measurement outcome to label a subensemble. In Ballentine, this is equation 9.28 and the statement preceding that this is the state of the sub-ensemble that is labelled by the previous measurement outcome. Ballentine does not call that state reduction, but that is only a matter of terminology since other people do call it state reduction or collapse. So yes, Ballentine does have collapse. So the question is can that be derived from unitary evolution and the Born rule alone?

 

[Feeble Wonk]

That seems to be very accurate!

Your position would seem to be the ultimate in the "shut up and calculate" school of thought regarding quantum physics. Does that mean that you believe it is impossible to interpret the findings of QT with regard to what it implies about the fundamental nature of physical existence? Or is it simply your preference on how to approach the subject.

 

[atyy]

Your position would seem to be the ultimate in the "shut up and calculate" school of thought regarding quantum physics.

Let's be clear the difference between vanhees71 and me is not "shut up and calculate". In fact in a sense my view is more shut up and calculate. If one shuts up and calculates, there is absolutely nothing wrong with collapse or state reduction.

I am simply defending traditional shut up and calculate :)

 

[vanhees71]

That is not accurate at all. The question is a mathematical statement. Collapse is the statement that after a measurement, we can use a classical measurement outcome to label a subensemble. In Ballentine, this is equation 9.28 and the statement preceding that this is the state of the sub-ensemble that is labelled by the previous measurement outcome. Ballentine does not call that state reduction, but that is only a matter of terminology since other people do call it state reduction or collapse. So yes, Ballentine does have collapse. So the question is can that be derived from unitary evolution and the Born rule alone?

The question is a physical statement. What Ballentine describes with his Eq. (9.28) is the Statistical operator after a von Neumann "filter measurement", which I'd consider a state-preparation procedure. For those who don't have available this excellent textbook, here's the statement

Suppose an ensemble of a quantum mechanical system is prepared such that it is discribed by the statistical operator $\hat{\rho}$. Then a measurement of an observable $R$. He takes an observable with a continuous spectrum. Let $|r,\beta \rangle$ denote the generalized eigenvectors, normalized to a $\delta$ distribution and $\beta$ labeling a possible degeneracy, which I take as a discrete set (you can also have a continuous set, but that doesn't change much). Then he considers a state-preparation procedure which filters out all systems out of the ensemble for which $R$ takes a value in an interval $\Delta_a$. The remaining sub-ensemble is then described by a new statistical operator, which is constructed as follows: We first define the projection operator
$$\hat{M}_R(\Delta_a)=\int_{\Delta_a} \mathrm{d} r \sum_{\beta} |r,\beta \rangle \langle r,\beta|$$
and then define
$$\hat{\rho}'=\frac{1}{Z} \hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a), \quad Z=\mathrm{Tr} \left [\hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a)
\right]$$
as the statistical operator of the corresponding sub-ensemble.

This is just a definition what you call preparation through an idealized von Neumann filter measurement. An example, where you can describe this entirely with quantum theory is the Stern-Gerlach experiment. In this example there's then no collapse but just the ignorance of all particles with the spin component not in the desired state. The "collapse" in the Copenhagen doctrine is nothing but the choice of the subensemble by the experimenter through filtering out all undesired complementary subensembles. There's no spooky non-quantum action-at-a-distance magic called "collapse"!

 

[atyy]

The question is a physical statement. What Ballentine describes with his Eq. (9.28) is the Statistical operator after a von Neumann "filter measurement", which I'd consider a state-preparation procedure. For those who don't have available this excellent textbook, here's the statement

Suppose an ensemble of a quantum mechanical system is prepared such that it is discribed by the statistical operator $\hat{\rho}$. Then a measurement of an observable $R$. He takes an observable with a continuous spectrum. Let $|r,\beta \rangle$ denote the generalized eigenvectors, normalized to a $\delta$ distribution and $\beta$ labeling a possible degeneracy, which I take as a discrete set (you can also have a continuous set, but that doesn't change much). Then he considers a state-preparation procedure which filters out all systems out of the ensemble for which $R$ takes a value in an interval $\Delta_a$. The remaining sub-ensemble is then described by a new statistical operator, which is constructed as follows: We first define the projection operator
$$\hat{M}_R(\Delta_a)=\int_{\Delta_a} \mathrm{d} r \sum_{\beta} |r,\beta \rangle \langle r,\beta|$$
and then define
$$\hat{\rho}'=\frac{1}{Z} \hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a), \quad Z=\mathrm{Tr} \left [\hat{M}_R(\Delta_a) \hat{\rho} \hat{M}_R(\Delta_a)
\right]$$
as the statistical operator of the corresponding sub-ensemble.

