I am getting into a mess with the HUP

[MHD93]

Now, I am lost, I am stuck with the Heisenberg principle..

Seriously, the more I read papers, standard textbooks, or any books on it, I find I am more lost, nearly all seem to contradict all.

*http://www.kevinaylward.co.uk/qm/ballentine_ensemble_interpretation_1970.pdf
In page 365,
This asserts that what we know as the HUP relation (which, in the context, is called dispersion principle) is not even closely related to a single measurement of position and momentum. ( I understand what it means about the standard deviation )

*A book called "Philosophy of Quantum Mechanics" claims that Heisenberg meant, by his principle, the measurement of one observable that ruins the measurement of the other.

*Wikipedia, then, says that HUP shouldn't be confused with observer effect, so it's different from what we said.

*Others say that position and momentum are meaningless terms unless the measurement defines them, so it's the HUP...

Another problem is that I'm being unable to trust the standard textbooks this while, because of the many writings that contradict them.

I hope you get me out of this..
Are Physicists actually not agreeing on this, or is it me who can't get the point?
I am interested in last view, it is the most meaningful for me, if you clear it up, I will appreciate it.

I ask you too to clarify the other ideas that some people think are the HUP, but they really are not, and whether their content is true or not.Best Wishes

 

[phinds]

*Wikipedia, then, says that HUP shouldn't be confused with observer effect ...

That is true. The HUP describes a fundamental physical relationship that is independent of any measurements we can make. It DOES mean that there are limits on simultaneous measurements, but the fundamental effect is not due to any limitation in our ability to make measurements to some degree of accuracy.

 

[Bill_K]

From the referenced paper by Ballentine:

A particle with known initial momentum p passes through a narrow slit in a rigid massive screen. After passing through the hole, the momentum of the particle will be changed due to diffraction effects, but its energy will remain unchanged. When the particle strikes one of the distant detectors, its y coordinate is thereby measure with an error δy. Simultaneously this same event serves to measure the y component of momentum, p_y = p sin θ, with an error δp_y which may be made arbitrarily small by making the distance L arbitrarily large. Clear the product of the errors need not have any lower bound, and so the common statement of the uncertainty principle given above cannot be literally true.

This does seem to contradict the statement of the HUP that is usually given.

 

[Jano L.]

Are Physicists actually not agreeing on this, or is it me who can't get the point?

I think so, it is quite messy subject. My advice: read Heisenberg first:-) and then others...
What HUP proposes is that at one instant a particle cannot have both position and momentum determined with arbitrary precision, and their simultaneous existence is purely hypothetical (not rejected).

Which of the pair is better determined will depend on the manner of measurement (experiment on the particle).

Others say that position and momentum are meaningless terms unless the measurement defnes them, so it's the HUP... messy subject.

No, this is not how it happened. In fact Heisenberg got the absence of simultaneous x and p from his theory of matrices, and then invented arguments why it should be so also in the measurements, so that his theory would fit measurements.

In my opinion, it is hard to show that every possible measurement will fail to measure x and p, so I do not think that HUP is rock solid. What can be said for sure is that uncertainty relations (not the principle!) can be derived for normalized wave functions. Then they say that statistical dispersion of x and p cannot be simultaneously zero for any such wave function. Similar relations hold also for the probability distribution of a Brownian particle, so I do not think they are exclusive characteristic of the quantum theory.

Hypothetically, if there existed state of particle with definite x and p, it couldn't be described by the wave function, and would require some extension of the quantum theory (one such extension already exists: the Bohmian mechanics).

 

[vanhees71]

I've the contrary advice. Don't read Heisenberg first but Bohr or at least a later Heisenberg than the very first publication about the principle, because Heisenberg's first interpretation of the uncertainty relation as a noise-disturbance relation due to measurement is definitely wrong. This has been proven experimentally only recently, but it has been clear from quantum theory all the time. I don't understand, why the wrong interpretation stuck. It's a miracle that the English Wikipedia got it right, although even many textbooks describe it wrong (the German Wikipedia is not as good).

