Exploring Heisenberg's Uncertainty Principle: Intuition & Explanations

[Demystifier]

BruceW you are right.

 

[vijayan_t]

I think we can not measure both at a time. see this

 

[Fredrik]

I think we can not measure both at a time.

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".

 

[BruceW]

That looks like an interesting thread. The idea is that an experiment which detects the position of a particle can also 'infer' the momentum, since the particle came from a small slit?

I think the answer is that until a particle is detected, its wave function is spread out over space and momentum-space (as we should expect). And then when we measure the position of the particle at the detector, it is not a measurement of that particle's momentum at that time or at an earlier time. In fact, at an earlier time, it wasn't a momentum eigenstate, so we can't really say it had momentum (the usual Copenhagen interpretation). So we're not strictly making a momentum measurement, even though we can establish a probability distribution, which would be useful to predict the outcome of a momentum measurement if we ever do decide to make one.

 

[Naty1]

I just noticed this discussion started on July 22, 2011...Today is Jan 7, 2012! It lasted way longer than Kim Kardashian's marriage!

Apparently I read parts of it last year and forgot, so imagine my surprise when I saw my own posts in the #150's !

Questioning "What did you mean by that?, and the like, is especially helpful since symantics seems to play an especially large role with QM.

In Ballentine's paper he even made a comment about Albert Messiah's book that underscored some of my own questions of interpretation and ambiguities when reading it. And Ballentine's paper does have a number of clearly explained standard QM principles which seem accurate and helped clarify my own understanding.

good job... even if lengthy!

 

[Naty1]

Well, now I am wondering if we have a consensus...I'll restate from my post #8:


--------------------------------------------------------------------------------

(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:

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

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...

...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.

And Fredrick: Have you changed your position from post#5 to
#283 above?

 

[Fredrik]

And Fredrick: Have you changed your position from post#5 to
#283 above?

Yes. I have changed my mind about this part of #5:

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).

This is the reason:

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).

 

[San K]

one way to look at it is:

in a single/double slit...

if we try to make the slit narrower, the spread of the wave increases

another analogy could be that of a water hose/pipe with water running under pressure

we are dealing with not simply a particle ...but with an entity with dual nature (wave, particle)...

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?

 

[Naty1]

Yes. I have changed my mind about this part of #5:

Originally Posted by Fredrik:

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).

Is your interpretation now different than Zappers:

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.

It seems to be and I take it you are not making any distinction between a single simultaneous measurement of a system versus repeated measurements of identically prepared ystems?


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

Time-Energy Uncertainty Relation

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.

Previously someone in these forums referenced some Course Lecture Notes, Dr. Donald Luttermoser,East Tennessee State Univerity, Spring Semester September 2005, 5th edition:

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...
QUOTE]



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".

 

[Naty1]

I forgot to check wikipedia: Some interesting perspectives:


http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle

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." [

Notice the "calculation" employed in determining momentum; the last paragraph seems aimed at measurement rather than inherent uncertainty and is attributed to Heisenberg in footnotes! I wonder if Heisenberg ever changed his view...

This is just too nutty...I'm leaving in frustration and take my Yorkies for a walk. THAT is
always enjoyable!

 

[Ken G]

Actually, I don't agree with the most literal interpretation of Zapper's remark, or with this citation from a few posts ago that appears to echo the same point:
"...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."

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), or if you do a direct momentum measurement, the time when it occurs must be uncertain. We all know that we can directly measure both the position, and the momentum, of a particle in subsequent measurements, but we don't call that a position and momentum measurement, because the two are not simultaneous. The same holds for the above measurements of position and momentum, it's just that the time difference is either indeterminate or otherwise being swept under the rug.

Now, this doesn't mean I'm completely disagreeing, call it a clarification. I agree the main point of the HUP is referring to predictions about what will happen next. If we are certain how a position measurement will come out, prior to doing it, then we are uncertain about how a momentum measurement will come out, this is the main point of the HUP. So it's about knowledge of a measurement that has not yet been done. But that is the exact same thing as knowledge about the particle, because that's what knowledge about a particle means-- knowledge about a measurement on the particle that has not been done. It does not refer to past measurements, any more than a momentum measurement that is followed by a position measurement still tells us about the current momentum of the particle. "Current" just means "if we did a measurement now, even though we haven't yet." 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. Hence, the HUP is indeed about limitations on the knowledge we can have about the particle, i.e., knowledge about the current state of the particle, not knowledge about the history of the particle. It can be argued that knowledge about history is not what we mean by knowledge about the particle, we can trace the history x(t) to arbitrary precision with repeated high-precision x measurements, the "momentum" of the particle refers to its current state. If we do a position measurement to infer a past momentum, as in that quote, what we have done is not the least bit different from doing a true momentum measurement, followed by a true position measurement, and no one thinks the HUP says that is impossible. The HUP refers to what we can know about the particle concurrently.

