Talk:Uncertainty principle/Archive 5

machine translation
There doesn't seem to be any English translation available except in an outrageously expensive anthology. The following OCR and machine translation may be good enough so that English speakers can puzzle it out:

http://www.scribd.com/doc/142071642/Heisenberg-Uber-den-anschaulichen-Inhalt-der-quantentheoretischen-Kinematik-und-Mechanik ... German (scans)

http://www.scribd.com/doc/142066606/Heisenberg-Uber-den-anschaulichen-Inhalt-der-quantentheoretischen-Kinematik-und-Mechanik ... German (OCR output)

http://www.scribd.com/doc/142068442/Heisenberg-Uber-den-anschaulichen-Inhalt-der-quantentheoretischen-Kinematik-und-Mechanik ... English (machine translation)

--75.83.76.23 (talk) 16:45, 17 May 2013 (UTC)

To motivate the principle?
In the introduction section of this page, the concluding sentence of the first paragraph uses the phrase "to motivate the pricniple". To the general reader this is meaningless, as a slightly knoweldgeble reader I can only guess that this is in reference to some facts not yet stated. Namely, that there are more than just these two ways to interprate the uncertaity principle and that there is a "motivation" for physists to use these two particular interpretations. But I am only guessing here and even the example I have givien is quite an obtuse use of the word. I have not changed this because I am not quite sure what the origanal author was attempting to say. Even if there is a good reason for using "motivate" here, it needs to be re-written for clarity. Dwightboone (talk) 20:18, 12 August 2013 (UTC)


 * The UP is both technical and counterintuitive. Before plunging into the core of the technical argument, the Introduction outlines, as a preamble, the basic motivation of why such a strange principle should be operative at all, and how one could vaguely imagine its necessity and role. As the following paragraphs indicate, in each formulation, the principle has its own logic and function, both quite well motivated, to my mind, and fleshed out technically below. (Of course, in yet other QM formulations, the UP has yet different presentations and motivation! What is covered here are the two best known ones, matrix and wave mechanics. But, then, e.g. in section 4.2, other formulations are covered, as well, which are too technical to be featured earlier in the article.)


 * But, of course, you could propose compelling improvements here. It is not clear if you are seeking more content, or if you misconstrued the established use of "motivate" in this context. I suspect you are keen to help the non-technical reader in Introduction_to_quantum_mechanics, not here. Cuzkatzimhut (talk) 21:02, 12 August 2013 (UTC)


 * Dwightboone brings the following collection of words into question:

"The following attempts to motivate the principle in these two interpretations."


 * It took me several attempts before I could construe the quoted material as a sentence, since I grouped two words together, "following attempts." I think it was supposed to be, to paraphrase, "The following discussion is an attempt to..." What the last part is intended to mean escapes me. I can guess, but only with the feeling of standing on the highest rung of a ladder while grasping a branch blowing in the wind. To me, "to motivate" means "to supply psychological pressure on somebody to get him/her to try to do something." "The CEO motivated senior staff members by offering a substantial bonus if certain goals could be met." The idea of "motive" is not one that applies to physics (unless we go back to Aristotle, perhaps). The writer is apparently trying to say that the discussion to follow will express something about the uncertainty principle as it applies to two interpretations of quantum mechanics. Is it like saying that the following discussions will, metaphorically speaking, give life to the uncertainty principle and show how it functions in relation to the two interpretations? I have no idea. I am only guessing. Good writing is, among other things, writing that avoids making readers come up with guesses about the intended meaning of an essay. P0M (talk) 07:41, 16 August 2013 (UTC)


 * P.S. Could you, Cuzmatzimhut that is, mean "justification" instead of "motivation"? There are reasons why heat needs to be explained in terms of motion. There are justifications for using the idea of atoms and/or molecules moving around to explain heat. Theories and narratives may have reasons behind them, and they may be justified by evidence and logical reasoning, but humans and other animals have motives for doing things. Sometimes humans are motivated to make certain claims in theory, history, law, etc. and they may be motivated to do so by psychological factors, e.g., the love of truth, the love of money, the desire for revenge or self-justification, etc. P0M (talk) 08:04, 16 August 2013 (UTC)
 * I'm against the use of 'motivation'. I can find no definition that would fit the context here, so I suspect it is a misuse of the word. I'm with POM in the use of something like 'justification' - or maybe 'give (good) reason for', or simply 'explain': 'The following attempts to explain the principle under(?) these two interpretations.' Myrvin (talk) 10:34, 16 August 2013 (UTC)

I'm leaving on vacation today, so I cannot be helpful. I supplanted "motivates" by a weak placeholder. I did not introduce that term, but it comports with physics texts, universally, Victorian-sounding as it might be. The OED specifies: "to provide or supply a motive to", as in "Goethe's art was not dramatic [...] he motivates too much for the stage." It is the answer to the question "Why bother with this?". But, indeed, the web is flooded with the meaning you all focus on. The broader meaning that was evidently being used basically amounts to "justify the plausibility of....", and is standard in physics texts: "To motivate the application of the Golden rule in this context, consider...". But if it confuses too many, well, a better substitute might serve. I hope you come up with a better one. Cuzkatzimhut (talk) 11:24, 16 August 2013 (UTC)
 * I am going to have to backtrack on this. Digging further, the OED says:

To provide or serve as a rationale for (some action, etc.); to justify.

Chiefly used in academic, esp. scientific, contexts.

1970  Nature 4 Apr. 44/1   The publisher motivates the slim size of these volumes by claiming it makes them more likely to be read.

1973  Physics Bull. Apr. 234/3  The demand for a relativistically acceptable version of momentum conservation is used to motivate the introduction of relativistic concepts of dynamics.

1988  Linguistics & Lang. Behavior Abstr. Dec. 1556/2  Three structural patterns are empirically motivated & theoretically accounted for.

