The HUP, simultaneous measurements, and eigenfunctions

[jmcelve]

Hi everyone,

I know this topic has been discussed quite a bit -- and in particular it's been done in this thread and this thread. But there are still some things I want to talk about in order to (hopefully) clarify my own thoughts.

One of the threads discusses this Ballentine article in which Ballentine presents an argument against the statement that the HUP requires that simultaneous measurements of conjugate variables have a lower bound in uncertainty. I find Ballentine's argument convincing, but in one of the threads, Demystifier makes a convincing argument that Balletine's experiment requires identifying the probability of the momentum with the probability of the position measurement |\langle y | \psi \rangle|^2 when it is experimentally the case the the momentum is identified with the Fourier transform |\langle p_y | \psi \rangle|^2 of the coordinate space wavefunction. My first question then is: where does Balletine's experiment fail?

Additionally, I see a lot of people make the argument that the HUP does not say anything about simultaneous measurements of conjugate variables, but that it is rather a statement about a given prepared state (which describes an ensemble of particles). I understand the argument and find it convincing since it doesn't make sense to talk about deviations for a single particle. But then what about the obvious manifestation of the HUP in single particle systems where simultaneous eigenfunctions of position and momentum cannot exist? Indeed, doesn't this suggest that the HUP *does* imply that, for a given particle, its position and momentum cannot be simultaneously identified up to arbitrary accuracy -- that there *is* a lower bound?

Many thanks,
jmcelve

 

[Jano L.]

But then what about the obvious manifestation of the HUP in single particle systems where simultaneous eigenfunctions of position and momentum cannot exist? Indeed, doesn't this suggest that the HUP *does* imply that, for a given particle, its position and momentum cannot be simultaneously identified up to arbitrary accuracy -- that there *is* a lower bound?

Hello jmcelve,

the important thing is to free oneself from the false idea that mathematical function \psi(\mathbf r) is a complete description of the particle. Unless this is done, the particle can never be thought to be in one place nor to have, in any situation, to have momentum. This is because there are no corresponding normalized \psi that would be able to describe the particle along Born's rule for |\psi|^2.

The natural thing to do is to expect that the particle has both position and momentum simultaneously, but this position and momentum are not properties of some wave function. Wave function can be naturally thought as describing probabilities, or frequencies in finite ensemble.

 

[Naty1]

not again! [LOL]

Good for you for reading those excellent and lengthy discussions. Consider taking notes for you own clarification.

Different people have different interpretations about identical mathematics...and some assign
preference to some math and others to other math. These have been going on for 90 years.
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My first question then is: where does Ballentine's experiment fail?

Fredrik: “Ballentine's argument in the article discussed in this thread seemed to prove that you could measure both [position, momentum] with accuracies Δx and Δp such that ΔxΔp is arbitrarily small. And I wasn't able to see what was wrong with it. But Demystifier was. I think that what he said here is a very good reason to not define QM in a way that makes what Ballentine described a "momentum measurement":

See post # 40 here:

https://www.physicsforums.com/showthread.php?p=3554463#post3554463

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"But then what about the obvious manifestation of the HUP in single particle systems where simultaneous eigenfunctions of position and momentum cannot exist?"

I don't think that assumption is correct. QM formalism: an observable is represented by a self adjoint operator on a Hilbert space, and a state, represented by a state operator [also called a statistical operator or density matrix]. The only values which an observable may take on are its eigenvalues and the probabilities of each of the eigenvalues can be calculated.

Fredrick said it this way: "It is possible to measure position and momentum simultaneously…a single measurement of a particle. In fact, we often measure the momentum by measuring the position and interpreting the result as a momentum measurement. (Check out figure 3 in this pdf). What we can't do is to prepare an identical set of states…. such that we would be able to make an accurate prediction about what the result of a position measurement would be and an accurate prediction about what the result of a momentum measurement would be….for an ensemble of measurements..."
and " "It's not true that every measurement puts the system in an eigenstate of the measured observable".Zapper says it this way: "...It is possible to measure position and momentum simultaneously…a single measurement of a particle. In fact, we often measure the momentum by measuring the position and interpreting the result as a momentum measurement. What we can't do is to prepare an identical set of states…. such that we would be able to make an accurate prediction about what the result of a position measurement would be and an accurate prediction about what the result of a momentum measurement would be….for an ensemble of measurements. from the prior post:

The natural thing to do is to expect that the particle has both position and momentum simultaneously

that is, if one is thinking classically; such is not the case with quantum mechanics! 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."

"I think we're closing in on an answer to my original question: There is no known argument or experiment that can completely rule out the possibility that particles have well-defined positions at all times, but we can rule out the possibility that the only significance of the wavefunction is to describe the statistical distribution of particles with well-defined positions." Here are my own summary notes from those earlier discussions: [Some will likely disagree with the descriptions I have chosen.]