This is just a definition what you call preparation through an idealized von Neumann filter measurement. An example, where you can describe this entirely with quantum theory is the Stern-Gerlach experiment. In this example there's then no collapse but just the ignorance of all particles with the spin component not in the desired state. The "collapse" in the Copenhagen doctrine is nothing but the choice of the subensemble by the experimenter through filtering out all undesired complementary subensembles. There's no spooky non-quantum action-at-a-distance magic called "collapse"!

The reason that this is not preparation only is that it is preparation based on the outcome of the preceding measurement. The new quantum state is prepared based on the previous outcome that $R$ takes a value in an interval $\Delta_a$, so it is a conditional preparation, and is a conditional state. By saying that the new state is prepared by conditioning on the preceding outcome, one has in effect generalized the Born rule for the preceding outcome to the new state.

 

[vanhees71]

Sure, it's a conditional preparation. That's what all preparations are. I think it's implied by the Born rule, but as I said, in the case of the Stern-Gerlach experiment, you can derive it from the dynamics. You just consider only a partial beam after one Stern Gerlach apparatus splitting up the incoming beam into partial beams with determined magnetic quantum number of its spin.

 

[kith]

atyy, my interest in this debate has receded a bit but in our recent discussion of the double slit you linked to a nice computer simulation which seems relevant to me.

A gaussian wave packet hits the double slit. There, a part of the wave packet is transmitted and propagated to the screen and a part is reflected. The reflected part never hits the screen. So for a position measurement at the screen, we get the same probabilities for two different states: (a) the superposition of the transmitted and the reflected part of the wave packet and (b) the transmitted part of the wave packet alone.

Would you say that QM needs a postulate for the case that we chose description (b) for calculations because we don't want to calculate things which aren't relevant for the measurement outcome?

 

[vanhees71]

No, you just only consider that part of particles hitting the screen. There's no addition to standard QT needed whatsoever to understand this experiment.

 

[atyy]

Sure, it's a conditional preparation. That's what all preparations are. I think it's implied by the Born rule, but as I said, in the case of the Stern-Gerlach experiment, you can derive it from the dynamics. You just consider only a partial beam after one Stern Gerlach apparatus splitting up the incoming beam into partial beams with determined magnetic quantum number of its spin.

That's the question: is it implied by the Born rule? In quantum theory in the minimal interpretation there are no sub-ensembles for a pure state before measurement, because sub-ensembles would mean definite trajectories. So at the moment of measurement, two sets of sub-ensembles appear (1) the measurement outcome (2) the quantum state, from which you can select and filter. The appearance of the sub-ensembles of quantum states and the link between the sub-ensembles of measurement outcomes and the sub-ensembles of quantum states is beyond the Born rule (without state reduction), because the Born rule only tells you about the sub-ensembles of measurement outcomes.

So the use of the conditional state to get joint probabilities between sequential measurements is beyond the Born rule.

Ballentine says on p248 "If the operators R and S do not commute, then (9.26) does not apply. We can still use (9.22) as a definition of the joint probability Prob(A&B|C), since the factors on the right hand side are both well defined, in principle." However it is not true that both factors are well-defined unless Eq 9.28 is taken as postulate for the conditional state.

 

[vanhees71]

That's not what I mean. A subensemble is just a partial ensemble the experimenter chooses with his setup. For me (9.28) is the definition in the quantum-theoretical formalism for this choice. Do you think, one would have to derive this somehow? If so from which assumptions? Also what should it have to do with the Born rule?

 

[atyy]

atyy, my interest in this debate has receded a bit but in our recent discussion of the double slit you linked to a nice computer simulation which seems relevant to me.

