Do particles have well-defined positions at all times?

[Fredrik]

I have always thought that this idea isn't even consistent with the standard version of QM, so I was really surprised when I found this quote in Ballentine's 1970 article "The statistical interpretation of quantum mechanics":

In contrast, the Statistical Interpretation considers a particle to always be at some position in space, each position being realized with relative frequency |\psi(\vec r)|^2 in an ensemble of similarly prepared experiments.​

Later in the article he admits that we don't know if this is really the case, but he insists that this view isn't inconsistent with QM. I would like to know if he's right.

Is there an argument that proves that particles don't have well-defined positions at all times? Aren't there experiments for which the assumptions "the particle is either here or there" and "the particle is in a superposition of here and there" give us different predictions?

 

[strangerep]

Aren't there experiments for which the assumptions "the particle is either here or there" and "the particle is in a superposition of here and there" give us different predictions?

But if you measure the position in the second case, the result will be either "here" or "there" anyway (assuming position eigenstates are mutually orthogonal).

So I don't see how these two assumptions are different.

 

[ueit]

Aren't there experiments for which the assumptions "the particle is either here or there" and "the particle is in a superposition of here and there" give us different predictions?

AFAIK there is no experiment to date to distinguish the two cases. In theory one could try to determine the mass distribution by measuring gravity but I doubt that such an experiment can be performed.

 

[zonde]

If we speak about photons maybe this experiment can help:
http://physics.nist.gov/Divisions/Div844/publications/migdall/psm96_twophoton_interference.pdf"
There HOM interference is recovered after photons have passed beamsplitter (have taken certain path).
But as usually with photon experiments you have to make fair sampling assumption.

 

[SpectraCat]

We have been discussing exactly this issue over on https://www.physicsforums.com/showthread.php?t=499451 ... Varon asked essentially the same question you raise. The answer seems to be that the Aspect experiments are the acid test (as expected). The statistical interpretation must rely on hidden variables to explain the Aspect experiments. Since Bell's theorem rules out local hidden variables, they must be non-local hidden variables that allow/facilitate superluminal signalling between the particles. Demystifier said that Bohmian mechanics can basically be seen as specific implementation of the general framework laid out by the statistical interpretation (see the more recent posts in the thread).

Note that Ballentine's statement in his 1970 paper to the effect that "the SI allows HV but doesn't require them", must be seen to be incorrect in light of the Aspect experiments. He addresses the Aspect experiments in his 1989 textbook, but I find his statements there to be more hand-waving and wishful thinking than any sort of satisfactory explanation.

 

[Hurkyl]

I have always thought that this idea isn't even consistent with the standard version of QM

Do you have an idea of "particles with well-defined positions" that does not include the particles of Bohmian mechanics?

 

[strangerep]

The statistical interpretation must rely on hidden variables to explain the Aspect experiments.

The Aspect experiments boil down to measuring statistical correlations. I don't see how SI (without HV) is incompatible with that. It doesn't matter if quantum correlations are nonlocal since correlation is not the same as causation.

 

[Demystifier]

Do you have an idea of "particles with well-defined positions" that does not include the particles of Bohmian mechanics?

I do. The ultimate reason why Bohmian mechanics works is not its specific law for particle trajectories, but the fact that, at any time, the statistical distribution of particle positions in an ensemble is given by |psi|^2. It is easy to write a different law for particle trajectories which is also compatible with |psi|^2. One may even conceive a stochastic theory of particle positions, where continuous trajectories are totally absent, in a manner compatible with |psi|^2. The reason why the specific Bohmian law for particle trajectories is preferred is the fact that other conceivable laws seem much more artificial. (Of course, many people find the Bohmian law itself artificial, which is why they don't like the Bohmian interpretation. Unfortunately, "artificiality" is a subjective category.)

 

[ueit]

The answer seems to be that the Aspect experiments are the acid test (as expected). The statistical interpretation must rely on hidden variables to explain the Aspect experiments. Since Bell's theorem rules out local hidden variables, they must be non-local hidden variables that allow/facilitate superluminal signalling between the particles.

Bell's theorem rests on the statistical independence assumption that is incompatible with the most important type of classical theories, field theories. The only lhv theories that are refuted are those without interaction between particles (Newtonian billiard balls-type of theories).

Once this assumption is put aside, those experiments only show that a different experimental setup leads to different results which does not conflict with a classical picture.

The same is true for interference experiments. If one assumes a classical field theory describing the motion of classical particles, then it is to be expected that a change in particle distribution (corresponding to opening or closing of a slit) will lead to a change in the particle's trajectory and a different result on the screen.

