What Intuitive Insights Explain Heisenberg's Uncertainty Principle?

[strangerep]

Fredrik,

In one of your early posts in this thread, you linked to Ballentine's 1970 paper
in reference to the joint position+momentum measurement controversy.

However, I can't seem to find a similar argument in his more recent textbook.
Do you know whether the argument is repeated there, and if so, where?

(I'm trying to find evidence for or against the possibility that Ballentine might
have modified his views between 1970 and 1998.)

 

[BruceW]

Am I right in thinking that for a single particle coming out of the slit, its total momentum is unknown since it is a superposition of plane waves, and therefore although this experiment gives us \frac{p_y}{p}, it doesn't actually give us p_y?

Edit: and does
\frac{p_y}{p}
commute with y? Because this would solve the paradox. Sorry if this has already been mentioned, but I didn't see it anywhere on this thread yet..

 

[BruceW]

This is my thinking on the single-slit experiment:

We can say that the wavefunction is spread over the space between the slit and the detector, and after some period of time, a particle is detected. So then collapse happens, i.e. the wavefunction goes to zero except for at the point it was detected. This effectively gives a measurement of position.

Then you could also go on to say that the collapse of the wavefunction happens retroactively, meaning that since we know the particle is a 'free' particle in the gap from slit to detector, it must have traveled in a straight line, so if we put the detector far from the slit, we effectively know the path that the particle must have taken to get to the detector. So in this way the experiment can be said to measure momentum.

But I think that by making this retroactive collapse, we effectively specify a wavepacket that moves with time, which is localised in z as well as y. (z being the direction perpendicular to the detector). Therefore, this wavepacket is not an eigenstate of z-momentum. The experiment gives us the angle of the particle's path, which gives us the ratio of y component and z component of momentum. But since we don't know the z component, the experiment doesn't give us the actual value of the y component of momentum.

 

[Fredrik]

Fredrik,

In one of your early posts in this thread, you linked to Ballentine's 1970 paper
in reference to the joint position+momentum measurement controversy.

However, I can't seem to find a similar argument in his more recent textbook.
Do you know whether the argument is repeated there, and if so, where?

(I'm trying to find evidence for or against the possibility that Ballentine might
have modified his views between 1970 and 1998.)

I don't remember seeing it when I skimmed the section about uncertainty relations a year ago. It's certainly possible that he has changed his mind about this since 1970.

 

[vanhees71]

In which respect should Ballentine have changed his mind since 1970. I can't see this from his marvelous book. To me it reads as an extended version of the RMP article. It's in Sect. 9.3 "The Interpretation of a State Vector" and very clearly written. This book convinced me about the unnecessity of the collapse postulate, and to favor the Minimal Statistical Interpretation (i.e., to just take Born's probability interpretation really seriously) which solves all quibbles with Einstein causality, which is solely caused by Copenhagen philosophy rather than needed elements of interpretation of quantum mechanics as a physical theory.

 

[Fredrik]

The specific detail we're thinking he might have changed his mind about is the question of whether it's appropriate to call what's going on in that single-slit thought experiment a "momentum measurement". If it is, then the argument presented in the article proves that you can measure position and momentum simultaneously, with margins of error that are much smaller than what a naive application of the uncertainty relation suggests. I'm thinking that if he hasn't changed his mind, then why isn't that argument in the book?

I think he might have changed his mind about one more thing actually. In the article, he is clearly assuming that a particle actually has a position and a momentum at all times, regardless of what its wavefunction looks like. This was discussed in another thread not too long ago. It doesn't seem to be possible to prove that assumption false, but in my opinion, it would be very strange to make it a part of a "minimal" interpretation. I don't recall seeing that assumption in his book, so he might have come to the same conclusion.

 

[BruceW]

In Ballentine's ensemble interpretation presented in the paper, he says that a state vector refers to an ensemble of identically prepared systems, not an individual system.
In his paper, he also says an individual particle has position and momentum.
But his ensemble interpretation in its most basic form doesn't seem to require that individual particles have exact position and momentum.
So I'd say in its most minimal form, the ensemble interpretation doesn't care if individual particles have exact momentum and position or not.

