# Is the uncertainty principle a property of elementary particles or a result of our measurement tools?

In many physics divulgation books I've read, this seems to be a commonly accepted point of view (I'm making this quote up, as I don't remember the exact words, but this should give you an idea):

Heisenberg's uncertainty principle is not a result of our lack of proper measurement tools. The fact that we can't precisely know both the position and momentum of an elementary particle is, indeed, a property of the particle itself. It is an intrinsic property of the Universe we live in.

Then this video came out: Heisenberg's Microscope - Sixty Symbols (skip to 2:38, if you're already familiar with the uncertainty principle).

So, correct me if I'm wrong, what we may claim according to the video is:

the only way to measure an elementary particle is to make it interact with another elementary particle: it is therefore incorrect to say that an elementary particle doesn't have a well defined momentum/position before we make our measurement. We cannot access this data (momentum/position) without changing it, therefore it is correct to say that our ignorance about this data is not an intrinsic property of the Universe (but, rather, an important limit of how we can measure it).

Please tell me how can both of the highlighted paragraphs be true or how they should be corrected.

• If a tree falls in the forest, but you never observe or find it, does it have a precise location? – Blackbody Blacklight May 24 '14 at 8:22
• @BlackbodyBlacklight Yes, it does :) – user3459110 May 24 '14 at 18:41
• @AwalGarg Well, in that case, you don't know how fast it's falling! :-D – David Richerby May 26 '14 at 11:08
• @DavidRicherby If I didn't observe it, how would I know? Its a matter of common sense ;) – user3459110 May 27 '14 at 6:04
• Thanks to all of you for your participation in this question/debate. The response has been amazing and I feel honored to have received so many good answers. I hope this will be useful to other people too. Regards :) – Saturnix May 28 '14 at 16:44

The first paragraph is basically right, but I wouldn't ascribe the uncertainty principle to particles, just to the universe/physics in general. You can no more get arbitrarily good, simultaneous measurements of position and momentum (of anything) than you can construct a function with an arbitrarily narrow peak whose Fourier transform is also arbitrarily narrowly peaked. Physics tells us position and momentum are related via the Fourier transform, mathematics places hard limits on them based on this relation.

The second paragraph is used to explain the uncertainty principle all too often, and it is at best misleading, and really more wrong than anything else. To reiterate, uncertainty follows from the mathematical definitions of position and momentum, without consideration for what measurements you might be making. In fact, Bell's theorem tells us that under the hypothesis of locality (things are influenced only by their immediate surroundings, generally presumed to be true throughout physics), you cannot explain quantum mechanics by saying particles have "hidden" properties that can't be measured directly.

This takes some getting used to, but quantum mechanics really is a theory of probability distributions for variables, and as such is richer than classical theories where all quantities have definite, fixed, underlying values, observable or not. See also the Kochen-Specker theorem.

This is really a footnote to Chris' answer but it got a bit long for a comment.

It sounds odd to claim that a particle has no position, but it's easier to understand if you appreciate that a particle is just an excitation in a quantum field. When Heisenberg was developing his ideas physicists still thought of particles as little billiard balls. With the development of quantum field theory we now understand that a particle is just an excitation in a quantum field. For example there is an electron quantum field that pervades all of spacetime. If you add a quantum of energy to this field it appears as an electron. Add a second quantum of energy and you have two electrons, and so on. Similarly, take a quantum of energy out of the field and an electron disappears. Incidentally this also explains how matter can turn into energy and vice versa.

This means the objects we call particles are altogether stranger than Heisenberg thought. They are certainly not tiny billiard balls, and they don't have the properties associated with tiny billiard balls like a precise position and momentum. However, your second paragraph verges on the truth when it points out that when the electron field exchanges energy with something else the exchange takes place at a (reasonably) well defined point, and we can think of this as the position of the electron.

• Is this actually correct, this intuitive field picture ? Or is there more to it than meets the eye and in fact, you can't actually picture a quantum field at all like a classical field ? – user37026 May 25 '14 at 17:02
• @L-L: what specifically are you questioning? Maybe you could post a new question with more detail. I didn't think I had claimed quantum fields were intuitively obvious - they aren't :-) – John Rennie May 25 '14 at 18:12
• So does that mean "we can't precisely know the position of a particle" is intended to mean roughly the same thing as "there is no single point when green becomes red in a color gradient" - the edges are fuzzy? – Izkata May 26 '14 at 3:57
• @Izkata: I'm not sure how well that analogy works. We can't precisely know the position of a particle because it doesn't have a position. – John Rennie May 26 '14 at 10:53

Although the uncertainty principle stems from the mathematical structure of QM, i.e., originates from the noncommutivity of some observable letting them behave as fourier transform pair as explained in another answer, I still think it is a statement on measurements, (i.e., imposes fundamental limits on measurements) since QM itself seems to be a theory of measurements (i.e., not of ontological reality).