2000  Speech Communication 32 187   The demiphone is motivated and experimentally supported.


 * So it's not a misuse of the term, especially in scientific contexts. But it is confusing for some - including me. Myrvin (talk) 13:22, 16 August 2013 (UTC)


 * We should be writing so as to minimize the overhead for readers trying to deal with something that is conceptually very difficult to begin with. The whole paragraph is badly written.
 * "The uncertainty principle can be interpreted in either the wave mechanics or matrix mechanics interpretation of quantum mechanics. In wave mechanics the principle is a more visually intuitive interpretation, whereas it was in matrix mechanics where it was first derived and is a more easily generalized interpretation. The following rationalizes the principle in these two interpretations."
 * "UP principle interpreted by interpretation," if we reduce the sentence to the bare bones. It suggests that people interpret things by using another interpretation. "UP principle is an interpretation, whereas it is another interpretation." Principles are not interpretations.
 * What was the writer of this paragraph really trying to say? A "principle" is something that rests on clear observational evidence. People can give an interpretation of the Uncertainty Principle in terms of what its practical consequence are in one context or another. If we begin with a wave picture of the quantum world, then we can show where the indeterminacy comes from by looking at the consequences of trying to shape a wave so that it has a clear location and also of trying to shape a wave so that it has an unambiguous frequency. If we begin with a matrix picture then the indeterminacy comes right out as a consequence of matrix math. So rather than simply stating the above two ideas dogmatically, the article will next demonstrate how uncertainty manifests itself by looking more closely at a wave description and a matrix description.
 * Have I got this right? Or have I made a muddle of my own that doesn't get at what the quoted text is trying to say?P0M (talk) 15:54, 16 August 2013 (UTC)
 * I intended the same meaning as "To provide or serve as a rationale for (some action, etc.); to justify. Chiefly used in academic, esp. scientific, contexts." To some, the uncertainty principle appears to be pulled out of thin air, so the introduction helps to motivate (i.e. justify) the principle.  If you prefer a different wording, then change it yourself.  I see some change has already been made.  In any case, I'm pleased to see the discussion has moved away from people linking to their self-published fringe views to more mundane problems like word choice. Teply (talk) 09:49, 17 August 2013 (UTC)
 * Thank you. I was definitely out of the loop on the meaning of "to motivate" used in this article. Since that understanding is not reflected in the New World Dictionary of the American Language that I have at hand, I have used Teply's most helpful paraphrase as a basis for understanding and reformulating the paragraph in question. The key need seems to be to allay the impression that the uncertainty principle "appears to be pulled out of thin air," so I have tried to work things around to where the uncertainty principle becomes justified to us because it shows itself and its key role in a couple of contexts that we can more easily understand. I hope this way of doing things is suitable. P0M (talk) 17:21, 17 August 2013 (UTC)

IP edit needs looking at
Could someone who understands the relivant equations check if this is legit:

https://en.wikipedia.org/w/index.php?title=Uncertainty_principle&diff=577478786&oldid=577299432

Geni (talk) 21:04, 16 October 2013 (UTC)


 * Looks sensible. Aren't the dimensions soundly matched now?Cuzkatzimhut (talk) 21:27, 16 October 2013 (UTC)

Robertson–Schrödinger uncertainty relation
Is self-adjointness really needed resp. where is it needed? Wouldn't it be fine, when the operators A and B are symmetric, i.e. satisfy "=" for f, g in the domain of A (which in general is something different than self-adjointness). -91.63.245.130 (talk) 02:35, 15 November 2013 (UTC)

References to anti-quantum pseudoscience
Could a kind editor please remove all the nonsense that has accumulated throughout this article about Heisenberg's original principle being misleading and the references to the completely wrong recent papers that claimed to circumvent it? The reason why none of this stuff is taught by the credible textbooks is that all this stuff is completely invalid and there is nothing wrong about the uncertainty principle.

As this article completely fails to explain, the principle is the true conceptual pillar of quantum mechanics, the formulation summarizing the key novelty of quantum mechanics relatively to classical physics, and whoever doesn't understand these basic points is lacking the competence that should be required for editors of articles about quantum mechanics. --Lumidek (talk) 10:29, 15 December 2013 (UTC)


 * This seems to me to be an overly sweeping statement. Perhaps you are just so convinced of the principle's foundational nature that you are not prepared to consider other things as being more fundamental, with the uncertainty relation as an emergent property? To me, even as a layman, it seems obvious that the wavefunction (from which the uncertainly principle can be derived) is far more fundamental. Indeed, I strongly doubt whether any inequality in quantum mechanics can be truly "fundamental" (i.e. not derivable from a more complete underlying model).
 * The lead as it stands seems to present the picture reasonably accurately, showing that care of interpretation is called for, and in particular that an interpretation associating it directly with measurement/observation is flawed. —Quondum 15:42, 15 December 2013 (UTC)


 * It's the very purpose of science to acquire and establish insights – sweeping statements are the best ones – and it is indeed the result of the research into these topics. Quantum mechanics works, it is fundamental, and all the proposed alternatives or attempts to make it non-fundamental have been ruled out. So it's just wrong to write texts in encyclopedias that obscure this scientific fact or that even downright contradict it. It is exactly as wrong as writing that the Earth is flat or that the species were created in 6 days. --Lumidek (talk) 17:57, 15 December 2013 (UTC)

QM is about expectations, and so is HUP
This article makes a serious error right up front. The principle does not apply to a single particle. It only applies to an ensemble. The relationship can only be derived by considering the standard deviations of multiple states. This is either multiple particles or the same particle multiple times. It is simple to show that Quantum Mechanics has nothing to say about a single particle. This thought experiment, for instance, is due to de la Pena: prepare a beam of electrons with precise momentum; this can be as precise as you like; then measure the position of one of the electrons as accurately as you like. You can easily beat the HUP relation. Jschlesinger (talk) 15:23, 16 July 2013 (UTC)


 * I moved this new discussion of Jschlesinger to the bottom of the page, here, as invited by the section heading. Please, try to stick to the house rules.


 * There is no serious error, as far as people with a minimum of physics background are concerned, and it is patently untrue that "Quantum Mechanics has nothing to say about a single particle". It has a lot to say about the probability of features of a single particle, normally probed and verified statistically by a population of similar particles. This is, of course, the point of quantum mechanics, namely that quantitative characterization of a particle's properties ("known" in the article's preamble) involves mere probabilities. The article clearly states right up front that the HUP is a statement on probabilistic expectations. How would you propose to improve the phrasing, in a meaningful way? (I adduced De La Peña's article in the mangled refs of stochastic quantization, but any mumbling in that direction would do this article, HUP, a major dis-service.)