Quantum mechanics doesn't say whether or not a particle has a position and a momentum at all times.
This is one way to describe what happens:
In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic, but statistically predictable way. After a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble.

http://en.wikipedia.org/wiki/Observa...ntum_mechanics

Synopsis: Is it possible to simultaneously measure the position and momentum of a single particle. Apparently not: The HUP doesn't say anything about whether you can measure both in a single measurement at the same time. That is a separate issue.

What we call "uncertainty" is a property of a statistical distribution. The HUP isn't about a single measurement and what can be obtained out of that single measurement. It is about how well we can predict subsequent measurements given the ‘identical’ conditions. The commutativity and non commutivity of operators applies to the distribution of results, not an individual measurement. This "inability to repeat measurements" is in my opinion better described as an inability to prepare a state which results in identical observables.

The uncertainty principle results from uncertainties which arise when attempting to prepare a set of identically prepared states. The wave function is associated not with an individual particle but rather with the probability for finding particles at a particular position.What we can't do is to prepare an identical set of states [that yields identical measurements]. NO STATE PREPARATION PROCEDURE IS POSSIBLE WHICH WOULD YIELD AN ENSEMBLE OF SYSTEMS IDENTICAL IN ALL OF THEIR OBSERVABLE PROPERTIES. [‘Identical’ state preparation procedures yield a statistical distribution of observables [measurements].]

The uncertainty principle restricts the degree of statistical homogeneity which it is possible to achieve in an ensemble of similarly prepared systems. A non-destructive position measurement is a state preparation that localizes the particle in the sense that it makes its wavefunction sharply peaked. This of course "flattens" its Fourier transform, so if the Fourier transform was sharply peaked before the position measurement, it isn't anymore.

The Uncertainty Principle finds its natural interpretation as a lower bound on the statistical dispersion among similarly prepared systems resulting from identical state preparation procedures and is not in any real sense related to the possible disturbance of a system by a measurement. The distinction between measurement and state preparation is essential for clarity.

A quantum state (pure or otherwise) represents an ensemble of similarly prepared systems. For example, the system may be a single electron. The ensemble will be the conceptual
(infinite) set of all single electrons which have been subjected to some state preparation technique (to be specified for each state), generally by interaction with a suitable apparatus. Albert Messiah, Quantum Mechanics, p119
“When carrying out a measurement of position or momentum on an individual system represented by psi, no definite prediction can be made about the result. The predictions defined here apply to a very large number [N] of equivalent systems independent of each other each system being represented by the same wave function [psi]. If one carries out a position measurement on each one of them, The probability density P[r], or momentum density, gives the distribution of the [N] results of measurements in the limit where the number N of members of this statistical ensemble approaches infinity.”
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Far from restricting simultaneous measurements of no commutating observables, quantum theory does not deal with them at all; it’s formalism being capable only of statistically predicting the results of measurements of one observable (or a commutative set of observables).

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[Fredrik]

Fredrick said it this way: "It is possible to measure position and momentum simultaneously…a single measurement of a particle. In fact, we often measure the momentum by measuring the position and interpreting the result as a momentum measurement. (Check out figure 3 in this pdf).

I think maybe you forgot what I said here.

That pdf is Ballentine's article. My claim that you can measure both at the same time was based on the validity of Ballentine's argument. But Demystifier's argument convinced me that what Ballentine considered a momentum measurement shouldn't have a distribution of results that are consistent with the predictions of QM.

Honestly, I'm still unsure about some of this. ZZ posted an article somewhere about a technique called ARPES that's used to measure momentum components. I still haven't tried to really understand the article, so I don't know what to make of it, but my first impression (based on things I no longer remember) was that this technique is similar enough to what Ballentine described, that Demystifier's argument might apply to it. But this is a technique that experimental quantum physicists use, and some of them should know what a momentum measurement is better than I do. So I have a feeling that there's something about this that I still don't understand.

 

[jmcelve]

It's nice to see that there's still some uncertainty (!) regarding these questions. I don't feel so alone in pondering them now.

I'll have to consider what you've said more thoroughly, Naty1, but thank you for the response.

The same thanks go to you Jano, though I think Naty1 is correct in his response to your statement:

The natural thing to do is to expect that the particle has both position and momentum simultaneously, but this position and momentum are not properties of some wave function.

I suppose the next place to start is to read Ballentine's discussion of measurements for a quantum system.

 

[strangerep]

I suppose the next place to start is to read Ballentine's discussion of measurements for a quantum system.

Probably better to read his more modern textbook instead of his much older paper on the statistical interpretation, imho.

