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-   -   What is it about position and momentum that forbids knowing both quantities at once? (http://www.physicsforums.com/showthread.php?t=516224)

 jeebs Jul22-11 05:18 PM

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

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?

 CJames Jul22-11 07:09 PM

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.

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.

 eaglelake Jul22-11 08:33 PM

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.

 Fredrik Jul22-11 08:49 PM

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.

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.

 CJames Jul22-11 10:18 PM

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...

 dlgoff Jul22-11 10:55 PM

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

Quote:
 Quote by Fredrik (Post 3416877) It is possible to measure position and momentum simultaneously.
I've always liked how ZapperZ explains this in is blog Misconception of the Heisenberg Uncertainty Principle.

 Fredrik Jul23-11 10:05 AM

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

Quote:
 Quote by nkadambi (Post 3416936) 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.
You have come to the right place. :smile: There are plenty of people here with all sorts of backgrounds.

Quote:
 Quote by dlgoff (Post 3417026) I've always liked how ZapperZ explains this in is blog Misconception of the Heisenberg Uncertainty Principle.
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. :smile:

 Naty1 Jul23-11 05:44 PM

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:

Quote:
 The HUP isn't about a single measurement and what can be obtained out of that single measurement. It is about how well we can predict subsequent measurements given the identical conditions.
and

Quote:
 What I am trying to get across is that the HUP isn't about the knowledge of the conjugate observables of a single particle in a single measurement. I have shown that there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single mesurement with arbitrary accuracy that is limited only by our technology. However, physics involves the ability to make a dynamical model that allows us to predict when and where things are going to occur in the future. While classical mechanics does not prohibit us from making as accurate of a prediction as we want, QM does!
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...

Quote:
 ...to measure a particle's momentum, we need to interact it with a detector, which localizes the particle. So we actually do a position measurement (to arbitrary precision). Then we calculate the momentum, which requires that we know something else about the position of the particle at an earlier time (perhaps we passed it through a narrow slit). Both of those position measurements, and the measurement of the time interval, can be done to arbitrary precision, so we can calculate the momentum to arbitrary precision. From this you can see that in principle, there is no limitation on how precisely we can measure the momentum and position of a single particle. Where the HUP comes into play is that if you then repeat the same sequence of arbitrarily precise measurements on a large numbers of identically prepared particles (i.e. particles with the same wave function, or equivalently particles sampled from the same probability distribution), you will find that your momentum measurements are not all identical, but rather form a probability distribution of possible values for the momentum. The width of this measured momentum distribution for many particles is what is limited by the HUP. In other words, the HUP says that the product of the widths of your measured momentum probability distribution, and the position probability distribution associated with your initial wave function, can be no smaller than Planck's constant divided by 4 times pi

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 :

Quote:
 Immediately after the operation of a measurement the system is in a dynamical state with the arbitrarily precise position measured; Such a state cannot be represented by a wave function (psi). The state psi in general corresponds to a probability distribution of finding some value x, not a precise value of x…..or of measuring momentum. The function psi does not give more than the statistics of positions…or momenta..... During the process of observation the measured system can not be considered as separate from the observed phenomena. The intervention of the measuring instrument destroys all causal connection between the state of the system before and after the measurement; this explains why one cannot in general predict with certainty in what state the system will be found after the measurement.

 Fredrik Jul23-11 06:17 PM

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.

 Naty1 Jul24-11 09:02 AM

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

Quote:
 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).

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

Quote:
 Where the HUP comes into play is that if you then repeat the same sequence of arbitrarily precise measurements on a large numbers of identically prepared particles (i.e. particles with the same wave function, or equivalently particles sampled from the same probability distribution), you will find that your momentum measurements are not all identical, but rather form a probability distribution of possible values for the momentum.

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?

 Fredrik Jul24-11 09:22 AM

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".

 Naty1 Jul24-11 03:25 PM

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....

 Naty1 Jul25-11 09:04 AM

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

eaglelake:

Quote:
 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.
BRAVO!!.....concise, well stated....

 Roo2 Jul25-11 10:42 PM

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

Quote:
 Quote by dlgoff (Post 3417026) I've always liked how ZapperZ explains this in is blog Misconception of the Heisenberg Uncertainty Principle.
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.

 CJames Jul25-11 11:08 PM

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

Is there something wrong with the link Fredrik posted earlier?

http://www.kevinaylward.co.uk/qm/bal...ation_1970.pdf

Page 365

I don't mean this to be sarcastic if it comes across that way.

 abaio Jul25-11 11:19 PM

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.

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