 * A lot of good it would do to one to have accurate measurements that could not be predicted or understood. QM takes you all the way to the collapse of the wave-function, but no further, which is where you insinuate it could go (?).  You certainly don't learn much new about a particle's QM properties after the collapse of its wavefunction, beyond making sure that the wf contemplated/calculated was, indeed,  the correct one—and normally the wf is probed statistically by a population of several particles in a repetitive collapse. Cuzkatzimhut (talk) 18:53, 16 July 2013 (UTC)


 * I think the main thing the OP is forgetting is that a "beam" of electrons (or photons, or anything else for that matter) does not have 100% uniformity with regard to all of its constituent particles. So while you would know the average momentum of all the electrons in the beam, you would not know the exact momentum of any single electron in the beam without making a separate measurement. — Preceding unsigned comment added by Hatster301 (talk • contribs) 12:35, 17 July 2013 (UTC)
 * You would think that, wouldn't you, forgetting the premise of the discussion, that the probability of momenta outside a range of that beam has already been established to be small. So, with overwhelming probability, the momentum of the individual particle observed (with virtually infinite precision in position) is pretty well known... The UP is a statement of probabilities of a single particle. I might as well quote directly from §3 of Dirac's classic, "The Principles of Quantum Mechanics", p.9:

"...the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. ... The new theory, which connects the wave function with probabilities for one photon, gets over the difficultay by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs." Cuzkatzimhut (talk) 08:05, 13 November 2013 (UTC)


 * This (OP's statement) isn't true, as others have already responded. I think whether the relationship between HUP and other aspects of QM (like wave-particle duality / self-interference) is fundamental or not is still open to argument, as is the role of expectation; but if you think QM has nothing to do with individual particles I'd be very happy to place a large wager at scant odds on a single measurement (free money is as useful to me as it is instructive to the person losing it). TricksterWolf (talk) 22:26, 23 December 2013 (UTC)

"known" vs "measured"
There's a common misconception among laypeople that the uncertainty principle is about human consciousness. The principle prevents humans from "knowing" certain attributes simultaneously. I'd like to suggest something similar to this edit, but I'd like to open the discussion generally just in case there's an even clearer way to avoid this issue.

> certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously

> certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be measured simultaneously

I've also seen "known in principle", to distinguish "known by someone" from "potentially knowable by anyone".

Sorenr (talk) 16:30, 1 March 2014 (UTC)


 * I suspect your discussion belongs to a physics forum, instead. The third paragraph already explains why "measurement" is deprecated, as it might lead to confusion about ingenuity of technology. "Knowing" in the technical sense has little to do with human consciousness and all this nonsense. In physics parlance, "known" means inferred by experimental means, direct or indirect, by anyone and everyone. This is an article on physics (if not Fourier analysis)  and not philosophy. The last thing this article needs here is flakey disquisitions on whether the moon is there when a mouse looks at it and all that. I would very strongly oppose the changes proposed, even though "in principle"s might  help. It might be helpful if you took a closer look at the unproductive circular discussions on this, archived in scary voluminousness, raging for years and resulting in repetitive damage.  Cuzkatzimhut (talk) 17:31, 1 March 2014 (UTC)

basic probability theory rules that i think should be taking into account
I thought about some anomaly stuff that exists in the basics of probability theory, that could have some implication in QM: Max Born probabilistic interpretation and Uncertainty principle. And it will insert some set theory based math to physics( measure theory which is basis of modern probability is based on set theory).

QM physics and any statistical theory presume strong law of large number holds all the time ( when n->inf average=mean ) But The strong law of large number holds only when the expected value of probability density function converges surely ( by the mean of Lebesgue integration), there are many probability density function that either the expected value or second moment does not hold this condition( they can converge by other types if integral such as improper reiman or gauge integral ( see here the status on integration definition in math http://www.math.vanderbilt.edu/~schectex/ccc/gauge/ gauge integral also has some connection with QM path integration)

so if some wave function has no first or second moment according to Lebesgue integration then what is the SD and expected value of position for example?

And if there is no such wave functions ( I don’t think it’s true because Cauchy/Lorentzian distribution is in use now), then such limiting conditions should be taken into account. ( adding to the demand that integral |Psy|^2 dx =1, integral |x|*|Psy|^2 dx < inf and integral x^2*|Psy|^2 dx < inf ) — Preceding unsigned comment added by Itaijj (talk • contribs) 12:56, 19 April 2014 (UTC)

Odd comma
Is the comma in the equation in box in the lead significant? It looks like it does something to the 2. Myrvin (talk) 13:56, 19 April 2014 (UTC)

I see that, later on, it isn't there. I've removed it. Myrvin (talk) 14:10, 19 April 2014 (UTC)
 * The comma is not part of the formula, but part of the sentence, and according to MOS:MATH, it should be there. I agree that it looks weird with the comma inside the box, but to avoid that I'd rather suggest to drop the box. The formula is outstanding enough as the only displayed formula in a long block of text, so the frame only makes it look a bit like an infobox or image, as if it was not really part of the text flow. &mdash;&thinsp; H HHIPPO  22:21, 19 April 2014 (UTC)
 * The original comma was too close to the 2, but this has now been fixed. It is not weird. It is only weird to readers who are unused to the formulas-are-text convention, subverted by poor editing. The box is essential. There are only a few truly important formulas in the article, and they should be boxed and otherwise highlighted.  Ideally, a reader looks at those first and appreciates what is involved, and decides whether to read anything else at all, or leave. Cuzkatzimhut (talk) 00:38, 20 April 2014 (UTC)
 * That works only for readers who can understand the formulae without reading the text, I'm not so sure that's a typical situation. The elements that should give you an overview of what the article is about are the lede and the TOC, not some boxes further down. But never mind, I'm happy to keep the boxes if everybody else likes them. &mdash;&thinsp; H HHIPPO  07:27, 20 April 2014 (UTC)

Masanao Ozawa
Hasn't the uncertainty principle recently been reformulated by Masanao Ozawa? See http://arxiv.org/abs/1402.5601

In which case shouldn't the reformulation be stated at the top of the article very clearly?88.203.90.14 (talk) 21:54, 15 May 2014 (UTC)