About the single-slit thought experiment, if you use the definition that the state vector only represents an ensemble of systems, then you can 'measure' momentum and position to arbitrary precision, because they are measurements made on separate systems.
But if you define the state vector as describing a single system, then you cannot have a state which is an eigenstate of both position and momentum.

 

[Fredrik]

About the single-slit thought experiment, if you use the definition that the state vector only represents an ensemble of systems, then you can 'measure' momentum and position to arbitrary precision, because they are measurements made on separate systems.
But if you define the state vector as describing a single system, then you cannot have a state which is an eigenstate of both position and momentum.

That's not right. The ensemble interpretation says that a particle's wavefunction represents the statistical properties of the ensemble of particles that participate in the experiment when you run it many times (assuming of course that the preparation is the same each time). Each time you run it, you're dealing with a single particle.

There's no state that has both a sharply defined position and a sharply defined momentum. The question of whether we can measure both at the same time doesn't depend on the interpretation (i.e. on what "thing in the real world" the wavefunction corresponds to).

Am I right in thinking that for a single particle coming out of the slit, its total momentum is unknown since it is a superposition of plane waves, and therefore although this experiment gives us \frac{p_y}{p}, it doesn't actually give us p_y?

Edit: and does
\frac{p_y}{p}
commute with y? Because this would solve the paradox. Sorry if this has already been mentioned, but I didn't see it anywhere on this thread yet..

Ballentine assumes that p is known. I'm not looking at the article right now, but I think the idea is that the particle the enters the slit is in an energy eigenstate, which gives it a well-defined p. According to Ballentine, that's not going to change when the particle goes through the slit.

 

[BruceW]

That's not right. The ensemble interpretation says that a particle's wavefunction represents the statistical properties of the ensemble of particles that participate in the experiment when you run it many times (assuming of course that the preparation is the same each time). Each time you run it, you're dealing with a single particle.

I was thinking of 'measurement' in terms of measurements on separate systems represented by the same state vector, since the ensemble interpretation only deals with ensembles of identical systems.

There's no state that has both a sharply defined position and a sharply defined momentum. The question of whether we can measure both at the same time doesn't depend on the interpretation (i.e. on what "thing in the real world" the wavefunction corresponds to).

Oh yeah, I didn't realize that. Thanks for pointing it out.

Ballentine assumes that p is known. I'm not looking at the article right now, but I think the idea is that the particle the enters the slit is in an energy eigenstate, which gives it a well-defined p. According to Ballentine, that's not going to change when the particle goes through the slit.

I'm hoping that Ballentine was wrong in his thought-experiment, but I'm not yet certain how he was wrong.
Edit: Although his thought experiment may be right for an ensemble of identical systems, but not for a single system.

 

[Fredrik]

I was thinking of 'measurement' in terms of measurements on separate systems represented by the same state vector, since the ensemble interpretation only deals with ensembles of identical systems.

OK, but then you just send one particle to a position measuring device and another to a momentum measuring device. Sure, you can do that at the same time, but that's not what "simultaneous measurement" or "joint measurement" means.

I'm hoping that Ballentine was wrong in his thought-experiment, but I'm not yet certain how he was wrong.

Atyy's suggestion seems the most promising, but we still have to find the derivation of the result that he and I discussed. See post #247 (including the quote).

 

[atyy]

Atyy's suggestion seems the most promising, but we still have to find the derivation of the result that he and I discussed. See post #247 (including the quote).

http://tf.nist.gov/general/pdf/1283.pdf and http://www.mpq.mpg.de/qdynamics/publications/library/Nature395p33_Duerr.pdf should give a lead into a paper which does it. Given the result, it should also be derivable from elementary quantum mechanics (Schroedinger), without going through the Wigner function.

A related hint is to look at Wikipedia's entry on Fraunhofer diffraction http://en.wikipedia.org/wiki/Fraunhofer_diffraction. The wave equation in question is not Schroedinger's, but it does seem that the far field amplitude pattern is the Fourier transform of the initial amplitude pattern. Obviously, this has to be redone using the wave equation we're interested in.