 

[Demystifier]

Is there an argument that proves that particles don't have well-defined positions at all times? Aren't there experiments for which the assumptions "the particle is either here or there" and "the particle is in a superposition of here and there" give us different predictions?

There is no such experiment. That's because, whatever you think you measure, ultimately you observe the POSITION of something (e.g. the position of a needle of the measuring apparatus.)

The classic example is measurement of spin. It is measured by the Stern-Gerlach apparatus
http://en.wikipedia.org/wiki/Stern-Gerlach_experiment
which actually measures the POSITION of the particle, which is then interpreted as spin up or spin down depending on whether the measured particle position is up or down.

There is actually a simple explanation why all measurements can ultimately be reduced to measurements of positions. This is because all measurements require a macroscopic apparatus, for which decoherence determines the preferred basis. But the decoherence-induced preferred basis is the position basis, due to the fact that interactions are local in the position space.

 

[zonde]

The statistical interpretation must rely on hidden variables to explain the Aspect experiments. Since Bell's theorem rules out local hidden variables, they must be non-local hidden variables that allow/facilitate superluminal signalling between the particles.

Considering that one of the earlier proponents of ensemble interpretation was Einstein seems strange to suggest that this interpretation allows superluminal signalling.
How about the idea that loophole free Bell tests will fail?

 

[Demystifier]

Considering that one of the earlier proponents of ensemble interpretation was Einstein seems strange to suggest that this interpretation allows superluminal signalling.

It's not strange at all, having in mind that Einstein died in 1955, while Bell discovered his result in 1964.

How about the idea that loophole free Bell tests will fail?

It's very very unlikely, but logically possible.

 

[CyberShot]

There is no such experiment. That's because, whatever you think you measure, ultimately you observe the POSITION of something (e.g. the position of a needle of the measuring apparatus.)

This is where I believe that the quantum theory, although it may have some merit, exaggerates the mathematics and tries to, erroneously, extend it to make claims about reality. Yes, according to your logic it's almost impossible in practice to simultaneously measure both momentum and position, but what grants us the reason to think that particles care about our technological difficulties? They still have those values, they are not ill-defined or smeared out over probability clouds. They exist. They are simply concealed to us. Forgive me, but I just don't see a way out.

 

[PhilDSP]

Isn't one consideration that, according to de Broglie's analysis, the momentum of a particle will have components that are not parallel to the direction of the particle's motion? So if you define position in terms involving momentum, even disregarding the HUP, it can't be definite.

Of course you don't necessarily need to define a particle's position such that it depends on momentum, do you?

 

[Fredrik]

But if you measure the position in the second case, the result will be either "here" or "there" anyway (assuming position eigenstates are mutually orthogonal).

So I don't see how these two assumptions are different.

For starters, they have different mathematical representations. The state vector

|\psi\rangle=\frac{1}{\sqrt{2}}\big(|\text{here} \rangle+|\text{there}\rangle\big)

represents a pure state of the "superposition of here and there" type. The corresponding state operator is

|\psi\rangle\langle\psi|=\frac{1}{2}\big(| \text{here}\rangle\langle\text{here}|+|\text{here} \rangle\langle\text{there}|+|\text{there}\rangle \langle\text{here}|+|\text{there}\rangle\langle \text{there}|\big)

but we would have to drop the two terms in the middle to get a state of the "either here or there" type. So doesn't Ballentine's claim imply that those two terms have absolutely no effect on any experiment that can be performed?

What about a double slit experiment with detectors that let the particle pass through at each slit? We know that this changes the interference pattern, and the only thing the detectors do is to change the state preparation from "superposition of left and right" to "either left or right". (Edit: For this to be accurate, we have to assume that we don't check which one of the detectors detected a particle).

Do you have an idea of "particles with well-defined positions" that does not include the particles of Bohmian mechanics?

I'm not sure I understand the question. I started this thread because I don't see how it makes sense to say that each particle prepared in a pure state like exp(-x^2) (times a normalization constant) has a definite position. So I guess the answer is "no I don't, but it seems that Ballentine does".

There is no such experiment. That's because, whatever you think you measure, ultimately you observe the POSITION of something (e.g. the position of a needle of the measuring apparatus.)

I agree with everything you said after the word "because", but I don't see how it has anything to do with what you said before the "because".

By the way, instead of saying that all measurements are really position measurements, I prefer to say that what a measurement does is to produce a record (that can be approximately described by classical physics) of the fact that a particular interaction has taken place. We can then infer the position and the time of that interaction from the position of the measuring device and the time when the record was created. This is the reason why all measurements are really position measurements, so it goes one step deeper.

