# The Uncertainty Principle **URL:** \ **Category:** Factual Questions\ **Created:** [October 7, 2004, 8:05am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997 "2004-10-07T08:05:46Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![gregongie](https://avatars.discourse-cdn.com/v4/letter/g/919ad9/32.png) [@gregongie](https://boards.straightdope.com/u/gregongie)\ **Post date:** [October 7, 2004, 8:05am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/1 "2004-10-07T08:05:46Z")
We are covering the uncertainty principle in my modern physics class, and I have a few questions regarding it. I think my professor presented it in a rather cumbersome way. Using Fourier analysis, he showed how if you try and make a more narrow probability function for momentum, you get a broad peak for position. While this mathematical treatment is all well and good, I feel like we skipped the lecture on the underlying reasoning of it. So, after a brief web search I found [this article](http://plato.stanford.edu/entries/qt-uncertainty/). It presents the historical development of the uncertainty principle, focusing on views of Heisenburg and Bohr. From what I can tell, Heisenburg’s argument is that at the quantum level, you cannot measure both position and momentum (or energy and time) with arbitrary precision because by taking a measurement you input energy into a system. This, and his argument of observing an electron in a microscope I can follow perfectly well. However, Bohr’s ideas, outlined in the same article, I can’t really make heads or tails out of. For example: > [@](#): > > [Bohr] pointed out that the uncertainties in the experiment did not exclusively arise from the discontinuities but also from the fact that in the experiment we need to take into account both the particle theory and the wave theory.\*\* It is not so much the unknown disturbance which renders the momentum of the electron uncertain but rather the fact that the position and the momentum of the electron cannot be simultaneously defined in this experiment.\*\* (See the “Addition in Proof” to Heisenberg’s paper.) Maybe I missed something, but the article doesn’t seem to explain _why_ they cannot be simultaneously defined. Or is that a meaningless question? If the answer to it is “just because”, I’m suppose I’m willing to accept that, but it seems like I’m missing out on a deeper understanding. So, in summary: How is the uncertainty principle a consequence of wave-particle duality? And how does Bohr’s view of it differ from Heisenburg’s? Thank you in advance. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 7, 2004, 8:33am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/2 "2004-10-07T08:33:04Z")
I’m not sure if this helps, or even if I’m right (great, eh?), but, energy inputs and measurement technologies notwithstanding, I see it like this: In order to absolutely measure the position of an object, you’d have to narrow down the period of observation to a single, dimensionless point of time (if that were possible), but within a zero-duration point in time, movement cannot be observed. ---
**Author:** ![Small\_Clanger](https://avatars.discourse-cdn.com/v4/letter/s/9fc348/32.png) [@Small\_Clanger](https://boards.straightdope.com/u/Small_Clanger)\ **Post date:** [October 7, 2004, 9:34am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/3 "2004-10-07T09:34:41Z")
Whoo hoo! ::sound of a can of worms opening:: > [@gregongie](#): > > you cannot measure both position and momentum (or energy and time) with arbitrary precision It’s not just a case of measurement. It is incorrect to talk of a quantum entity _having_ an exact postition or momentum. I think the Bohr approach was that all you should consider is the result of your experiment or calculation, trying to mentally model what is “really going on” is looking for trouble. > [@](#): > > How is the uncertainty principle a consequence of wave-particle duality? I’m as confused as you are. The “duality” is something you impose on (say) an electron when you measure something about it. It’s not sitting there in the electron making it uncertain about where it is 🙂 As usual I step back and wait to be shot down by a real physicist. Said _real_ physicist will probably just say “Shut up and do the maths”. Quantum theory makes NO SENSE so don’t waste time thinking about it. ---
**Author:** ![muttrox](https://avatars.discourse-cdn.com/v4/letter/m/a8b319/32.png) [@muttrox](https://boards.straightdope.com/u/muttrox)\ **Post date:** [October 7, 2004, 9:35am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/4 "2004-10-07T09:35:15Z")
Heisenberg was wrong about one thing – the uncertainty principle is unrelated to measurement. Measurement is a way of trying to explain a tough concept that many people can understand. However, it holds whether you measure it or not. As for why – there is no why. One of the frustrating things about quantum mechanics is that many of the phenomona are completely counter-intuitive, and there doesn’t seem to be a clear mutually agreed upon way to describe the phenomona outside of mathematics. The Uncertainty Principle is like that – once you get outside the math, the why’s don’t really make sense. ---
**Author:** ![sleestak](https://avatars.discourse-cdn.com/v4/letter/s/919ad9/32.png) [@sleestak](https://boards.straightdope.com/u/sleestak)\ **Post date:** [October 7, 2004, 9:39am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/5 "2004-10-07T09:39:22Z")
I think the issue that Bohr is concerned about is the duality of electrons. IIRC, and I very well could be wrong, electrons someimes behave like a wave and sometime behave like a particle. I know that light does this. The difficulty arise sin trying to pinpoint an exact position for a wave. But I could be wrong. Slee ---
