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BruceW you are right.
It looks like you're right about that, but it takes a more sophisticated argument than a blurry photo of a moving car. Ballentine's argument in the article discussed in this thread seemed to prove that you could measure both with accuracies Δx and Δp such that ΔxΔp is arbitrarily small, and I wasn't able to see what was wrong with it. But Demystifier was. I think that what he said here is a very good reason to not define QM in a way that makes what Ballentine described a "momentum measurement".vijayan_t said:I think we can not measure both at a time.
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.
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!
...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
Yes. I have changed my mind about this part of #5:Naty1 said:And Fredrick: Have you changed your position from post#5 to
#283 above?
This is the reason:Fredrik said: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).
Fredrik said:I think Demystifier's argument for why the position measurement in Ballentine's thought experiment shouldn't be considered a momentum measurement was convincing. He actually posted it in another thread, here. (See my posts and his, in the 35-40 range. The main point is in post #40).
jeebs said: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, [tex] \Delta x \Delta p \geq \hbar/2 [/tex]
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?
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 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.
Position-momentum uncertainty relations originate from the fact that the momentum is defined, to within a constant, as the characteristic wave number of a plane wave, and that, rigorously speaking, a plane wave extendeds over all space; to localize the momentum at an exact point of space has no more meaning than to localize a plane wave. Just as momentum is a wave number and cannot be localized in space, so energy is a frequency and cannot be localized in time...In the position momentum uncertainty relations, the positiion and momentum play exactly symmetrical roles; they both can be measured at a given time t. ..In the (energy-time) relation on the other hand, the energy and time play fundamentally different roles: the energy is a dynamical variable of the system, whereas the time t is a parameter.
The HUP strikes at the heart of classical physics: the trajectory. Obviously, if we cannot know the position and momentum of a particle at t0 we cannot specify the initial conditions of the particle and hence cannot calculate the trajectory...Due to quantum mechanics probabilistic nature, only statistical information about aggregates of identical systems can be obtained. QM can tell us nothing about the behavior of individual systems. Moreover, the statistical information provided by quantum theory is limited to the results of measurements...
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Roger Penrose in THE ROAD TO REALITY had very little to say about the HUP[!] but did mention this: (which is a bit different than 'simultaneous' measurements)
A measurement of a particle's momentum would put it into a momentum state, corresponding to some classical value P, and any subsequent measurement of the momentum in this state would yield the same result P. However if the state were instead subjected to a subsequent position measurement following an initial measurement of momentum, the result would be completely uncertain, and anyone result for the position would be as likely as any other.
[This seems to relate some earlier discussion here.] Penrose does have several pages of math contrasting position states and momentum space descriptions..."above my paygrade".
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
It is impossible to determine both momentum and position by means of the same measurement, as indicated by Born above...
"Assume that its initial momentum has been accurately calculated by measuring its mass, the force applied to it, and the length of time it was subjected to that force. Then to measure its position after it is no longer being accelerated would require another measurement to be done by scattering light or other particles off of it. But each such interaction will alter its momentum by an unknown and indeterminable increment, degrading our knowledge of its momentum while augmenting our knowledge of its position. So Heisenberg argues that every measurement destroys part of our knowledge of the system that was obtained by previous measurements." [
My interpretation of the uncertainty relations is the same as his. I do however disagree with the specific words quoted above. His argument is the same as the one in Ballentine's article, so I guess we have a different interpretation of the single slit experiment described there. After the discussion with Demystifier, I came to the conclusion that what happens in Ballentine's thought experiment isn't a momentum measurement. It has nothing to do with the fact that momentum is "inferred" rather than "directly measured". It's just that the results won't be distributed as described by the restriction of the function ##\vec p\mapsto|\langle\vec p,\psi\rangle|^2## to the y axis. This means that a theory that says that this is a momentum measurement is significantly worse at making predictions about results of experiments than one that says that this is not a momentum measurement.Naty1 said:Is your interpretation now different than Zappers:
...there's nothing to prevent anyone from knowing both the position and momentum of a particle in a single mesurement...
Fredrik said: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.
I am. I don't consider a series of measurements to be a single measurement. This quote from another thread explains what I mean:Naty1 said:I take it you are not making any distinction between a single simultaneous measurement of a system versus repeated measurements of identically prepared ystems?
Fredrik said:If you run this experiment over and over on electrons that were all prepared in the same spin state, then you can figure out how the first pair of magnets were aligned. This isn't what one would normally consider a "measurement" in QM. A measurement is an interaction between the system and the measuring device that puts a component of the measuring device, that I'll call "the indicator component" here, into one of many possible final states labeled by numbers. The indicator component must appear as a classical object to a human observer, and its possible final states must be distinguishable. Otherwise, it wouldn't be of any use as an indicator. The number corresponding to the final state is considered the "result" of the measurement.
I'm a bit puzzled by what Messiah said, because he starts out saying roughly that there's no function that has a constant absolute value and is very sharply peaked (duh), and then the words "they both can be measured at a given time t" appear out of nowhere. Maybe he just meant that either of them can be measured, not that a joint measurement is possible.Naty1 said:Here are some statements which seem to me supportive of Zapper's. But these are hardly "crystal clear".