 

[Fredrik]

Regarding the definition of "the statistical interpretation", I would like to draw everyone's attention to this quote from the article:

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).
​

The quote I included in post #1 is from section 1.3, which is supposed to define the interpretation. This quote is from section 1.2. In section 5, he says that we don't know if particles have definite positions at all times.

So it seems that he defines the statistical interpretation as QM without the unnecessary assumption that a state vector represents all the properties of a single system, and with the unnecessary assumption that particles have well-defined positions at all times. My own view, which is that QM is just an assignment of probabilities to possible results of experiments, would then be a different interpretation than Ballentine's statistical interpretation (because I don't assume that particles have well-defined positions at all times), but it would be an interpretation that Ballentine doesn't want to distinguish from his own.

I wish he hadn't included that stuff I quoted in #1 in section 1.3, because now the "statistical interpretation" is ambiguously defined. Alternatively, he could have chosen to talk about "statistical interpretations", plural. Different statistical interpretations would then have been distinguished by the unnecessary assumptions they make.

 

[SpectraCat]

Regarding the definition of "the statistical interpretation", I would like to draw everyone's attention to this quote from the article:

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).
​

The quote I included in post #1 is from section 1.3, which is supposed to define the interpretation. This quote is from section 1.2. In section 5, he says that we don't know if particles have definite positions at all times.

So it seems that he defines the statistical interpretation as QM without the unnecessary assumption that a state vector represents all the properties of a single system, and with the unnecessary assumption that particles have well-defined positions at all times. My own view, which is that QM is just an assignment of probabilities to possible results of experiments, would then be a different interpretation than Ballentine's statistical interpretation (because I don't assume that particles have well-defined positions at all times), but it would be an interpretation that Ballentine doesn't want to distinguish from his own.

I wish he hadn't included that stuff I quoted in #1 in section 1.3, because now the "statistical interpretation" is ambiguously defined. Alternatively, he could have chosen to talk about "statistical interpretations", plural. Different statistical interpretations would then have been distinguished by the unnecessary assumptions they make.

I completely agree with that ... I said before that I wished he would have just said "particles *may* have well-defined positions at all times, or they may not ... the statistical interpretation does not require one condition or the other to be true." From what I have read, that statement is actually consistent with pretty much everything else in the SI ... does anyone know of a facet of SI that requires particle positions to be well-defined at all times?

 

[Demystifier]

I started this thread because I don't see how it makes sense to say that each particle prepared in a pure state like exp(-x^2) (times a normalization constant) has a definite position.

A more correct saying would be "each particle is prepared such that we don't KNOW its position exactly, but only probabilistically with a probability density exp(-x) (times a normalization constant)". Would that be more compatible with particles having definite positions?

 

[Fredrik]

A more correct saying would be "each particle is prepared such that we don't KNOW its position exactly, but only probabilistically with a probability density exp(-x) (times a normalization constant)". Would that be more compatible with particles having definite positions?

I wouldn't say it like that, because the phrase "we don't know its position" strongly suggests that it has a position. If we knew for sure that particles have positions, then it would make sense to say something like that. (In my example, the probability density would be exp(-2x^2) times the square of the normalization constant).

Define P=\int_S |\psi(\vec x)|^2 d^3x.

If particles have well-defined positions:

P is the probability that the particle is in the region S.​

If they don't:

P is the probability that a detector covering the region S will produce a record of a detection.​

If it's unknowable, I would choose the latter statement and avoid statements that suggest that particles have positions.

 

[DrChinese]

This is where I believe that the quantum theory, although it may have some merit, exaggerates the mathematics and tries to, erroneously, extend it to make claims about reality. Yes, according to your logic it's almost impossible in practice to simultaneously measure both momentum and position, but what grants us the reason to think that particles care about our technological difficulties? They still have those values, they are not ill-defined or smeared out over probability clouds. They exist. They are simply concealed to us. Forgive me, but I just don't see a way out.

This was the EPR line of reasoning, 1935. After Bell and Aspect, it is clear this perspective does not survive in the form you give it. Technology is not the issue at all.

 

[Varon]

This was the EPR line of reasoning, 1935. After Bell and Aspect, it is clear this perspective does not survive in the form you give it. Technology is not the issue at all.

You are saying that Bell and Aspect experiment is enough to refute that particles have well-defined positions at all time?

Maybe not so. Why can't we say particles have well defined positions at all times and they were superluminally connected (without any omnicient wave function like in Bohmian's). Since no information is tranfered, then it doesn't really violate the spirit of special relativity.
Why is this not possible?