 

[BruceW]

I think even the statistical interpretation runs into problems if an individual particle has exact position and momentum.

Think about the ground-state hydrogen atom. If the electron actually had exact position, then the hydrogen atom would have an electric dipole moment.
If we imagine a statistical ensemble of hydrogen atoms, each would have a different, non-zero electric dipole moment. So the state representing this ensemble would be a superposition of different dipole moment eigenstates.

But this can't be true, since the correct state representing such an ensemble of ground-state hydrogen atoms is an eigenstate of zero electric dipole moment. (Which is verified by experiment). (i.e. every time electric dipole moment is measured, we get zero).

So even in the statistical interpretation, an individual particle cannot always have exactly defined position. Does all of this sound right? I guess this might be why Ballentine changed his mind about particles having well-defined position and momentum?

 

[Fredrik]

I think even the statistical interpretation runs into problems if an individual particle has exact position and momentum.

Think about the ground-state hydrogen atom. If the electron actually had exact position, then the hydrogen atom would have an electric dipole moment.

But it would be only be detectable in experiments with a very high resolution of time. In particular, it might influence the result in scattering experiments involving "fast" particles but not in scattering experiments that involve "slow" particles. And if the particles are fast enough, wouldn't this force us to abandon the picture of an atom as elementary anyway? I suck at this type of calculations, but I would imagine that we would describe a "fast" incoming particle as being scattered off the nucleus or off the electron, regardless of whether we believe that particles "actually" have positions or not.

I'm not saying that your argument definitely fails, only that I'm not convinced by this version of it.

So even in the statistical interpretation, an individual particle cannot always have exactly defined position.

That's what I used to think, but I'm not so sure anymore. I started a thread about it here, but no one who participated in it knew any convincing arguments against the idea that all particles have positions. Post #51 shows what I was thinking at the end of the discussion.

Anyway, this is off topic for this thread, so if you want to discuss arguments for or against particles having well-defined positions at all times (regardless of their wavefunctions), then I suggest that you either resurrect that thread or start a new one.

 

[Fredrik]

But I though the whole point here is that we are trying to generalize some kind of "measurement" as an inference, from the picture outlined in Ballentine.

Yes, but in a generalized sense (as you were the one seeking to define new measurements).

I just want to make one thing clear: I'm not trying to generalize quantum mechanics, or the concept of "measurement". The way I see it, a theory hasn't been fully defined until we have specified how to interpret the mathematics as predictions about results of measurements. So a full definition of "quantum mechanics" must specify what interactions we are to think of as "measurements" and what numbers to think of as "results of measurements". I'm just trying to figure out how those specifications should be made.

In other words, I'm trying to find a proper definition of QM, not a generalization or an improvement.

Partly, but I think we can do a lot better.

Just because something is tradition doesn't make it satisfactory.

To address your example, it's the notion of falsifiability that needs to be developed. In particular what happens when a theory IS falsified. Then an extrinsic theory simply fails as there is not rational mechanism for using the information that cause the falsification to evolve the theory.

So my proposal is that we should abandom the descriptive picture of a theory which in poppian spirit is simply either corroborated or Wrong with a picture where a theory is an interaction tool. Where beeing wrong is in fact an essential part of hte learning curve that that we should add some analysis into the induction part, how a new theory is induced from a falsified teory. This is the completely ignored part in the descriptive view.

The idea that we might be able to further weaken the concept of falsifiability and still have something that deserves to be called a "theory" (because it improves our understanding of reality) is interesting, but very far from the topic of this thread. I think it's actually far from the topic of any thread about QM, since QM is (statistically) falsifiable. I think it could make an interesting thread in the philosophy forum, but I don't think it's appropriate to bring it into every discussion about QM (or even into any discussion about QM).

You did however succeed at making your views a bit clearer to me.

 

[BruceW]

But it would be only be detectable in experiments with a very high resolution of time. In particular, it might influence the result in scattering experiments involving "fast" particles but not in scattering experiments that involve "slow" particles. And if the particles are fast enough, wouldn't this force us to abandon the picture of an atom as elementary anyway? I suck at this type of calculations, but I would imagine that we would describe a "fast" incoming particle as being scattered off the nucleus or off the electron, regardless of whether we believe that particles "actually" have positions or not.

