# What is it about position and momentum that forbids knowing both quantities at once?

1. ### jeebs

326
You are no doubt familiar with Heisenberg's uncertainty principle, putting a limit on the accuracy with which we can measure a particle's position and momentum, $$\Delta x \Delta p \geq \hbar/2$$
On my course I was shown the derivation, it popped out of a few lines of mathematics involving the Cauchy-Riemann inequality.

However, I've been wondering if there is any reason to intuitively expect difficulties when trying to simultaneously know both quantities. What I mean is, is there anything about the nature of "position" and "momentum" that hints that we should not be able to know both simultaneously?

One explanation I heard was that if you, say, bounced a photon off an atom to measure its position, then the recoil would affect its momentum, thus giving rise to the uncertainty - this seems straightforward enough. However, I have also been told that this is apparently not a valid explanation, although I do not understand why.
Can anyone shed any light on this for me?

2. ### CJames

355
Re: what is it about position and momentum that forbids knowing both quantities at on

The intuitive explanation you present is called Heisenberg's microscope. There are two primary problems with it. The first is that it only results in an approximate expression of the equation that you cite.

The second is that it attacks its own premises. The thought experiment first assumes that the electron has a definite location and momentum, and then demonstrates why such a thing can't exist, which invalidates its own premises.

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Re: what is it about position and momentum that forbids knowing both quantities at on

Here is a very general answer: From the axioms of QM and the math that is used to build observables and states of systems, it turns out that position and velocity (and also momentum, because momentum p = mv) are what are called "canonical conjugates", and they cannot be both be "sharply localized". That is, we cannot measure them both to an arbitrary level of precision. It is a mathematical fact that any function and its Fourier transform cannot both be made sharp.

This is a purely a mathematical fact and so has nothing to do with our ability to do experiments or our present-day technology. As long as QM is based on the present mathematical theory, it cannot be done using the mathematics we have.

4. ### eaglelake

128
Re: what is it about position and momentum that forbids knowing both quantities at on

There is nothing in Classical theory that prevents us from knowing both the momentum and the position of a particle with "certainty". i.e. we can repeat the classical experiment many times and always get the same result for momentum and the same result for position.

But in quantum mechanics, position and momentum are linear operators in a Hilbert space and, most importantly, the momentum operator and the position operator do not commute. This means that there is no wavefunction that is a common eigenfunction of both momentum and position. Further, we must know the wavefunction in order to calculate the uncertainties. For practice, make up a (simple) wavefunction and do the caculations for $$\Delta x$$ and $$\Delta p$$ and then take their product to convince yourself that the uncertainty principle is satisfied. I am trying to emphasize that this is quantum mechanics and not classical physics.

Bouncing a photon off an atom tells us nothing about any uncertainties. We must bounce many identically prepared photons off like atoms in order to get the statistical distributions of atomic position measurements and atomic momentum measurements. What we call "uncertainty" is a property of a statistical distribution. You cannot determine an uncertainty from a single measurement. I hope this helps.
Best wishes.

5. ### Fredrik

10,251
Staff Emeritus
Re: what is it about position and momentum that forbids knowing both quantities at on

Good post nkadambi, but I must point out an inaccuracy in what you said. I didn't really understand this myself until recently. It is possible to measure position and momentum simultaneously. In fact, we often measure the momentum by measuring the position and interpreting the result as a momentum measurement. (Check out figure 3 in this pdf). What we can't do is to prepare a state such that we would be able to make an accurate prediction about what the result of a position measurement would be and an accurate prediction about what the result of a momentum measurement would be.

jeebs: Mathematically, the "uncertainty" is derived from the axioms of QM, and is only non-zero if the commutator of the two operators is non-zero. Physically, I think the problem is always that a device that prepares a state with a sharply defined value of one of the observables would interfere with a device that prepares a state with a sharply defined value of the other observable. So... non-zero uncertainty = non-commutativity = the state preparation devices would interfere with each other.

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Re: what is it about position and momentum that forbids knowing both quantities at on

Thanks for the post, Fredrik. And thanks for the paper: I'll check it out.

I am new to this forum (I just registered couple hours ago!) I mainly have a pure math background, and only just starting into mathematical physics. Are you a grad student or professor? At my school there is hardly anyone who ventures into math physics. I am looking to make a few friends online with similar background.

7. ### CJames

355
Re: what is it about position and momentum that forbids knowing both quantities at on

Wow Fredrik, that is incredibly helpful. Just when I thought I was starting to understand something...

8. ### dlgoff

3,074
Re: what is it about position and momentum that forbids knowing both quantities at on

I've always liked how ZapperZ explains this in is blog Misconception of the Heisenberg Uncertainty Principle.

9. ### Fredrik

10,251
Staff Emeritus
Re: what is it about position and momentum that forbids knowing both quantities at on

You have come to the right place. There are plenty of people here with all sorts of backgrounds.

Yes, ZZ understood this a long time before I did. I would have understood it much sooner if I had read his posts in these threads more carefully. I was naive enough to think I already understood these things.

Last edited: Jul 23, 2011
10. ### Naty1

Re: what is it about position and momentum that forbids knowing both quantities at on

(sorry this is so long but I have just been struggling through the same concepts.)