I want to quote here the viewpoint of Asher Peres from his precious book, Quantum Theory: Concepts and Methods:

$$\Delta x\,\Delta p\ge \hbar/2\,.\tag{4.54}$$

An uncertainty relation such as (4.54) is not a statement about the accuracy of our measuring instruments. On the contrary, its derivation assumes the existence of perfect instruments (the experimental errors due to common laboratory hardware are usually much larger than these quantum uncertainties). The only correct interpretation of (4.54) is the following: If the same preparation procedure is repeated many times, and is followed either by a measurement of $x$, or by a measurement of $p$, the various results obtained for $x$ and for $p$ have standard deviations, $\Delta\,x$ and $\Delta\,p$, whose product cannot be less than $\hbar/2$. There never is any question here that a measurement of $x$ “disturbs” the value of $p$ and vice-versa, as sometimes claimed. These measurements are indeed incompatible, but they are performed on different particles (all of which were identically prepared) and therefore these measurements cannot disturb each other in any way. The uncertainty relation (4.54), or more generally (4.40), only reflects the intrinsic randomness of the outcomes of quantum tests.

The Heisenberg uncertainty principle is simply an algebraic consequence: the algebra of observables is not commutative $\square$.

• The "badass answer"! Good one. – Fattie May 26 '14 at 5:58

Understanding the uncertainty principle really only involves accepting the idea that, at small scales, elementary particles behave like waves. The uncertainty principle is a well-known property of waves.

One of the consequences of this idea is that position and wavelength cannot be measured to an infinite precision simultaneously with one another. Imagine, first, that you have a wave which looks like a bell curve (which quickly rises and falls around a particular location). If I were to ask you where the wave is, you would quickly point to the peak. However, the wavelength is slightly less well defined, so you could only tell me what the width of the peak is, which is only a rough estimate of a wavelength. Similarly, if you had a periodic wave (a sinusoid, for example). It would be easy to tell me what the wavelength of the wave is by simple inspection, but, since it is infinite in extent, you would be hard pressed to tell me that it has anything that you or I would call a position.

• Your first sentence is a bit dangerous. It could be interpreted the way that on small scales elementary particles can be understood in terms of classical waves. That's of course not true. – taupunkt May 24 '14 at 19:45

the only way to measure an elementary particle is to make it interact with another elementary particle: it is therefore incorrect to say that an elementary particle doesn't have a well defined momentum/position before we make our measurement.

No it is not incorrect to say that "an elementary particle doesn't have a well defined momentum/position" , either describing before or after any measurement, because quantum mechanics is about probabilities. The electron around the nucleus is in an orbital, not an orbit. A probability locus.

We have done this single particle scattering experiment without a microscope over and over again in elementary particle interactions by scattering particles on each other and detecting their track in our chambers.We measure the crossections for the scattering, i.e. how probable it was. We have found that particles do not behave like billiard balls, which is the ultimate expectation of classical scatterings with no uncertainty. In classical mechanics we can go back and find the (x,y,z) to any accuracy also for an equally accurate momentum , not so in the microcosm of particles scattering on each other.

All this microscope business is beside the point, since the Heisenberg Uncertainty has been incorporated in the Quantum Mechanics theory: all observables correspond to operators and for some observables the commutators do not commute, these are the canonical commutation relations. .The predictions of the theory of Quantum Mechanics, which are based on the canonical commutations relations, have been fully validated and are being continuously validated. Quantum mechanics is the underlying framework of nature from which classical mechanics emerges. This Heisenberg microscope is an unnecessary excursion to the time when experimental verification was scarce .

We cannot access this data (momentum/position) without changing it, therefore it is correct to say that our ignorance about this data is not an intrinsic property of the Universe (but, rather, an important limit of how we can measure it).