I'm not saying that your argument definitely fails, only that I'm not convinced by this version of it.

I don't see why the charge distribution would only be detectable in experiments with high time resolution.
For example, the effect of putting the ground-state hydrogen atom in an external electromagnetic field and seeing how it is affected would show that the electron cannot have a specific position.
Or another example, the selection rules which specify the wavelength of photon that can be absorbed would be affected by the position of the electron in a given atom.

That's what I used to think, but I'm not so sure anymore. I started a thread about it here, but no one who participated in it knew any convincing arguments against the idea that all particles have positions. Post #51 shows what I was thinking at the end of the discussion.

Anyway, this is off topic for this thread, so if you want to discuss arguments for or against particles having well-defined positions at all times (regardless of their wavefunctions), then I suggest that you either resurrect that thread or start a new one.

It seems on topic to me, since if particles have well-defined positions at all times, then the OP's question would have a definite answer.

 

[Fredrik]

I don't see why the charge distribution would only be detectable in experiments with high time resolution.

I meant that if the electron's trajectory doesn't favor any particular "side" of the nucleus, then the atom will "spend about the same amount of time on each side", and the atom would appear to have no magnetic moment at all. This is of course a pretty naive argument, but I haven't seen a reason to try to find a more sophisticated one yet.

For example, the effect of putting the ground-state hydrogen atom in an external electromagnetic field and seeing how it is affected would show that the electron cannot have a specific position.

What would the effect be? How long does it take to observe it? Wouldn't the electron have time to visit positions on all sides of the nucleus before that time has passed?

Or another example, the selection rules which specify the wavelength of photon that can be absorbed would be affected by the position of the electron in a given atom.

Why? The wavefunction is still a solution of the same Schrödinger equation, and the photon isn't going to have a well-defined position.

It seems on topic to me, since if particles have well-defined positions at all times, then the OP's question would have a definite answer.

What would that answer be? I don't see how the assumption that particles have positions at all times implies any kind of answer to the OP's question. We would still have difficulties finding out what those positions are (and we wouldn't be able to do it without changing the state). It also wouldn't tell us anything about what sort of interactions we should think of as "measurements" (which is what most of the discussion has been about).

 

[BruceW]

I meant that if the electron's trajectory doesn't favor any particular "side" of the nucleus, then the atom will "spend about the same amount of time on each side", and the atom would appear to have no magnetic moment at all. This is of course a pretty naive argument, but I haven't seen a reason to try to find a more sophisticated one yet.

The reason I chose the ground state hydrogen atom is because the electron has zero orbital angular momentum, so it is not in orbit around the nucleus.

Why? The wavefunction is still a solution of the same Schrödinger equation, and the photon isn't going to have a well-defined position.

The possible energy levels of an atom will depend on the charge distribution.
Maybe you will say that the behaviour of the atom depends only the state of the ensemble of different systems it is represented by. But in this case, there is no physical meaning to the exact position of the electron in an individual atom, since it does not affect that atom.

I guess you could say that the position of the electron is exact, yet its charge distribution is spread out. But then what is the physical meaning of the electron's position, if it doesn't correspond to the location of charge?

I suppose the electron could have exact position, as long as the definition of position has no physical meaning. So I should change my answer from 'particles don't always have position' to 'the position of a particle doesn't always have a physical meaning'.

What would that answer be? I don't see how the assumption that particles have positions at all times implies any kind of answer to the OP's question. We would still have difficulties finding out what those positions are (and we wouldn't be able to do it without changing the state). It also wouldn't tell us anything about what sort of interactions we should think of as "measurements" (which is what most of the discussion has been about).

You're right, it doesn't directly answer the OP's question. Although it is central to an explanation of position and momentum in quantum mechanics.
Part of my answer to the OP would be: "Unless a particle is in an eigenstate of position, exact position does not have physical meaning. An eigenstate of position and momentum is not allowed, so a particle cannot have physically meaningful position and momentum simultaneously".