I hope the essence of Zapper's HUP explanation is here:

and

Somebody in the recent past posted this....my boldface.. (I did not record the poster, maybe even Zapper??..was a trusted source here.) I'm posting this to confirm that it is an equivalent description, that it matches Zappers blog...

So what I think these mean is that you can get precise but not necessarily ACCURATE simultaneous measurements...that is, you cannot REPEAT the exact measurement results as is possible to arbitrary precision in classical measurements. What had me confused, and I hope I understand better, was that commutativity and non commutativity of operators applies to the distribution of results, not an individual measurement.

In Quantum Mechanics, Albert Messiah provides an interpretation for the inability to repeat the measurements :

11. ### Fredrik

10,251
Staff Emeritus
Re: what is it about position and momentum that forbids knowing both quantities at on

This "inability to repeat measurements" is in my opinion better described as an inability to prepare a state with the desired properties (or as the non-existence of such a state in the mathematical part of QM). Since you measure the momentum by measuring the position, you can measure both with an accuracy that's only limited by the size of the detector.

Last edited: Jul 23, 2011
12. ### Naty1

Re: what is it about position and momentum that forbids knowing both quantities at on

I'm surprised, if I understand what you posted: Zapper's blog which I quoted above seems to me a bit different:

But I haven't quite been able to figure out exactly what "prepare a state" means which Messiah in QUANTUM MECHANICS also mentions but doesn't explain. Where is Zapper?

13. ### Fredrik

10,251
Staff Emeritus
Re: what is it about position and momentum that forbids knowing both quantities at on

It's not different. The "desired" properties are precisely those properties that would ensure that the results of all the momentum measurements (on different members of an ensemble of identically prepared systems) are essentially the same, and that the results of all the position measurements (on different members of the same ensemble) are essentially the same. ZZ's statement explains what my statement means.

I just don't like the phrase "repeated measurements", because it sounds like it might be referring to something you do repeatedly to the same particle (without re-preparing it between measurements) rather than to the members of an ensemble of identically prepared particles.

To prepare a state is just to bring a particle on which we intend to do a measurement to the measuring device. Different ways of doing that may give us different average results. Two ways of doing it (two preparation procedures) are considered equivalent if no series of measurements can distinguish between them (i.e. if they give us the same wavefunction, or more generally, the same state operator/density matrix). These equivalence classes are often called "states".

Last edited: Jul 24, 2011
14. ### Naty1

Re: what is it about position and momentum that forbids knowing both quantities at on

Fredrik: thanks for the assistance....I have a bit more thinking to do, but I "get" the last two of your three paragraphs....

"It's not different". well, THAT's a relief!!!!!! maybe wording semantics got in the way...

Your explanation of "to prepare a state" clarifies what that means....I sure do not like that terminology, but maybe that's just me....

15. ### Naty1

Re: what is it about position and momentum that forbids knowing both quantities at on

eaglelake:

BRAVO!!.....concise, well stated....

16. ### Roo2

41
Re: what is it about position and momentum that forbids knowing both quantities at on

Is there a peer-reviewed article which presents this argument (that the limit of accuracy for the measurement of a single electron is technology rather than HUP)? It makes sense to me intuitively and mirrors what I was taught, but I'd like to see it addressed formally rather than via blog.

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18. ### abaio

12
Re: what is it about position and momentum that forbids knowing both quantities at on

Great informoation. Thanks. It would have been awsome to be there when the Quantum giants were discussing and racing to find new discoveries in the new mysterious quantum world. Bohr vs. Einstein was a great duel...kind of that like Edison vs. Tesla.

19. ### Roo2

41
Re: what is it about position and momentum that forbids knowing both quantities at on

Thanks! I was skimming this thread at work and didn't notice the link. That's exactly what I was looking for. I was actually amused to find this thread; I had a discussion with my room mate a couple weeks ago about HUP. Both of us are college undergrads, and our experience with quantum mechanics is limited to one semester of physical chemistry apiece (at different universities). I mentioned that I had been taught that HUP applied to sets of measurements of p and r and didn't apply to the position and momentum of a single particle. He replied that since the particle is itself a wave, and a wave has no defined absolute position, then HUP applies to a single particle. I don't have the physics background to respond to that, so I let it be. That paper is clearing up the picture for me. Forgive me for asking again for references, but I don't have the physics education to know off the bat... have there been any revisions to the interpretation of the statistical model in regard to a single particle measurement in the 40 years since this paper was published?

20. ### fuesiker

49
Re: what is it about position and momentum that forbids knowing both quantities at on

The most intuitive way, imho, to view the uncertainty principle is to recognize that the wavefunction of some system in momentum space is the Fourier transform of the wavefunction of that same system in position space. You can arrive at this by honoring the fact that the momentum operator is the propagator of translation, and that is how you can relate both wavefunctions to each other to arrive at the equation showing they are related by the Fourier transform.

Now that we are convinced of this, we can look at the properties of the Fourier transform. If a function K is the Fourier transform of a function F, then the sharper F is, the broader is K.

The uncertainty principle has nothing to do with observation. It does not arise due to observation. It is the nature of things that there are non-commuting observables (time and energy, momentum and position) that cannot be determined simultaneously to arbitrary precision. We find this unintuitive because in our universe, Planck's constant is way too small for quantum effects to have become intuitive to our senses, just like special relativisitic effects are also counter-intuitive to our senses since c is so big in our world. Imagine if we lived in a world where c is 30 mph ;)