This is incorrect. In classical mechanics we expect our theoretical formulas to work to any accuracy and thus any uncertainty is attributed to measurement errors. In quantum mechanics our formulas do not predict exact space and time and energy/momentum variables. They predict a probability for finding the particle with the specific value of the variables. It is an intrinsic , part of the theory , property, it has been fully validated and tells us that the underlying framework is probabilistic.

The Heisenberg Uncertainty principle does not need thought experiments since all experiments we do in the microcosm do not deviate from it.

The short answer to your question would probably be that the first interpretation is now accepted by most physicists and the second interpretation is mainly a thought experiment to get a better feel for the implications of the uncertainty principle.

The longer answer might start with the statement that your question is basically the question about the interpretation of quantum mechanics, i.e. the 'meaning' of the wave function. The interpretation of quantum mechanics was heavily debated during the founding years of quantum mechanics and there are still some discussions about it. The discussions between Bohr and Einstein are a classics in physics.

There is a famous thought experiment by Einstein, Podolski and Rosen, the so called Einstein-Podolski-Rosen (EPR) Paradox, that tries to find an experimental situation where you can distinguish between the two interpretations. Einsteins idea was, that quantum mechanics is great to calculate the outcome of experiments, but that it is incomplete, i.e. there must be additional parameters to 'fully' describe the system. Such attempts to further develop quantum mechanics are known as hidden-variable theories.

Such hidden-variable theories do exist, the best known probably is Bohmian Mechanics. In such theories the uncertainty principle is, to my knowledge, indeed 'merely' a result of the measurement process. However, such theories are typically plagued with other serious problems, e.g. it seems notoriously difficult to construct relativistic hidden-variable theories.

In that context you will come across Bell's inequalities. Bell managed to find a way to experimentally distinguish between quantum mechanics and types of hidden-variable theories. Such experiments were all in favour of quantum mechanics. This sets large constraints on any attempt to construct such theories.

Heisenberg's Uncertainty Principle is said to be a conjecture, according to recent updates in the field of Physics. According to Earle Kennard inequality or the generalization of Howard Robertson

The fluctuation exists regardless whether it is measured or not, and the inequality does not say anything about what happens when a measurement is performed.

Ozawa's inequality is the new version of Earle Kennard inequality.

The first paragraph is wrong according to Heisenberg's own version of the principle, and the second paragraph is a consequence of that version, which is also wrong. The truth is, first paragraph is true according to Ozawa's inequality and not according to Heisenberg's own version.

To understand it, read the following extracted paragraph from Scientific American page:

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.

Led by Yuji Hasegawa, the team prepared a stream of neutrons and measured two spin components simultaneously for each, in direct violation of Heisenberg's version of the principle.

Heisenberg uncertainty principle, as stated by Heisenberg himself in his book "Physics and Philosophy", is the property of non-commuting observables/variables to not be able to be measured simultaneously at an arbitrary level of precision and ultimately a property of nature itself.

The uncertainty between momentum and position, for example, is directly analogous to Planck's constant.

And, as other people mentioned, this also manifests in Fourier Transforms as the inability to localize a measurement or signal both in time and frequency domains.

Heisenberg (and the whole Copenhagen interpretation) states exactly that this is a property of nature and not an inability of the experimenter nor of measuring apparratus.

The previous paragraphs are exactly the "prevailing/official" Copenhagen interpretation.

However this is not where all end. One should hold in mind two things:

1. The uncertainty is directly related to the (current) Quantum Formalism (as in the example of Fourier Transforms). So it is possible another formalism will have different aspects regarding this formulation of uncertainty. There are many no-go theorems (like Bell's theorem) about hidden-variables theories etc.. However this is an area still open in many respects (related to uncertainty). Locality and Realism are important aspects as others also.

2. Leaving aside the formalism, the interpretation itself (whether with the current or another formalism) is another point of departure in order to acquire a (another) physical (realist) picture of quantum mechanics. This is not necessarily connected to the 1st point, it can be done independantly or in unison.

In the final analysis, one should be able to understand the difference between a formalism and its relations to (physical) measurement, from the act of interpretation of this formalism (etc..)

Thank you

UPDATE:

One should also keep in mind the "quantum collapse of the wavefunction" during a measurement. After a (quantum) measurement has been made, any further measurement will provide the same result ("quantum superposition" related to some aspects of uncertainty) is removed.

## protected by Qmechanic♦May 24 '14 at 14:17

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