 * Please see section 4.3. The coverage is appropriate and in a suitable position. It is a technical sideshow for experts and certainly has no conceptual impact on the way one understands quantization. Cuzkatzimhut (talk) 00:17, 16 May 2014 (UTC)

Proof of the Kennard inequality using wave mechanics confusion
The proof part with the integration by parts in confusing because of the the two x's that are used are different, I suggest Chi for one of them. Also, should we include a note that tells the reader that the last step is via Dirac-deltas? Andy Jiang (talk) 04:32, 24 September 2014 (UTC)


 * ??? Appears like standard textbook material and standard Fourier analysis. What do you propose, explicitly, to make it better? what two x's are you referring to?Cuzkatzimhut (talk) 14:53, 24 September 2014 (UTC)


 * Unless I'm mistaken, one of the x's is for the function g as input and the other is a dummy variable. Otherwise, in the second last step, it would look as if you could cancel terms. I changed it now, tell me if you agree.Andy Jiang (talk) 23:20, 24 September 2014 (UTC)


 * I see your point.Agreed. The dummy variable, integrated over, should indeed be something different. The eye slid over the dual use all too easily. Constructive move. Cuzkatzimhut (talk) 23:30, 24 September 2014 (UTC)

Question about your diagram of Heisenberg's microscope: How are the incoming photons directed?
Does your diagram of the Heisenberg microscope have the incoming light coming from the right direction? See --guyvan52 (talk) 22:44, 15 February 2015 (UTC)
 * File:Heisenberg_gamma_ray_microscope.svg
 * File:Heisenberg microscope with wavefronts and electron scatter.svg
 * Talk:Heisenberg's_microscope

The uncertainty principle as a force?
I recently had a discussion at the Science Refdesk about white dwarfs. It seems - as our article on electron degeneracy pressure sort of explains - that since electrons are subject to the Pauli exclusion principle, and measuring the mass of the star means we know it has n electrons, and we know each of those states is distinguishable, which is to say, the difference between them can be measured, it follows that the actual distribution of the electron momenta is forced to have values that differ by the amount of the uncertainty. There are some details that are less clear, like why the math at the link above is based on h whereas here it would seem to be h/4 pi - though there we are speaking of the actual allowed difference between particles and here we're speaking of the standard deviation...

My understanding is incomplete, but it seems amazing that the uncertainty principle can apparently hold up the surface of a star. It would be really nice if someone can go through it here, and if possible list other examples where the principle is doing something more 'active' than just blunting our measurements. Wnt (talk) 02:36, 26 March 2015 (UTC)


 * I only causally watch this page, but the "experts" are likely to find that this conversation takes us too far into quantum mechanics proper. Loosely speaking, the uncertainty principle requires a higher uncertainty in the momentum, and hence more kinetic energy if the white dwarf's electrons if the star is too small.  You can "guess" a lot of truth from the equation $$\Delta x\Delta p \approx \hbar$$, but such insights might not belong in this Wikipedia article.  Someday we will get to this on Wikiversity, but right now I am up to my ears in freshman-level concepts.--Guy vandegrift (talk) 05:01, 26 March 2015 (UTC)


 * Indeed, you should not be on this talk page, and I disagree that the article needs any mention of pseudo-forces, or Pauli pressures, which, at the very worst, might belong to de Broglie wavelength, or properly, Exchange force, or even  Degenerate matter. I suspect you are misusing the precise term "uncertainty principle" as a loose placeholder term for the inverse linkage between wavelengths and momenta in QM Fourier analysis. Cuzkatzimhut (talk) 10:44, 26 March 2015 (UTC)
 * The article on electron degeneracy pressure mentions the Heisenberg uncertainty principle already, and redlinks "Heisenberg speed"... it's just really confusing about it. Can you make that text clearer without introducing any technical inaccuracy that I might? Wnt (talk) 23:49, 27 March 2015 (UTC)


 * Apologies, I don't have the time to do a half-decent job. As it stands, the paragraph is meaningless, and the bizarre "unfunded mandate" redlink on "Heisenberg velocity" sheer flakey nonsense. Frankly, that stub should not exist, but, instead, be a more organized section of the main article, degenerate matter. The Pauli exclusion principle and the many-body exchange interactions are far more relevant, but of course, as always in Fourier analysis, there are inverse correlations in position and momentum spreads in these statistical multiparticle distributions quite analogous to what is being discussed in this, UP, article for one particle. I can't easy salvage a flawed paragraph, but you might try... Adducing a mainstream reference might help. All I could do is decrimson the hapless "Heisenberg velocity" bluff and link to the main article.   Unfortunately, the main article itself is flawed when it comes to the hand-waving poetry of the UP, cf Talk:Degenerate matter/Archive 1, a conversation I would rather not get drawn into.... Of course, I am broadly siding with the critics there...  Cuzkatzimhut (talk) 01:27, 28 March 2015 (UTC)

Number of states as 2σxσp/ħ ?
Is the two in the denominator represent the fact that p can be plus or minus? Can I just multiply by the std deviation percentage &sigma;=0.341 squared to count states in a given x*p like this below? If the number of states I always here about is this, then it would be worth mentioning on the page.

number states $$ = \frac{(\sigma_x)\cdot(2\sigma_p)}{\hbar}$$ ?

I was able to derive the entropy equation from using Shannon's H for information, but it required the above to be true so that I could count states in a square box instead of spherical phase space volume of states. Ywaz (talk) 12:45, 28 January 2016 (UTC)


 * Hi. I do not think so.   The Fourier transform of a wave packet gives
 * $$\Delta x \, \Delta k \ge 1$$
 * The factor 2 comes from Kennard. He found that, for a gaussian wave packet,
 * $$\Delta x = \sqrt{2} \, \sigma_x$$
 * $$\Delta k = \sqrt{2} \, \sigma_k$$
 * Therefore
 * $$\sigma_x \, \sigma_k \ge 1/2$$
 * Since
 * $$\sigma_p = \hbar \, \sigma_k $$
 * Kennard finally obtained the inequality:
 * $$\sigma_x \, \sigma_p \ge \hbar/2$$


 * Stiglich (talk) 13:42, 29 January 2016 (UTC)

Systematic and statistical errors: adding the Fujikawa relation to the Ozawa one

 * PRESENT VERSION:

Systematic error
The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation. Heisenberg's original version, however, was interested in systematic error, incurred by a disturbance of a quantum system by the measuring apparatus, i.e., an observer effect. If we let $$\epsilon_{\mathcal{O}}$$ represent the error (i.e., accuracy) of a measurement of an observable $$\mathcal{O}$$ and $$\eta_{\mathcal{O}}$$ represent its disturbance by the measurement process, then the following inequality holds:

In fact, Heisenberg's uncertainty principle as originally described in the 1927 formulation mentions only the first term. Applying the notation above to Heisenberg's position–momentum relation, Heisenberg's argument could be rewritten as
 * $$\cancel{\epsilon_x \eta_p \sim \frac{\hbar}{2}} \, \, $$ (Heisenberg).