A gaussian wave packet hits the double slit. There, a part of the wave packet is transmitted and propagated to the screen and a part is reflected. The reflected part never hits the screen. So for a position measurement at the screen, we get the same probabilities for two different states: (a) the superposition of the transmitted and the reflected part of the wave packet and (b) the transmitted part of the wave packet alone.

Would you say that QM needs a postulate for the case that we chose description (b) for calculations because we don't want to calculate things which aren't relevant for the measurement outcome?

In the simplest version of the double slit, there is only one measurement, which is registered when the particle hits the screen. Collapse is only needed for calculating the joint probabilities of two or more sequential measurements. So there is no collapse in the simplest version of the double slit.

 

[atyy]

That's not what I mean. A subensemble is just a partial ensemble the experimenter chooses with his setup. For me (9.28) is the definition in the quantum-theoretical formalism for this choice. Do you think, one would have to derive this somehow? If so from which assumptions? Also what should it have to do with the Born rule?

For just choosing a sub-ensemble, and calculating the probabilities of future measurements on the sub-ensemble, there is no need for collapse. In this case, it is indeed fine to consider the procedure one does as simply state preparation.

However, for calculating the probability of one measurement $a$, selecting a sub-ensemble based on the first measurement outcome then doing measurement $b$, collapse is needed to calculate $P(A,B)=P(B|A)P(A)$. What collapse does is define $P(B|A)$ via $P(\rho'|A)$.

 

[vanhees71]

I still don't understand this argument. In the example of the Stern-Gerlach experiment with measuring the magnetic spin quantum number of a neutral particle, which is the paradigmatic filter measurement, you just have the following:

(a) a unpolarized beam of neutral silver atoms, originating from a little oven by letting it out of a little hole in this oven
(b) the beam is directed through an inhomogeneous magnetic field with a large homogeneous component in $z$ direction, which entangles the magnetic spin quantum number $\sigma_z \in \{\pm 1/2 \}$ with the position of the silver atom. The magnetic field is taylored such that you get two well-separated partial beams with defined $\sigma_z$. This is described by unitary time evolution.
(c) The partial beam with, say, $\sigma_z=1/2$ is directed through a 2nd magnetic field with a large homogeneous component in $x$ direction and so $\sigma_z$ is measured via the resulting position-$\sigma_z$ entanglement. Again this is described by unitary time evolution.

The only non-unitary thing in (9.28) is the renormalization in the denominator, but that's simply, because you want to consider only the partial beam with $\sigma_z=+1/2$ prepared with step (b). That's just a calculational convenience and has nothing to do with a physical collapse. You could as well choose to normalize the total probability to 42, which is never done in practice, because we define probabilities to add up to 1.

At least for such an idealized von Neumann Filter measurement (which I'd rather call preparation for clarity), you don't need a collapse, and only for such a type of filter measurement you need it in that flavor of the Copenhagen doctrine.

In most other cases, you don't care about what happens after the measurement. You don't think much about the switched off proton or heavy-ion beam at the LHC ending quite abruptly in the beam dump. In principle you "measure" position of the particles in the beam, because you know it ends somewhere at the beam dump, but you cannot say that afterwards the particles have a somehow "collapsed" new state localized in the beam dump, because most of them are just gone by some annihilation process or something else. I consider the collapse hypothesis as quite empty and more confusing than helpful, because it has very serious issues with relativistic causality, as was pointed out by EPR in their famous paper (although Einstein himself didn't like it too much, and he has written a much better version by himself alone (in German):

A. Einstein, Quanten-Mechanik und Wirklichkeit, Dialectica 2, 320 (1948)
http://dx.doi.org/10.1111/j.1746-8361.1948.tb00704.x

 

[kith]

In the simplest version of the double slit, there is only one measurement, which is registered when the particle hits the screen.

There's also the preparation by the double slit. You often associate collapse with Ballentine's equation 9.28 which can be applied here: after the wave packet hit the slit, we project on the transmitted part because we don't care about the reflected part. Since you think that equation 9.28 needs to be derived or postulated, I'd like to know what you think about this concrete example.