 

[Fredrik]

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...

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.

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. I mentioned this in #5:

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 take it you are not making any distinction between a single simultaneous measurement of a system versus repeated measurements of identically prepared ystems?

I am. I don't consider a series of measurements to be a single measurement. This quote from another thread explains what I mean:

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.

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

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.

Dr. Donald Luttermoser

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.

Roger Penrose

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.

 

[Fredrik]

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 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 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),

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.

or if you do a direct momentum measurement, the time when it occurs must be uncertain.

This is an interesting comment, but is there a device that can do that?


*) I can't make myself call it a "principle". To me, a "principle" is a loosely stated idea that might help you guess an appropriate definition of a new theory.

 

[Ken G]

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.

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.

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.

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.

This is an interesting comment, but is there a device that can do that?

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.

 

[Naty1]

Fredrik: Thanks for your reply in #292...

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. ...

good.

What I liked about Luttermosers comment:

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...
QUOTE]

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?

 

[Naty1]

My post #290 quotes an introductory portion the the Wikipedia discussion on HUP.

http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle

That excerpt by itself does not do justice to the overall article:

I don't understand it all yet...[might never!]... but the article offers some worthwhile insights. Among other things the "experts" have differed on their interpretations and their views seem to have changed over time. And Kennard and Popper's interpretations say nothing which prohibits a single measurement to arbitrary precision.

 

[Ken G]

And Kennard and Popper's interpretations say nothing which prohibits a single measurement to arbitrary precision.

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.

 

[mikel137]

Early experiments in optics suggested an elusive quantity existed in light. Measurements of variables such energy and wavelength were possible with calorimeters and prisms. The speed of light was slowly (over a period of centuries) measured with increasing accuracy, and with various interference devices, wavelengths were also determined.

In the 1600's it was also found that something related to energy existed in invisible radiation beyond the red and violet ends of the spectrum. Because it was energy, ratios were possible with other quantities such as wavelength and frequency, which was inferred from wavelength and speed.

The "something related to energy" was found to have particular dimensions. They had been understood during the 1700's in classical mechanics (horses, cannons, pulleys, hoists and other macroscopic physical systems) as ACTION, which has dimensions closely related to energy. The dimensions of the time rate of change of action are the same as the dimensions of energy. Conversely, action is the path integral of energy in time.

Even stranger was this: the action in the ratio between energy and frequency is a very small amount of action. It was named the quantum of action. Now, as to why light waves have a small quantity of action in them, I haven't a clue. Light wave that were emitted billions of years ago arrive with exactly the same quantity as light waves from an LED flashlight or the screen you are looking at.

Inverting the ratio this is stated as" The product of energy and time (in light, the wavetime) is a very small constant which has the dimensions of action. It took in sum, thousands of years to figure it out. Yet we are literally made of molecular bonds and spectra in which the action quantum is figurative in every one. Life didn't know itself in that much detail until the microscope was invented.

Please read that carefully. The ratio of energy to frequency (same thing as product of energy and wavetime) was found to be a very small amount of action. That very small amount of action was given the name word quantum. It has dimensions of mass or energy, length * length or area, and frequency or the inverse of time. The mass and energy relation is the E=m*c^2 Einstein equation.

Connecting cosmology from small quantum phenomena to the vast fields of general relativity is just about where research is today. Hopefully, a starship will emerge from it. Personally, I think it won't be soon, but may happen eventually.

 

[salvestrom]

Just going to boldy step in here and see if I can lay this down in layman's terms.

You're all discussing the extent of the HUP. There are apparently three general sides to this:

You can know the past, a bit about the present, but not know anything of the future.

You can know the past and present, but not know anything about the future.

You can't know anything except the past.

I've always understood, or accepted, the HUP to mean the first. There's a lot of talk in favour of the second. The third seems largely a philosophical extension of the second.

Let me know how this turns out!

 

[San K]

Does this analogy work, somewhat:

Think of it as a lever with position and momentum at each end of the lever. If we try to make the movement of one end shorter, the movement of the other end becomes larger.

notes:
- the length of the lever is fixed (planks lenght)
- Position and velocity/momentum are (at the) ends of the same lever, canonical conjugates