Such a formulation is both mathematically incorrect and experimentally refuted. It is also possible to derive a similar uncertainty relation combining both the statistical and systematic error components. Nevertheless, with sufficient care, Heisenberg's intuitive observation may be formulated and proven in a mathematically consistent manner.


 * PROPOSED MODIFICATIONS:

Systematic and statistical errors
The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation $$\sigma$$. Heisenberg's original version, however, was interested in systematic error, incurred by a disturbance of a quantum system by the measuring apparatus, i.e., an observer effect.

If we let $$\epsilon_A$$ represent the error (i.e., inaccuracy) of a measurement of an observable A and $$\eta_B$$ the disturbance produced on a subsequent measurement of the conjugate variable B by the former measurement of A, then the inequality proposed by Ozawa — encompassing both systematic and statistical errors — holds:

In fact, Heisenberg's uncertainty principle as originally described in the 1927 formulation mentions only the first term. Applying the notation above to Heisenberg's position–momentum relation, Heisenberg's argument could be rewritten as
 * $$\cancel{\epsilon_x \, \eta_p \,\simeq\, \frac{\hbar}{2}} \, \, $$ (Heisenberg).

Such a formulation is both mathematically incorrect and experimentally refuted. Nevertheless, with sufficient care, Heisenberg's intuitive observation — regarding the systematic error alone — may be formulated and proven in a mathematically consistent manner.

It is also possible to derive an uncertainty relation that, as the Ozawa's one, combines both the statistical and systematic error components, but keeps a form very close to the Heisenberg original inequality. By adding Robertson and Ozawa relations we have
 * $$\epsilon_A \eta_B + \epsilon_A \sigma_B + \sigma_A \eta_B + \sigma_{A}\sigma_{B} \geq \left|\langle[\hat{A},\hat{B}]\rangle \right| .$$

The four terms can be written as:
 * $$(\epsilon_A + \sigma_A) (\eta_B + \sigma_B) \geq \left|\langle[\hat{A},\hat{B}]\rangle \right| .$$

Defining:
 * $$\bar \epsilon_A \equiv (\epsilon_A + \sigma_A)$$

as the inaccuracy in the measured values of the variable A and
 * $$\bar \eta_B \equiv (\eta_B + \sigma_B)$$

as the resulting fluctuation in the conjugate variable B, Fujikawa established an uncertainty relation similar to the Heisenberg original one, but valid both for systematic and statistical errors:


 * Stiglich (talk) 00:06, 24 January 2016 (UTC)

After a week with no critical remarks, I'm going to implement the proposed modifications in the Wiki page.

Stiglich (talk) 17:31, 31 January 2016 (UTC)

Systematic and statistical errors: adding Heisenberg two experimental situations with moder notation

 * PRESENT VERSION:

In fact, Heisenberg's uncertainty principle as originally described in the 1927 formulation mentions only the first term. Applying the notation above to Heisenberg's position–momentum relation, Heisenberg's argument could be rewritten as
 * $$\cancel{\epsilon_x \, \eta_p \,\simeq\, \frac{\hbar}{2}} \, \, $$ (Heisenberg).

Such a formulation is both mathematically incorrect and experimentally refuted. Nevertheless, with sufficient care, Heisenberg's intuitive observation — regarding the systematic error alone — may be formulated and proven in a mathematically consistent manner.


 * PROPOSED MODIFICATIONS:

Heisenberg uncertainty principle, as originally described in the 1927 formulation, mentions only the first term of Ozawa inequality, regarding the systematic error. Using the notation above to describe the error/disturbance effect of sequential measurements (first A, then B), it could be written as

The formal derivation of Heisenberg relation is possible but far from intuitive. It was not proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years. Also, it must be stressed that the Heisenberg formulation is not taking into account the intrinsic statistical errors $$\sigma_A$$ and $$\sigma_B$$. There is increasing experimental evidence that the total quantum uncertainty cannot be described by the Heisenberg term alone, but requires the presence of all the 3 terms of the Ozawa inequality. Using the same formalism, it is also possible to introduce the other kind of physical situation, often confused with the previous one, namely the case of simultaneous measurements (A and B at the same time):

The two simultaneous measurements on A and B are necessarily unsharp or weak.


 * Stiglich (talk) 17:49, 12 February 2016 (UTC)

After one week with no critical remarks, I'm going to implement the proposed modifications in the Wiki page.

Stiglich (talk) 00:33, 19 February 2016 (UTC)

Is it correct to use Kennard inequality to introduce Heisenberg's one ?
I have the impression it's misleading to use Kennard inequality

to introduce Heisenberg's one. They relate to different kinds of quantum uncertainty: Systematic errors (Heisenberg); statistical errors (Kennard).

Also, the way to derive Kennard inequality is straightforward from Schwarz inequality in Hilbert spaces. On the opposite, the derivation of Heisenberg relation is possible but far from intuitive. It was not proposed by Heisenberg, but formulated in a mathematically consistent way only in recent years.

I suggest to use a modern formulation of Heisenberg inequality as a starting point for the article, namely:

for the error/disturbance correlation of sequential measurements (x first, then p).

With the proposed formulation, it's possible to introduce right away the other kind of physical situation, often confused with the previous one, namely the case of simultaneous measurements (x and p at the same time):

At that point, the explanation of the difference between systematic and statistical errors would naturally lead to the Kennard inequality:

Stiglich (talk) 23:38, 11 February 2016 (UTC)


 * I would disagree: Most textbooks and physics education venues use what you call the "statistical" definition as Heisenberg's relation, regardless of history or the finer conceptual points covered further down in the article. Quantum theory is a statistical theory ---a direct quote from Moyal, the "owner" of the racket! Practicing physicists actually using the relation and not perorating on "meanings" use Kennard--and, in fact, the entropic uncertainty further down. It hardly matters to them who found it and how, for the purposes of the top half. They need the stuff.