 

[mrspeedybob]

If a particle has a well defined position at all times how can it interfere with itself? It seems to me that the double slit experiment alone is enough to dispell this idea.

 

[Varon]

If a particle has a well defined position at all times how can it interfere with itself? It seems to me that the double slit experiment alone is enough to dispell this idea.

Ballentine explained it clearly in the same paper:

"As in any scattering experiment, quantum theory predicts the statistical frequencies of the various angles through which a particle may be scattered. For a crystal or diffraction grating there is only a discrete set of possible scattering angles because momentum transfer to and from a periodic object is quantized by a multiple of delta p = h/d, where delta p is the component of momentum tranfer parallel to the direction of the periodic displacement d. This result, which is obvious from a solution of the problem in momentum representation, was first discovered by Duane (1923), although this early paper had been much neglected until its revival by Lande (1955, 1965). There is no need to assume that an electron spreads itself, wavelike, over a large region of space in order to explain diffraction scattering. Rather it is the crystal which is spread out, and the electron interacts with the crystal as a whole through the laws of quantum mechanics."

Is it true?

 

[Fra]

Regarding the definition of "the statistical interpretation", I would like to draw everyone's attention to this quote from the article:

Although there are many shades of interpretation (Bunge, 1956), we wish to distinguish only two:

(I) The Statistical Interpretation, according to which a pure state (and hence also a general state) provides a description of certain similarly prepared systems, but need not provide a complete description of an individual system.

(II) Interpretations which assert that a pure state provides a complete and exhaustive description of an individual system (e.g. an electron).
​

What exactly does "complete" mean in Ballentines use? If it means sometihng like the EPR usage

"In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of physical quantity is the possibility of predicting it with certainty without disturing the system"

Then Ballentine seem to subscribe to a certain form of realism? Because the question is then, how do you KNOW which quantities that can be predicted without disturbing the system? This disturbs me with this interpretation.

I think the more sensible meaning of this would be to use a different definition of Complete, that's different from Einstiens use. I propose to repalce this EPR statement

"A sufficient condition for the reality of physical quantity is the possibility of predicting it with certainty without disturing the system"

with

"A sufficient condition for the reality of physical quantity is the possibility of inferring an expectation from available information"

The realism part is removed but would probably correspond to something like

"The reaction form the system, on the ractional action following he inferred expectation would not cause a revision of the expectation"

This latter thing corresponds to assuming that "expectations" are laws. Something which from the point of view of inference is quite unfounded, and moreover unnecessary in order to play the game, because yoiu always make one move at a time anyway.

Then we could say it's a "complete description" of individual system in the sense that it contains all rationally inferrable statements. But this would be observer dependent (unless we constrict ourselves to the ensemble of small subsystem. And would use less realism. It's a "completeness" term without resorting to realism.

/Fredrik

 

[strangerep]

I figure I should repeat the following point in this thread...

Ballentine does not say that a particle has a well-defined position
(unless prepared in a position eigenstate). In general it has "some"
position, by which he means that there's a probability density function
given by the usual square modulus of the wave function.

I think you guys are reading extra things into Ballentine's words which
aren't really there.

 

[Varon]

I figure I should repeat the following point in this thread...

Ballentine does not say that a particle has a well-defined position
(unless prepared in a position eigenstate). In general it has "some"
position, by which he means that there's a probability density function
given by the usual square modulus of the wave function.

I think you guys are reading extra things into Ballentine's words which
aren't really there.

I just bought physicist Victor Stenger book in Kindle a while ago called "The Fallacy of Fine-Tuning: Why the Universe is Not Designed for Us". In the section 15.6 The Statistical Interpretation. It is mentioned:

"This empircal result supports that convensional interpretation of the wave function as associated not with individual particles but rather with the probability for finding a particle at a particular position. In this interpretation, the object always is a particle, not a wave, and the wave aspect is a mathematical abstraction used in the model to make probability calculations"

(back to me)
But in Statatistical Interpretation. The object is always a particle. Isn't it. And just like Bohmian Mechanics. Something that always has position has automatically well-defined position, only we don't know what it is (tell me if this is wrong and why). This is the gist of the argument. Because in Bohr original postulate. Before there is measurement to determine its position, the electron has no position. In Statistical Interpretation. An electron always has position. I think this is what defines the Statistical Interpretation, isn't it? where a particle always has position? Because if they admit a particle is a wave too. Then one particle is enough to contribute to interference.. hence no need to create the Statistical interpretation.