 

[atyy]

I still don't understand this argument. In the example of the Stern-Gerlach experiment with measuring the magnetic spin quantum number of a neutral particle, which is the paradigmatic filter measurement, you just have the following:

(a) a unpolarized beam of neutral silver atoms, originating from a little oven by letting it out of a little hole in this oven
(b) the beam is directed through an inhomogeneous magnetic field with a large homogeneous component in $z$ direction, which entangles the magnetic spin quantum number $\sigma_z \in \{\pm 1/2 \}$ with the position of the silver atom. The magnetic field is taylored such that you get two well-separated partial beams with defined $\sigma_z$. This is described by unitary time evolution.
(c) The partial beam with, say, $\sigma_z=1/2$ is directed through a 2nd magnetic field with a large homogeneous component in $x$ direction and so $\sigma_z$ is measured via the resulting position-$\sigma_z$ entanglement. Again this is described by unitary time evolution.

You are missing the first measurement. You must make a direct or indirect measurement of the spin after the first apparatus, before it enters the second apparatus. If that is not part of your experiment, there is only one measurement, and collapse is not needed.

 

[atyy]

There's also the preparation by the double slit. You often associate collapse with Ballentine's equation 9.28 which can be applied here: after the wave packet hit the slit, we project on the transmitted part because we don't care about the reflected part. Since you think that equation 9.28 needs to be derived or postulated, I'd like to know what you think about this concrete example.

You must make a first measurement of the position (or some other observable) at the slits, otherwise there is only one measurement, not the minimum of two measurements needed to discuss sequential measurement. The important point about Eq 9.28 is not Eq 9.28 alone, but that it is conditional on the previous measurement outcome.

 

[kith]

You must make a first measurement of the position (or some other observable) at the slits, otherwise there is only one measurement, not the minimum of two measurements needed to discuss sequential measurement. The important point about Eq 9.28 is not Eq 9.28 alone, but that it is conditional on the previous measurement outcome.

Do you consider a polarization filter or a single slit to be an appropriate measurement device?

 

[atyy]

Do you consider a polarization filter or a single slit to be an appropriate measurement device?

Do you get a definite measurement outcome at a particular time after using these devices?

 

[kith]

Do you get a definite measurement outcome at a particular time after using these devices?

What do you mean by outcome? I do get a state of definite polarization resp. position immediately after the filter resp. slit. Regarding the time of the measurement, the situation is analogous to the measurement at the screen.

 

[vanhees71]

You must make a first measurement of the position (or some other observable) at the slits, otherwise there is only one measurement, not the minimum of two measurements needed to discuss sequential measurement. The important point about Eq 9.28 is not Eq 9.28 alone, but that it is conditional on the previous measurement outcome.

No, why? The 1st SG device separates the particles spatially according to $\sigma_z$. I just place the 2nd SG device in the partial beam providing particles with determined $\sigma_1=1/2$. Then I count the particles with determined $\sigma_x=\pm 1/2$ by just setting counters at the appropriate places. There the particles are absorbed and the detector counts them. The result, according to QT, is of course $P(\sigma_x=1/2|\sigma_z=1/2)=P(\sigma_x=1/2|\sigma_z=1/2)=1/2$, and that's what comes out from actual measurements as far as I know.

 

[vanhees71]

Do you consider a polarization filter or a single slit to be an appropriate measurement device?

In my opinion both are both measurement devices (for polarization or position, respectively) and filter-preparation devices (for the corresponding quantities).

 

[atyy]

No, why? The 1st SG device separates the particles spatially according to $\sigma_z$. I just place the 2nd SG device in the partial beam providing particles with determined $\sigma_1=1/2$. Then I count the particles with determined $\sigma_x=\pm 1/2$ by just setting counters at the appropriate places. There the particles are absorbed and the detector counts them. The result, according to QT, is of course $P(\sigma_x=1/2|\sigma_z=1/2)=P(\sigma_x=1/2|\sigma_z=1/2)=1/2$, and that's what comes out from actual measurements as far as I know.

If one assigns the particle a definite spin or position without making a measurement then one is assuming these quantities exist without measurement. It is analogous to assuming that particles have definite trajectories, which are hidden variables.

 

[vanhees71]

It exists due to the preparation procedure, because after the 1st SG apparatus $\sigma_z$ and position are entangled. There's a 100% correlation between the value of $\sigma_z$ and the partial beams coming out of the 1st SG apparatus!