 

[Naty1]

I finally found Zappers blog...on HUP:

Misconception of the Heisenberg Uncertainty Principle.

http://physicsandphysicists.blogspot.com/2006/11/misconception-of-heisenberg-uncertainty.htmlFredrik: "I think maybe you forgot what I said here...

yeah, well that was post #287! [LOL]
and my notes are so long I never got that far this time around...[I amended my notes this time!]

Fredrik: " Honestly, I'm still unsure about some of this."

Well, please make up your mind! [LOL] When you do, your comments are quite clear. If you keep changing your mind it makes quoting you considerably less helpful!

But I did already post: "...Different people have different interpretations ...going on for over 90 years" so at least I got that part right.

 

[Naty1]

jmcleve:

I'll have to consider what you've said more thoroughly, Naty1, but thank you for the response.

Just note I was quoting from earlier discussions; I'm lucky when I THINK I understand what
the experts are posting.

Not only are QM conceptual interpretations elusive and subtle, but QM also suffers from people drawing different conclusions about a single statement. These forms are rife with
QM statements being dissected...as,say, three different people drawing three different interpretations from a single statement.

I even bought a huge Albert Messiah 'Quantum Mechanics' combined two volumes hoping to find carefully edited explantions that would be clear and unambiguous...no such luck!
People here objected to some wording there,too! Glad it was used and cheap!

I still like the 'Shut up and calculate.' explanation.

 

[Naty1]

While I was skimming my notes to update based on Fredrik's comment above, I came across this statement from the previous discussion:

The wave function describes an ensemble of similarly prepared particles rather than a single scattering particle. A wave function with a well defined wavelength must have a large special extension, and conversely a wave function which is localized in a small region of space must be a Fourier synthesis of components with a wide range of wavelengths. We cannot measure them both to an arbitrary level of precision. A function and its Fourier transform cannot both be made sharp. This is a purely a mathematical fact and so has nothing to do with our ability to do experiments or our present-day technology. As long as QM is based on the present mathematical theory an arbitrary level of precision cannot be achieved.

I'm not sure I interpret this as did the original poster...nor the same as when I recorded it. I read this to say, paraphrasing, 'Even if you could prepare an ensemble of identically prepared, precisely the same, particle states [which you can't] the nature of Fourier synthesis prevents one from make simultaneous sharp measurements.

Anybody else see it this way??

 

[Jano L.]

from the prior post: The natural thing to do is to expect that the particle has both position and momentum simultaneously

that is, if one is thinking classically; such is not the case with quantum mechanics! 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."

Of course, but you deviated from the original question of jmcelve. The question was

But then what about the obvious manifestation of the HUP in single particle systems where simultaneous eigenfunctions of position and momentum cannot exist? Indeed, doesn't this suggest that the HUP *does* imply that, for a given particle, its position and momentum cannot be simultaneously identified up to arbitrary accuracy -- that there *is* a lower bound?

I have tried to explain that the uncertainty relation based on \psi has no bearing on the individual particle. I can try to explain better, if my argument was not clear.

I did not say that the statistical interpretation REQUIRES that the electron HAS position and momentum simultaneously; there is no point on insisting what electron HAS or IS.

However, it is natural to think r,p can be used to describe the electron, for many reasons. The statistical interpretation in the sense of Bohm or Ballentine showed that this is not in contradiction to experiments or the rest of the probabilistic theory based on the wave function.

The expression "statistical interpretation" easily gets unclear. In 30's Heisenberg would say that the Copenhagen interpretation is the statistical interpretation and deny applicability of simultaneous position and momentum to one particle. But, after works of Einstein, Bohm and Ballentine we have every reason to believe that the results of the theory do not rest on the denial of existence of r,p for particles. Nowadays, I think it is more appropriate to understand the expression "statistical interpretation" as referring to a theory in which \psi(\mathbf r) gives probability in coordinate space but says nothing about the r,p of the individual particle.

 

[Naty1]

I have tried to explain that the uncertainty relation based on ψ has no bearing on the individual particle. I can try to explain better, if my argument was not clear.

nope...the only part I tohught MIGHT need clarification I commented upon. In fact my
last post matches yours, I think, nicely.

 

[Jano L.]

I almost agree with your last post, apart from the sentence

the nature of Fourier synthesis prevents one from make simultaneous sharp measurements.

What can be measured depends on theory one adopts. If one denies r,p in theory, of course in that theory one cannot measure them. This is the case when we adopt the idea "the particle is a wave and since the wave has neither position nor momentum the particle does not have them either".

But if we assume that the wave function is just an auxiliary function (lives in configuration space!) used to describe particles, and allow r,p for particles, they may be measured and no nature of Fourier transformation can prevent this.