**Author:** ![Dr.Lao](https://avatars.discourse-cdn.com/v4/letter/d/2bfe46/32.png) [@Dr.Lao](https://boards.straightdope.com/u/Dr.Lao)\ **Post date:** [October 7, 2004, 11:05am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/6 "2004-10-07T11:05:29Z")
> [@sleestak](#): > > I think the issue that Bohr is concerned about is the duality of electrons. IIRC, and I very well could be wrong, electrons someimes behave like a wave and sometime behave like a particle. I know that light does this. Actually, all particles have wave functions associated with them. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 7, 2004, 11:42am UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/7 "2004-10-07T11:42:59Z")
> [@muttrox](#): > > Heisenberg was wrong about one thing – the uncertainty principle is unrelated to measurement. Measurement is a way of trying to explain a tough concept that many people can understand. However, it holds whether you measure it or not. > > As for why – there is no why. One of the frustrating things about quantum mechanics is that many of the phenomona are completely counter-intuitive, and there doesn’t seem to be a clear mutually agreed upon way to describe the phenomona outside of mathematics. The Uncertainty Principle is like that – once you get outside the math, the why’s don’t really make sense. That’s wrong, the uncertianty principle is certainly about measuremnt; on one side of the inequality is the product of the root-mean-square deviations of two observables for a set of repeated measurements. Infact quantum mechanics only tells us what happens when we make a measuremnt, it doesn’t tell us what goes on between measuremnets (though not everyone would agree with that). To answer the OP, you have to rmeber that when Heisenberg first derived the HUP all he did was to prove that the unceratinty between postion and momentum was approximately equal to h (in the precise form it is greater than or equal to h-bar/2) and it’s certainly possible to derive this approximate relationship by considering optics and the wave-particle duality of photons (this as you’ve probaly guessed is related to Fourier transforms) there are many simplified versions of Heisenberg’s orginal argument on the web. Unfortunatley I really wouldn’t know which way to go to derive the precise form, except by considering the fundamnetal postulates of QM and Schwarz’s inequality ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 7, 2004, 1:54pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/8 "2004-10-07T13:54:16Z")
> [@gregongie](#): > > I think my professor presented it in a rather cumbersome way. Using Fourier analysis, he showed how if you try and make a more narrow probability function for momentum, you get a broad peak for position. > \ > Maybe I missed something, but the article doesn’t seem to explain _why_ they cannot be simultaneously defined. Because if you try and make a more narrow probability function for momentum, you get a broad peak for position. Really, the functional analysis is the best way of doing it, since it makes no mention whatsoever of what experimental technique is under consideration. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 7, 2004, 2:04pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/9 "2004-10-07T14:04:25Z")
> [@Complex Conjugate](#): > > That’s wrong, the uncertianty principle is certainly about measuremnt; on one side of the inequality is the product of the root-mean-square deviations of two observables for a set of repeated measurements. Infact quantum mechanics only tells us what happens when we make a measuremnt, it doesn’t tell us what goes on between measuremnets (though not everyone would agree with that). This is a fundamental divide between those who interpret the Copenhagen Interpretation ontologically and those who interpret it epistemologically. That’s a Great Debate, not a General Question. > [@](#): > > To answer the OP, you have to rmeber that when Heisenberg first derived the HUP all he did was to prove that the unceratinty between postion and momentum was approximately equal to h (in the precise form it is greater than or equal to h-bar/2) and it’s certainly possible to derive this approximate relationship by considering optics and the wave-particle duality of photons (this as you’ve probaly guessed is related to Fourier transforms) I don’t see where optics enter into it. The Fourier analysis enters into it because the Fourier transform is a change-of-basis in the Hilbert space of states. Expand in eigenstates of position (the operator f -\> x\*f), interpret components as probabilities and calculate the standard deviation of a position observation. Alternately, expand in eigenstates of momentum (f -\> idf/dx) and calculate the standard deviation of a momentum observation. Where does wave/particle enter into it? Simple: particles are position eigenstates (it’s _here_), while plane waves are momentum eigenstates (it’s moving _that way_). ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [October 7, 2004, 2:08pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/10 "2004-10-07T14:08:20Z")