 * These should dominate the article,which is not really a forum of settling finer points, so it would be a terrible idea to confuse the novice, for whom this is written right up front. ... Have you followed the raging wars in the past discussions on this very page? I would adamantly believe in starting from Kennard and leaving any ε talk to much much later in the article, assuming the hapless reader has not been chased away! Cuzkatzimhut (talk) 23:57, 11 February 2016 (UTC)

Along with Cuzkatzimhut, I also completely disagree. In logical accordance with the points made, these are not "finer points" -- it is simply making up a contrived historical narrative that did not previously exist. Wikibearwithme (talk) 19:28, 27 August 2016 (UTC)

Acceptable viewpoint. Thanks. Stiglich (talk) 16:21, 12 February 2016 (UTC)


 * Thanks for your understanding and good work on section 4.3. I hope I did not come on too strongly, but the article has had a turbulent and cyclic history of rewrites, as you can see from the archived discussions, until it stabilized here... Just a thought: However nice it may be to have "everything under one roof" here, if you were an enthusiast of section 4.3 issues, you might start a separate stub on them, overlapping them, and also linked here, of course....  Cuzkatzimhut (talk) 16:50, 12 February 2016 (UTC)

Hi. Do not worry. I used to write on Wikipedia in another language. By comparison, this is a gentlemen's British club... Stiglich (talk) 00:28, 15 February 2016 (UTC)

Does The Heisenberg Uncertainty Principle Imply that Spacetime is Quantised?
This may prove to be far from a simple question, and potentially a stupid one. The Uncertainty Principle implies that we can only know a particles position with a certain accuracy (which has to be traded off with knowledge of the particle's momentum). Does this imply that the particles position (and hence space in general) is itself quantised?

I ask this question, as the issue of whether the Heisenberg Principle implies that Space is Discretised might Possibly be of importance when considering the Weyl Tiling Argument.

It seems to me that the standard deviation COULD be associated with a probability density of something (which, presumably, is what the position & momentum ARE - some probability density, classically this would be the Dirac Delta function, but I'm not too sure in quantum mechanics). ASavantDude (talk) 15:18, 5 September 2016 (UTC)

historical "revisionists" have compromised this article with new-age fluff
The statements about the original uncertainty principle, being, infact, an "observer effect" is absolute bunk. Certain editors are using this article as an excuse for promoting their own self-serving (and self-deluded) versions of history (note all the contemporary - 2012, etc- papers that purportedly change everything).

The uncertainty principle, per se, as a statistical uncertainty was well-known before Heisenberg. The quantum uncertainty principle, as a fundamental physical (real) constraint in quantum systems, was then absolutely REQUIRED, by first-principle derivation, simply by the assumed validity of Planck's constant (it falls right out of basic wave mechanics - such first principles derivations have nothing to do with the experimental data-treatment concept of "standard deviation," though someone here would apparently like to reinvent that concept, as well, to fit their narrative). Heisenberg merely noted this already apparent reality (Dirac correspondence refers to it beforehand) by formalizing it and coining the phrase.

If, in contrast, somebody wants to argue that the entire physics community has not been thinking clearly on this for the last century, then this great new pioneer should find their own stub on which to preach their new version of history. Wikibearwithme (talk) 06:30, 27 August 2016 (UTC)


 * Your point is not altogether meritless, but venting is rarely salutary: propose, instead, specific, preferably incremental, changes that would help and pass muster. Cuzkatzimhut (talk) 15:52, 20 September 2016 (UTC)

Atomic vibration
Is there possible to track the atoms in vibration. I think according Heisenberg's uncertainty principle, we unable to track the motion of Atom in vibration. What are you thinking about this ? HARIPRASHAD RAVIKUMAR (talk) 12:22, 8 October 2016 (UTC)

Landau-Pollak relation?
Should I add the Landau-Pollak relation, which caps the probability of localization to a region and thus bars the the uncertainty from being mostly pushed out to improbable but faraway tails, to the article? Or is Landau-Pollak already subsumed by one of the listed relations, or otherwise not an important aspect of the uncertainty principle? Rolf H Nelson (talk) 15:41, 30 December 2016 (UTC)


 * You might start an independent stub or an article (like entropic uncertainty), and only if it climbed up in quality and grew  clear or compelling, link it or excerpt it here, like the entry uncertainty. Cuzkatzimhut (talk) 16:00, 30 December 2016 (UTC)

The principle of uncertainty at the Planck scale.
The article should be mentioned on the ratio of uncertainties $$\sigma_{r_s}\sigma_{r}\ge\ell^2_{P}$$, where $$r_s$$ - the gravitational radius, $$r$$ - radial coordinate, $$\ell_{P}$$ - Planck length. This is another form of Heisenberg's uncertainty principle between the momentum and the coordinate at the Planck scale. Indeed, this relation can be written as follows: $$\sigma_{(2GM/c^2)}\sigma_{r}\ge G\hbar /c^3$$, where $$G$$ - gravitational constant, $$M$$ - body mass, $$c$$ - speed of light, the $$\hbar$$ - Dirac's constant. Cutting left and right of the same constants, we arrive at the relation of the Heisenberg uncertainty $$\sigma_{(M c)}\sigma_{r}\ge\hbar/2$$. Installed uncertainty relation predicts the emergence of virtual black holes (quantum foam) at the Planck scale.