Have a look at the following, I wrote to summarize this here in the forum some time ago. Perhaps, this is another chance to get some discussion on the different interpretations of the uncertainty relation, which indeed refers to the preparation of a system in a (pure or mixed) state (i.e., a projective or ideal von Neumann measurement as this is put more precisely nowadays) but not necessarily on the influence of the measurement of one variable on another non-compatible one. So here's the link to this posting, I wrote a while ago:

https://www.physicsforums.com/showthread.php?t=664972

 

[strangerep]

From the referenced paper by Ballentine:
[...]
This does seem to contradict the statement of the HUP that is usually given.

It's better to read his more recent textbook "QM -- A Modern Development" than that rather old paper.

I like Ballentine's emphasis (in his textbook) on what vanhees71 already mentioned, i.e.,

the uncertainty relation, which indeed refers to the preparation of a system

(my emboldening).

 

[Avodyne]

I just want to point out that there is no dispute among physicists about the mathematics of quantum mechanics, or how to apply it to make (probabilistic) predictions for results of experiments, once the experimental setup has been described with sufficient precision.

 

[strangerep]

I just want to point out that there is no dispute among physicists about the mathematics of quantum mechanics, or how to apply it to make (probabilistic) predictions for results of experiments, once the experimental setup has been described with sufficient precision.

Good point. Maybe there's even a useful principle lurking in the words I've underlined.

Something like:
"All disputes involving QM interpretations are resolved by describing the experimental setup with greater precision, and expressing the predicted results probabilistically".

 

[Naty1]

In a previous discussion in these forums here

https://www.physicsforums.com/showthread.php?t=516224
what is it about position and momentum that forbids knowing both quantities at once?
There are some 300 posts..and I, at least, reached some definite conclusions based on the comments of experts here reflected in my notes below.

the answer was determined to be : nothing prevents an arbitrarily accurate single reading. The better your scientific apparatus, the better your readings. But there are interesting details [below].

My notes regarding your referenced paper:

good paper: wrong conclusion: The Statistical Interpretation of Quantum Mechanics

http://www.kevinaylward.co.uk/qm/ballentine_ensemble_interpretation_1970.pdf

This was the concsensus, I believe, of several experts from these forums [advisors,mentors,etc].

My synopsis notes from the discussion:

Is it possible to simultaneously measure the position and momentum of a single particle. 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.

NO STATE PREPARATION PROCEDURE IS POSSIBLE WHICH WOULD YIELD AN ENSEMBLE OF SYSTEMS IDENTICAL IN ALL OF THEIR OBSERVABLE PROPERTIES.
/////////////////////
Further explanatory comments:

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’ initial conditions. The commutativity and non commutivity of operators applies to the distribution of results, not an individual measurement. This "inability to repeat identical measurement results" 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…from identical initial conditions. The wave function is not associated 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. [instead, 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.”

 

[phinds]

That is true. The HUP describes a fundamental physical relationship that is independent of any measurements we can make. It DOES mean that there are limits on simultaneous measurements, but the fundamental effect is not due to any limitation in our ability to make measurements to some degree of accuracy.

Naty, I have clearly been mistaken about this (the above is incorrect about simultaneous measurements) forever and I appreciate the excellent discussion you have provided. I will stop misleading people in the future if/when I contribute to answers to similar questions.

 

[Naty1]

Here are a few other insights, some from that same very long discussion, some not...

you may want to read that discussion and pick out descriptions that make sense to you...
these gave me some insights I found helpful...

..."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."

There was a mathematical reference but I don't recall seeing a word interpretation:
Heisenberg's Uncertainty from Dirac's Brackets

John Baez
http://www.cbloom.com/physics/heisenberg.html

Somewhere in the physicsforums discussion I posted above is a reference to Zapper's blog explanation...I'll post it here if I can find it...it is consistent with my prior posts, I think above, and very well done.