Alber Messiah, Quantum Mechanics, Two Volumes in One, 1999, page 135
Not sure what part of his quote you find interesting. If it's the "cannot know the position and momentum of a particle" part, this is just his way of saying what I said about state preparation above.Naty1 said:Dr. Donald Luttermoser
This too is roughly what I said about state preparation above, plus the fact that a non-destructive position measurement is a state preparation that localizes the particle in the sense that it makes its wavefunction sharply peaked. This of course "flattens" its Fourier transform, so if the Fourier transform was sharply peaked before the position measurement, it isn't anymore.Naty1 said:Roger Penrose
I think this statement should say "prepare" where it says "measure", because a measurement can destroy the system, and I don't think it's clear from the uncertainty theorem* that a joint measurement that destroys the particle is impossible.Ken G said:The HUP says it is impossible to have arbitrary concurrent knowledge of both the position and the momentum, so it is also impossible to measure the position and momentum, to arbitrary precision, concurrently.
I would say that if you have inferred it, you have measured it. Maybe you should be talking about "preparation" rather than "measurement" here too.Ken G said:The problem with this argument is that you either don't measure the momentum, you merely infer it (and get the average momentum for some previous time period, not the current momentum, so that is not a momentum measurement),
This is an interesting comment, but is there a device that can do that?Ken G said:or if you do a direct momentum measurement, the time when it occurs must be uncertain.
Yes, if we use "prepare", we are certainly safe. But I think we can go a step further, and really ask what a "measurement" is. Maybe there are measurements of two different types, those that destroy the state, and those that prepare the state. Usually, when one talks about "measurement theory" or "the measurement problem", one is talking about the latter-- a la Schroedinger's cat. Destroying a cat is not the same thing as preparing a dead cat-- and one cannot have simultaneous knowledge about various aspects of a particle that doesn't exist any more. What's clear is that the HUP is about knowing position and momentum at the same time-- we all know there is no HUP for a momentum measurement followed by a position measurement, though that is effectively the same situation as the argument cited in that quote above that claimed (erroneously I claim) to specify the momentum and position at the same time.Fredrik said:I think this statement should say "prepare" where it says "measure", because a measurement can destroy the system, and I don't think it's clear from the uncertainty theorem* that a joint measurement that destroys the particle is impossible.
The key issue is not a distinction between what you can infer is true and what you can measure is true, it is between what you can know is true versus what you can know was true. Again the issue is around the simultaneity of the knowledge, crucial to the HUP.I would say that if you have inferred it, you have measured it. Maybe you should be talking about "preparation" rather than "measurement" here too.
One way to directly measure momentum is to measure a Doppler shift and infer velocity. But if you analyze such a measurement, you will find that although you can get the Doppler shift to arbitrary precision, to use it as current knowledge of the momentum requires that the recoil from the interaction be negligible, so such a measurement requires that the photon must have an energy that is very definite and very low. Together, that means that neither the time nor the location or the interaction can be certain, so the particle cannot be localized in the momentum measurement. These are all measurements of the "preparation" variety, but note that "preparation" is generally used in the context of an intial condition, not the final state that tests some theory. Here we are testing the theory and doing the measurement at the end, yet it is a measurement that is attempting (and failing) to convey simultaneous position and momentum knowledge of the particle.This is an interesting comment, but is there a device that can do that?
I do however agree with ZZ's main point (and probably everything in that blog post except the words quoted above), which is that the uncertainty theorem is about the statistical distribution of the results of future measurements. The theorem doesn't say anything about whether you can measure both at the same time. That is a separate issue. (I believe you can't measure both at the same time, but I haven't seen a proof of that). What it actually says is that you can't prepare a state that has both a sharply defined position and a sharply defined momentum. ...
The HUP strikes at the heart of classical physics: the trajectory. Obviously, if we cannot know the position and momentum of a particle at t sub 0 we cannot specify the initial conditions of the particle and hence cannot calculate the trajectory...Due to quantum mechanics probabilistic nature, only statistical information about aggregates of identical systems can be obtained. QM can tell us nothing about the behavior of individual systems. Moreover, the statistical information provided by quantum theory is limited to the results of measurements...
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was the "trajectory part" I, too, interpretated as did you:
...this is just his way of saying what I said about state preparation above.but had not thought of in the way Luttermoser expressed it although it has been alluded to in this thread (I think) in different terms.
You also said:I'm a bit puzzled by what Messiah said
yeah, well now there are at least three of us saying that! you, me, and Ballentine! I read
some of his explanations in QUANTUM MECHANICS repeatedly (without success) trying to figure his intent.
I wonder if the OP feels his/her question was answered?
Yet I still have not seen any examples of a "single measurement" that could possibly be interpreted as yielding simultaneous knowledge, to arbitrary precision, of position and momentum. I feel this is the crucial issue that is being overlooked-- of course you can do one measurement, then the other, or equivalently, do one measurement that gives both the previous momentum and the current position, and have both be to arbitrary precision. But the HUP refers to knowing both at the same time. That means there has to be a time when both refer to the current state of the particle. None of the examples given above do that. This is related to Fredrik's point about "preparation" of the system, but I would stress the issue of simultaneity of the knowledge-- that's why there is such an important difference between preparing a particle versus inferring something about its history. We must get away from the classical assumption that a particle is what it was, because that overlooks how changing what can be said about the particle changes the particle.Naty1 said:And Kennard and Popper's interpretations say nothing which prohibits a single measurement to arbitrary precision.