If you consider your particle as a wavefunction, then you can precisely denote its wavelength (which is proportional to momentum), but then you have no idea of the particle location. A sine wave has no unique location in space. If you try to pin down location by describing the particle as a superposition of waves, you can localize it better and better the more terms you add. But each term is a different wavelength, so in the intermediate case you have a “wave packet” that is made up of a mix of wavelengths (and therefore a spread of momentum), and is narrowedc down to a certain region in space, so neither position nor wavelength/momentum are exactly defined. The extreme case is when you have precisely located a position in space (a delta function in position). But in order to construct such a wavefunction you need an infinite range of wavelengths. So if you know exactly where the particle is, you have no idea of its wavelength/momentum. And all of this is independent of any measurement. You simply can’t even construct a theoreetrical waveform which simultaneously has a well-defdined wavelength and position. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [October 7, 2004, 2:10pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/11 "2004-10-07T14:10:12Z")
[It’s a joke.](http://boards.straightdope.com/sdmb/showthread.php?p=5156480&highlight=Heisenberg#post5156480) ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 7, 2004, 3:13pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/12 "2004-10-07T15:13:58Z")
> [@Mathochist](#): > > This is a fundamental divide between those who interpret the Copenhagen Interpretation ontologically and those who interpret it epistemologically. That’s a Great Debate, not a General Question. The quantuj formalism itaself only speaks about the results of mearuemnts it says nothing about what ‘physically’ happens inbetween measurements (of course it does tell us about the time evoltuion of the wavefunction inbetween measuremnts, but the quantum formalism delibrately and sensibly does not attatch any physcial interpretation to the wavefunction), though some would certainlycontend that it is not impossible to say what happens inbetween measurements. > [@](#): > > I don’t see where optics enter into it. The Fourier analysis enters into it because the Fourier transform is a change-of-basis in the Hilbert space of states. Expand in eigenstates of position (the operator f -\> x\*f), interpret components as probabilities and calculate the standard deviation of a position observation. Alternately, expand in eigenstates of momentum (f -\> idf/dx) and calculate the standard deviation of a momentum observation. Optics certainly did entire into it for the original (but imprecise) derivation which was based on physical arguments specifically from optics (see Hesienberg’s _The Physical Principles of the Quantum Theory_ 1930, which includes the orginal physical arguments from his 1927 formulation of the HUP). ---
**Author:** ![John\_Mace](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/john_mace/32/185_2.png) [@John\_Mace](https://boards.straightdope.com/u/John_Mace)\ **Post date:** [October 7, 2004, 3:42pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/13 "2004-10-07T15:42:21Z")
Actually, the Fourier analysis method is the most elegant way to explain it. Since position space maps to momentum space thru a Fourier transform, the HUP drops out of that like a lead brick. What’s the FT of a delta function? QED. I’m not sure it makes sense to talk about “what really happens” as opposed to what the equations predict. If we knew “what really happens” we’d change the equations to reflect that. Our mathematical represtentation is the best approximation we have of “what really happens”. ---
**Author:** ![Small\_Clanger](https://avatars.discourse-cdn.com/v4/letter/s/9fc348/32.png) [@Small\_Clanger](https://boards.straightdope.com/u/Small_Clanger)\ **Post date:** [October 7, 2004, 4:12pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/14 "2004-10-07T16:12:47Z")
> [@John Mace](#): > > Our mathematical represtentation is the best approximation we have of “what really happens”. It was just a matter of time. Please note that I did use scare quotes in my post too. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [October 7, 2004, 4:27pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/15 "2004-10-07T16:27:06Z")
> [@John Mace](#): > > I’m not sure it makes sense to talk about “what really happens” as opposed to what the equations predict. If we knew “what really happens” we’d change the equations to reflect that. Our mathematical represtentation is the best approximation we have of “what really happens”. But unfortunately with QM there is no obvious way to associate the mathematical theory with ‘what really happens’, except when a measuremnt is made. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [October 7, 2004, 4:49pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/16 "2004-10-07T16:49:24Z")