178.120.122.41 (talk) 10:52, 4 March 2017 (UTC)
 * I agree it should be mentioned. Try to find published sources that discuss this. El_C 10:58, 4 March 2017 (UTC)
 * With a a strong emphasis on reputable sources. The discussion is normally a warren of fringe science. Cuzkatzimhut (talk) 11:59, 4 March 2017 (UTC)
 * I've seen it here and here and here. 178.120.122.41 (talk) 14:02, 4 March 2017 (UTC)

I am relieved to notice that @User:‎Isambard Kingdom reverted this borderline COI contri. The OP's strategic self-reversion with Minsk IP succor is something I have not encountered yet, but it smacks of creativity in mooting the 3R rule. Personally, I am not as keen as I should be to fuss about the marginal Virtual black holes here, but showcasing this stuff on a high traffic and high quality article such as this is too much. A quick review of this here (UP) article's history is a reminder of incessant, unrelenting,  attempts to showcase marginal projects. Cuzkatzimhut (talk) 17:49, 11 March 2017 (UTC) As evidenced by today's unsocked activity, the campaign to self-promote the content of the virtual black hole article has not abated. The educated reader would immediately appreciate there are literally dozens of such rough-and-ready trivial applications of the UP to the Planck scale context, and choosing this one as an illustration of such applications is not exactly an informative act. Cuzkatzimhut (talk) 14:36, 17 March 2017 (UTC)
 * Stephen Hawking put forward the concept of virtual black holes (1995). Do you think that Hawking is a marginal? 178.120.93.51 (talk) 05:28, 20 March 2017 (UTC)


 * Tendentious. Hawking is not arguing in support of this marginal addition .Cuzkatzimhut (talk) 10:17, 20 March 2017 (UTC) In particular, capturing the eyeballs of this article's readers for self promotion is highly inappropriate.  I tried to stay out of the flagrant misuse of the virtual black holes article, but evidently this was misconstrued. Please heed the not a forum spec at the top of this very page. Cuzkatzimhut (talk) 13:59, 20 March 2017 (UTC)

In accordance with the uncertainty relation $$\sigma_{r_s}\sigma_{r}\ge\ell^2_{P}$$ or $$r_s\sim\ell^2_{P}/r$$, the metric tensor in the Schwarzschild solution (on the Planck scale) has the form $$g_{00}=1-r_s/r\approx 1-\ell^2_P/r^2\approx 1-\sigma_g$$, where $$\sigma_g\sim\ell^2_P/r^2$$ are the fluctuations of the metric tensor.


 * "For the space-time region with size $$r$$, the uncertainty of the metric tensor is of the order of $$\sigma_g\sim\ell^2_P/r^2$$", (see Regge T., Gravitational fields and Quantum mechanics, Nuovo Gimento, 7, 215 1958).


 * The equation for the fluctuations of the metric tensor $$\sigma_g$$ agrees with the Bohr-Rosenfeld uncertainty relations $$\sigma_g(\sigma_r)^2\ge 2\ell^2_P$$ or $$\sigma_g\sim 2\ell^2_P/r^2$$, (see Hans-Jürgen Treder, Ansprachen und Vorträge auf den Festveranstaltungen des Einstein-Komiees bei Akademie der Wissenschaften, vom 28.2. bis 2.3, 1979 in Berlin) 178.120.141.104 (talk) 08:31, 21 March 2017 (UTC)

You appear confused. My point is not that the obvious relation does not hold. It is that it is one of hundreds of evident applications of the uncertainty principle--God knows there is a plethora in atomic and condensed matter, nuclear, particle, physics, etc... They need not be listed in this article. This is absolutely the wrong venue for this. The reader has no need for it, and has to be protected from it. This is not a promotional soapbox for endless specialized side-discussions. Cuzkatzimhut (talk) 16:13, 21 March 2017 (UTC)

What's a "timelimited signal"?
This article mentions that "a function cannot be both time limited and band limited", but "time limited" is currently referencing an article about the concept of deadline, which makes no sense. — Preceding unsigned comment added by 187.39.123.82 (talk) 20:18, 16 June 2017 (UTC)


 * Band limited means that a signal only has frequencies over a given, finite width, band. The spectral intensity is zero outside. On the other hand, time limited means it is only non-zero over a finite time interval. There is a not so obvious exception for periodic signals, which have similar properties to time limited signals. Note that the (time vs. frequency) Fourier transform has an integral from t=-infinity to t=+infinity. This question comes up more often in terms of sampling theory and Nyquist sampling, but it also applies here. Gah4 (talk) 16:30, 24 June 2017 (UTC)