 

[Fra]

There are traits of realism thinking that doesn't ever seem to wash away. This is what disturbs me:

"This empircal result supports that convensional interpretation of the wave function as associated not with individual particles but rather with the probability for finding a particle at a particular position. In this interpretation, the object always is a particle, not a wav
...
(back to me)
But in Statatistical Interpretation. The object is always a particle. Isn't it. And just like

It seems this reasoning presumes that particle or wave are an exchaustive list of possibilities. Both have flavours of realism.

If you look strictly upon statistical description of what we actually measure. Say counter states etc. Then it really does not matter what the system "IS" in some sense of realism. All that matters is that the information/knowledge we have (ie. the statistics) is what it is.

Both trying to say that "behind statistics there is particles, or waves" are really unnecessary and in fact just confusing as it has too much realism flavour to it. Something that really ISN't available in the original picture. Because I think our predictions strictly speaking is just about the future counter states anyway.

I think this even supports the duality. Maybe you can if you want, extrapolate the raw info at hand to be the shadow of an idea of particles, or a picture of waves. But does it make a difference how we label the patterns in the information?

If we instead focus on the information, and how to infer from it a rational expectation of the change of the same pattern, then I think we will see that the issue of particles or waves is a non-issue.

/Fredrik

 

[strangerep]

But in Statatistical Interpretation. The object is always a particle. Isn't it.

No, not necessarily. The word "particle" is bandied around far too loosely, imho.
I find the "field" picture generates fewer inconsistencies.

BTW, (also imho), modern reading of the statistical interpretation should be taken together with the relational ideas of Rovelli, e.g.,

quant-ph/9609002 and quant-ph/0604064

and also a dose of Mermin's emphasis on correlations

quant-ph/9801057 and quant-ph/9609013

If you look strictly upon statistical description of what we actually measure. Say counter states etc. Then it really does not matter what the system "IS" in some sense of realism. All that matters is that the information/knowledge we have (ie. the statistics) is what it is.

Yes. Maybe I'll go a bit further and say that all we do is establish correlations through interactions (between system and apparatus, etc, etc).

 

[Varon]

There are traits of realism thinking that doesn't ever seem to wash away. This is what disturbs me:



It seems this reasoning presumes that particle or wave are an exchaustive list of possibilities. Both have flavours of realism.

If you look strictly upon statistical description of what we actually measure. Say counter states etc. Then it really does not matter what the system "IS" in some sense of realism. All that matters is that the information/knowledge we have (ie. the statistics) is what it is.

Both trying to say that "behind statistics there is particles, or waves" are really unnecessary and in fact just confusing as it has too much realism flavour to it. Something that really ISN't available in the original picture. Because I think our predictions strictly speaking is just about the future counter states anyway.

I think this even supports the duality. Maybe you can if you want, extrapolate the raw info at hand to be the shadow of an idea of particles, or a picture of waves. But does it make a difference how we label the patterns in the information?

If we instead focus on the information, and how to infer from it a rational expectation of the change of the same pattern, then I think we will see that the issue of particles or waves is a non-issue.

/Fredrik

Bad start for particle physicist Victor Stenger, isn't it. He was supposed to debunk Fine Tuning in his book and he already made the first wrong assumption, how could people believe him? He should have at least consulted you or this forum first.

 

[Fredrik]

I figure I should repeat the following point in this thread...

Ballentine does not say that a particle has a well-defined position
(unless prepared in a position eigenstate). In general it has "some"
position, by which he means that there's a probability density function
given by the usual square modulus of the wave function.

I think you guys are reading extra things into Ballentine's words which
aren't really there.

What makes you say that? I don't see how the text I quoted in #1 can be interpreted any other way. I think it shows clearly that he thinks particles have well-defined positions at all times, and since he's saying it in the section (1.3) that's supposed to define the interpretation, it's hard to argue that he didn't intend it to be a part of the statistical interpretation. (As I mentioned above, section 1.2 contradicts that, but section 1.3 is still the section that's supposed to define the interpretation).

He also claims repeatedly that this idea isn't inconsistent with QM. For example, in section 3.2, he says that the claim that particles don't have perfectly well-defined values of position and momentum at all times is "easily seen to be unjustified". (I didn't find his argument convincing). In section 5, he says "quantum theory is not inconsistent with the supposition that a particle has at any instant both a definite position and a definite momentum, although there is a widespread folklore to the contrary".

Even if I'm wrong about his intention to include that supposition as part of the definition of the interpretation, it's undeniable that he claims (over and over) that it's consistent with QM. I really don't see how that claim can be correct. In particular, I don't see how the supposition that particles have well-defined positions at all times can be consistent with the predictions of QM about interference patterns in double-slit experiments.