Blokhintsev (1968) : “If the wave function were a characteristic of a single particle it would be of interest to perform such a measurement which would allow us to determine its own individual wave function. No such measurement is possible.”

Note carefully ...That seems to be the source of single measurement issues.

Born’s postulate is that the square of the wave function psi represents a probability density function.

The different interpretations of quantum theory are most sharply distinguished by
their interpretations of the concept of state.

(a) The statistical interpretation according to which a pure state and hence a general state provides a description of certain statistical properties of an ensemble of similarly prepared systems, but need not provide a complete description of an individual system,
(b) Interpretations which assert that a pure state provides a complete description of an individual system, e.g., an electron. This second hypothesis is unnecessary for quantum theory and leads to serious difficulties.

[I did not record this source, but it sound like Messiah's QUANTUM MECHANICS:]

Physical systems which have been subjected to the same state preparation will be similar in some of their properties but not all of them. ... So it is natural to assert that a quantum state represents an ensemble of similarly prepared systems, but does not provide a complete description of an individual system...For example, a single scattering experiment consists of shooting a single particle at a target and measuring its angle of scatter. Quantum theory does not deal with such an experiment but rather with the statistical distribution of the results of an ensemble of similar results... The wave function describes not a single scattering particle but an ensemble of similarly accelerated particles. Quantum theory predicts the statistical frequencies of the various angles through which a particle may be scattered.

edit: one more,possibly from Fredrik:

[Whoever wrote this, it's about the clearest and most concise explanation I can find.]

It is possible to measure position and momentum simultaneously…a single measurement of a particle... 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.

edit: that last one is my abbreviated interpretation of a longer post.

 

[Naty1]

phinds:

Naty, I have clearly been mistaken about this...

welcome to the club![LOL]...

 

[Naty1]

Not to beat this to death, but when trying to decipher all this myself, I found different [but consistent] perspectives really helpful.

I read Vanhees link [posted earlier in which he discusses Heisenberg-Robertson uncertainty...

I especially liked this part: [his post happens to be about neutron spins but that's incidental here]

...the entire meaning of the [quantum] states is to provide probabilities for the outcome of measurements……A quantum mechanical state vector thus refers not to properties of single neutron spins but to ensembles of neutrons with equally prepared spins…We do not perform a joint measurement of both observables on one single neutron but do statistics about independently prepared ensembles of such single neutrons performing independent measurements of each observable separately. That's the meaning of the Robertson uncertainty relation: It describes a relation between the uncertainties of two observables…. due to the preparation of a quantum state….. measuring the observables separately on equally prepared ensembles.

which serves as a great introduction to this one [which I finally found in my notes]

PAllen: If you are measuring position and momentum of the 'same thing' at two different times, the measurements are necessarily timelike. The measurements occur at two times on the world line of the thing measured. This order will never change, no matter what the motion of the observer is. If, instead, they occur for the same time on the "thing's" world line, they are simultaneous for the purposes of the uncertainty principle.

to measure a particle's momentum, we need to interact with it via a detector, which localizes the particle. So we actually do a position measurement (to arbitrary precision). Then we calculate the momentum, which requires that we know something else about the position of the particle at an earlier time (perhaps we passed it through a narrow slit). Both of those position measurements, and the measurement of the time interval, can be done to arbitrary precision, so we can calculate the momentum to arbitrary precision. From this you can see that in principle, there is no limitation on how precisely we can measure the momentum and position of a single particle.

Where the HUP comes into play is that if you then repeat the same sequence of arbitrarily precise measurements on a large numbers of identically prepared particles (i.e. particles with the same wave function, or equivalently particles sampled from the same probability distribution), you will find that your momentum measurements are not all identical, but rather form a probability distribution of possible values for the momentum. The width of this measured momentum distribution for many particles is what is limited by the HUP. In other words, the HUP says that the product of the widths of your measured momentum probability distribution, and the position probability distribution associated with your initial wave function, can be no smaller than Planck's constant divided by 4 times pi.