Observer effect vs Heisenberg uncertainty principle
I came to this page to remind myself of the precise articulation of the HUP, and was quite intrigued to read on the page that it is distinct from, and often confused with, the observer effect in physics. However, in checking the first reference provided for that statement (the Sci Am article), I read: "Yet the uncertainty principle comes in two superficially similar formulations that even many practicing physicists tend to confuse. Werner Heisenberg's own version is that in observing the world, we inevitably disturb it. And that is wrong, as a research team at the Vienna University of Technology has now vividly demonstrated." This would seem to indicate that Heisenberg's "own version" was the observer effect. Can someone knowledgeable in the field clarify (not just for my edification, but in the article since it seems secondary sources might talk about different "versions" of HUP where this article takes a POV that there is one HUP)? And I'd also recommend adding a back-link to HUP in the observer effect article if relevant. Martinp (talk) 13:02, 23 June 2017 (UTC)
 * Things are well-explained in the very first paragraph of this article, to a thoughtful reader; it is quite unclear what you believe is missing. That Sci Am article is a bit inflammatory in its discussion of history. Heisenberg utilized the observer effect to reassure us that the HUP cannot be beat; the first reaction of the outsider, and the physics community at the time is/was: "But, why can't I beat the HUP? Surely, I'll just....". Cuzkatzimhut (talk) 13:16, 23 June 2017 (UTC)
 * Cuzkatzimhut, thanks for the reply. Apologies, I seem to have stepped on some toes in how I formulated my question, not intended. Speaking in layman's terms (since I last studied any physics 25 years ago, I'm afraid I can't do any better...), I'm familiar with a formulation of HUP, in the lead of the article, as the simultaneous unknowability of position and momentum, or more generally/loosely precise present state as well as a precise prediction of future motion. This is stated more precisely via the Kennard inequality, also right at the start of the article. I've also heard HUP stated as "observing the world necessarily disturbs it", a version which nonphysicists are fond of quoting as a philosophical observation of greater applicability, e.g. in social systems. The implication of the 3rd para of this article is that this latter formulation is mislabeled as being HUP, that it is actually something else, namely the observer effect, which merely "has been confused with it". Being intrigued by this, I clicked the first reference, the SciAm article, and it appears to take a broader terminological approach, namely that there are 2 formulations of HUP, namely Kennard's and "Heisenberg's original formulation", which it would seem was the original one.
 * I'm therefore left trying to reconcile terminology. This article, saying "HUP=A. By the way it's confused with A', which is not really HUP but related at the quantum level." Or SciAm, apparently saying "There are 2 formulations of HUP, HUP1=A and HUP2=A' ". In my field, one of the biggest challenges is that terms have both narrow technical and related broader philosophical meanings, and frankly Wikipedia articles in my field are a mishmash of which of these meanings are described. So it set off my spidey senses that something similar may be going on here. Now you or others can reassure me that common practice in physics has become indeed that HUP=A, not A', and the SciAm article happens to in this specific area take a niche/nonmainstream view. Or you can tell me I'm just misunderstanding things. Or it's possible that this article as currently written is taking a specialized, narrow view, which is not wrong, but might make sense to broaden the aperture at the start and reflect if there is a terminological divergence, that some say HUP=Kennard only, and others that HUP=Kennard or observer effect -- if that is indeed the case. That's where I lack sufficient knowledge. Martinp (talk) 23:58, 23 June 2017 (UTC)
 * Yes, one did get a little spooked. If you looked at the voluminous Archived past volumes of this page, this has been a contentious issue with increased unsound intrusions swept out periodically. Indeed, A'="observing the world necessarily disturbs it", is the observer effect in QM, as described in that article, and mindlessly confused with A, and far more mysterious than A. I strongly disagree with the sensationalist and self-serving habit of pinning A' to Heisenberg's somehow misunderstanding his own principle, with forensic chapter-and-verse unsound history of science (note the weasel words "...many physicists, probably including Heisenberg himself, have been under the misapprehension...". WH's original paper is quite hard-nosed and transforms Gaussians, for crying out loud! It is A). A is a basic fact of Fourier analysis, and so the wave nature of matter, which Heisenberg tried to argue for and defend using A'... and was thus posthumously smeared with misunderstanding his own principle. I do wish to reassure you that "common practice in physics has been indeed that HUP=A".  In fact, WH's gedanken-microscope used classical optics, his sad thesis exam Achilles heel, not wavefunction collapse subtleties. QM is weird, and both A and A' are very counterintuitive in their particulars; but more absurd philosophical battles are waged on A' and the problem of measurement, "collapse of the wave function", and interpretations of QM these days than poor technical, unremarkable A, the subject of this article. Cuzkatzimhut (talk) 03:16, 24 June 2017 (UTC)


 * Seems to me that it eventually gets back to Interpretations of quantum mechanics. Some interpretations make it easier to answer a certain kind of question, and in some cases the observer effect is important.  While I believe that the Copenhagen interpretation is now shown to be false (that is, does not always agree with nature), it still works well enough, often enough, to be useful.  (Consider Newtonian mechanics, still useful though proven false by both relativity and QM.)  The observer effect works often enough, and is often easier to show, but uncertainty is still there, even without observers.  At some point, it gets down to the statement of interaction between quantum and classical systems, but there aren't any classical systems. It is easier for us to analyze, to understand, if we assume that there are classical systems, but that is just an approximation. Within that approximation, some explanations work. But in reality, our measuring devices are also quantum systems. Gah4 (talk) 16:41, 24 June 2017 (UTC)


 * This is part of my point. Such discussions have no place in this article, as per not a forum warning on top of this page. They might end up in the article wikilinked above. This is an article about a feature of Fourier analysis that surprised physicists 90 years ago. Not all aspects of QM are still up for conceptual grabs.Cuzkatzimhut (talk) 17:25, 24 June 2017 (UTC)


 * As well as I remember it (from a long time ago), Heisenberg got it from matrix mechanics. Now we know that uncertainty is fundamental to QM, but they didn't know that, as they were still trying to understand it. The observer effect is one way to understand it, and also a common one, and mostly goes along with the Copenhagen interpretation. And yes, the connection to Fourier analysis is more fundamental. Still, there is a quote, I believe from Feynman: nobody understands quantum mechanics.  We do experiments, and come up with interpretations of quantum mechanics, the latter mostly to make it easier for people to understand and explain. I suspect that Heisenberg used the observer effect, like many today, to make it more explainable to others, and maybe even to himself. But that is all it is: a way to explain it. QM would work just fine without observers, and did for many years. Gah4 (talk) 19:01, 19 March 2018 (UTC)

DFT uncertainty principle
The uncertainty principle for the discrete fourier transform is only stated in the article Discrete Fourier transform. I think it should also be included in this article in the section "Harmonic analysis". Fvultier (talk) 09:57, 11 July 2017 (UTC)
 * I would disagree strongly. The DFT uncertainty principle is covered well in that article and showcased prominently in the "also see" section of this one. It is an obvious transcription of the continuous UP of this article, after the discrete transcription rules are established, done in that article. There is no good reason to overburden this already stressed and problematic article with extra tangents and footnotes. This article has repeatedly (cf. the archived discussions) been abused as an "eyeball harvesting" vehicle, and should be defended against "this too, me too" infiltrations. At the very most, you might choose to insert a short sentence in the harmonic analysis section linking to that article, if, for some obscure reason, you imagined the motivated reader would nevertheless fail to read on to find the perfectly noticeable wikilink in the "also see" section. Of course, the DFT article's UP section may be improved substantially.Cuzkatzimhut (talk) 13:06, 11 July 2017 (UTC)
 * I agree that this article contains to much information already. The section about the UP in harmonic analysis however does not state the very simple formula $$N\leq \|x\|_0 \cdot \|X\|_0.$$. Fvultier (talk) 21:02, 11 July 2017 (UTC)
 * It is not that surprising, either. But, if you insisted, a minimal parenthetical one liner with the suitable wikilink and the evocative formula meant to just make people curious, if they were not familiar with it, might be innocuous... Cuzkatzimhut (talk) 21:16, 11 July 2017 (UTC)


 * This is tricky. Heisenberg, as well as I know it, discovered uncertainty with matrix mechanics, which is discrete, but infinite. The DFT, on the other hand, is discrete and finite. For that reason, I believe that it doesn't need to go here, besides that the DFT page describes it well enough. Gah4 (talk) 07:51, 26 March 2018 (UTC)