While using position and momentum to try to understand complementary may help to visualize it, there is always a danger that you come to reduce complements to just this one example. You shouldn’t because it’s much more basic than that. There are an infinite number of “cannonically conjugate” qualities that the uncertainly principle represents. Go back to Schrödinger’s orignal statement on the subject as quotes in this article on Quantum Entanglement and Information from the [Stanford Encyclopedia of Philosophy"](http://plato.stanford.edu/entries/qt-entangle/): > [@](#): > > In the original EPR article, two particles are prepared from a source in a certain quantum state and then move apart. There are ‘matching’ correlations between both the positions of the two particles and their momenta: a measurement of either position or momentum on a particular particle will allow the prediction, with certainty, of the outcome of a position measurement or momentum measurement, respectively, on the other particle. These measurements are mutually exclusive: either a position measurement can be performed, or a momentum measurement, but not both simultaneously. Either correlation can be observed, but the subsequent measurement of momentum, say, after establishing the position correlation, will no longer yield any correlation in the momenta of the two particles. It is as if the position measurement disturbs the correlation between the momentum values. The puzzle is that the quantum state of the particle pair is inconsistent with any assignment of precise position and momentum values to the particles separately. These values would be the common cause of the correlations, and would provide an explanation of the correlations in terms of the initial correlations between the properties of the two systems at the source. EPR concluded that the quantum state was incomplete. > > Here is how Schrödinger put the puzzle in the first part of his two-part article (Schrödinger, p. 559): > > \*Yet since I can predict either x1 or p1 without interfering with the system No. 1 and since system No. 1, like a scholar in an examination, cannot possibly know which of the two questions I am going to ask first: it so seems that our scholar is prepared to give the right answer to the first question he is asked, anyhow. Therefore he must know both answers; which is an amazing knowledge; quite irrespective of the fact that after having given his first answer our scholar is invariably so disconcerted or tired out, that all the following answers are ‘wrong.’ \* > > What Schrödinger showed was that if two particles are prepared in a quantum state such that there is a matching correlation between two ‘canonically conjugate’ dynamical quantities — quantities like position and momentum whose values suffice to specify all the properties of a classical system — then there are infinitely many dynamical quantities of the two particles for which there exist similar matching correlations: every function of the canonically conjugate pair of the first particle matches with the same function of the canonically conjugate pair of the second particle. Thus (p. 559) system No. 1 ‘does not only know these two answers but a vast number of others, and that with no mnemotechnical help whatsoever, at least with none that we know of.’ > > Schrödinger coined the term ‘entanglement’ to describe this peculiar connection between quantum systems (Schrödinger, p. 555): > > _When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled._ > > He added (Schrödinger, p. 555): > > \*Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being ‘best possibly known,’ i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself. \* ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 7, 2004, 8:29pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/17 "2004-10-07T20:29:10Z")
> [@Exapno Mapcase](#): > > While using position and momentum to try to understand complementary may help to visualize it, there is always a danger that you come to reduce complements to just this one example. You shouldn’t because it’s much more basic than that. There are an infinite number of “cannonically conjugate” qualities that the uncertainly principle represents. And all of them are related through something akin to a Fourier transform. Yet another score for the “rather cumbersome” way of doing things. ---
**Author:** ![gregongie](https://avatars.discourse-cdn.com/v4/letter/g/919ad9/32.png) [@gregongie](https://boards.straightdope.com/u/gregongie)\ **Post date:** [October 7, 2004, 9:20pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/18 "2004-10-07T21:20:45Z")
Forgive me. I see now that the Fourier transform is a rather elegant way of showing the relationship between two canonically conjugate quantities. I was just stuck with Heisenburg’s original assumption, that the uncertainty principle is a consequence of measurement. I guess there was comfort in the fact that I could visualize it. Anyway, I’m certain now, QM is one big elaborate Zen koan. ---
**Author:** ![InvidiousCourgette](https://avatars.discourse-cdn.com/v4/letter/i/e47c2d/32.png) [@InvidiousCourgette](https://boards.straightdope.com/u/InvidiousCourgette)\ **Post date:** [October 7, 2004, 9:31pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/19 "2004-10-07T21:31:08Z")
[New Scientist](http://www.newscientist.com/hottopics/quantum/inthebeginning.jsp) has a great series of pages about this stuff. ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [October 7, 2004, 9:46pm UTC](https://boards.straightdope.com/t/the-uncertainty-principle/267997/20 "2004-10-07T21:46:46Z")
I have nothing to add, except to point out that **Mathochist** is an apt username. > [@Mathochist](#): > > the Fourier transform is a change-of-basis in the Hilbert space of states. Expand in eigenstates of position (the operator f -\> x\*f), interpret components as probabilities and calculate the standard deviation of a position observation. Alternately, expand in eigenstates of momentum (f -\> idf/dx) and calculate the standard deviation of a momentum observation. [Next page](https://boards.straightdope.com/t/the-uncertainty-principle/267997.md?page=2)