 

[strangerep]

I didn't see an answer to this earlier post, so here's my $0.02 ...

Am I right in thinking that for a single particle coming out of the slit, its total momentum is unknown since it is a superposition of plane waves, and therefore although this experiment gives us \frac{p_y}{p}, it doesn't actually give us p_y?

[...] and does
\frac{p_y}{p}
commute with y?

I presume you mean
\frac{p_y}{|p|}

where |p| := \sqrt{p_x^2 + p_y^2 + p_z^2} is the magnitude of the 3-momentum.

If that's what you meant, then assuming commutation relations of the form:
<br /> [x_j , p_k] ~=~ i \delta_{jk} \hbar <br />
it can be shown by induction that
<br /> [x_j , f(p)] ~=~ i \hbar \; \frac{\partial f(p)}{\partial p_j}<br />
For your case,
<br /> f(p) ~=~ \frac{p_y}{|p|} ~=~ \frac{p_y}{\sqrt{p_x^2 + p_y^2 + p_z^2}}<br />
and differentiating wrt p_y gives
<br /> \frac{\partial f(p)}{\partial p_y}<br /> ~=~ \frac{1}{|p|} - \frac{p_y^2}{|p|^3}<br />
So the answer is that p_y does not commute with f(p).

 

[strangerep]

In which respect should Ballentine have changed his mind since 1970. I can't see this from his marvelous book. To me it reads as an extended version of the RMP article.

But some items from Ballentine's 1970 RMP article don't appear in his 1998 textbook, afaict. In the RMP article, look at Fig 3 and the associated discussion (starting near the bottom right of p365 and continuing onto the next page). Ballentine explains that the p_y "measured" in this way involves some geometric inferences and an assumption that linear motion in a free-field region (Newton's first law) remains valid in QM. He also points out that these techniques are "universally employed" in scattering experiments.

Some might say that y, p_y have not been "measured simultaneously", because the value of p_y is calculated based partly on data at earlier times. ISTM, this just highlights the importance of being clear on what is being assumed, what is being literally measured, and how much counterfactual thinking one is silently employing.

It's in Sect. 9.3 "The Interpretation of a State Vector" and very clearly written. This book convinced me about the unnecessity of the collapse postulate, and to favor the Minimal Statistical Interpretation (i.e., to just take Born's probability interpretation really seriously) which solves all quibbles with Einstein causality, which is solely caused by Copenhagen philosophy rather than needed elements of interpretation of quantum mechanics as a physical theory.

I agree with all of that.

 

[strangerep]

The specific detail we're thinking [Ballentine] might have changed his mind about is the question of whether it's appropriate to call what's going on in that single-slit thought experiment a "momentum measurement". If it is, then the argument presented in the article proves that you can measure position and momentum simultaneously, with margins of error that are much smaller than what a naive application of the uncertainty relation suggests. I'm thinking that if he hasn't changed his mind, then why isn't that argument in the book?

As mentioned in my previous post, after re-reading the specific section of Ballentine's RMP article, I now think the question is more subtle, and that one must be very clear about the places where counterfactual thinking is introduced. Just as in Bell experiments where some of the paradox is resolved my focussing only on correlations which exclude counterfactual artefacts, perhaps some related thinking is appropriate here. I.e., we "measure" p_y only by "correlating" various other raw data.

In this context, Ballentine's treatment of measurement and apparatus is section 9.2 of his textbook increases in importance. His emphasis there is that measurement of an "object" by an "apparatus" really consists only arranging an interaction such as to produce correlations between object initial state and apparatus final state.

I.e., correlations are what's critically important in QM, not correlata.

 

[BruceW]

I didn't see an answer to this earlier post, so here's my $0.02 (...) So the answer is that p_y does not commute with f(p).

Ah, thanks for that. I guess the answer to Ballentine's thought-experiment isn't as simple as I thought :(

 

[Fredrik]

As mentioned in my previous post, after re-reading the specific section of Ballentine's RMP article, I now think the question is more subtle, and that one must be very clear about the places where counterfactual thinking is introduced. Just as in Bell experiments where some of the paradox is resolved my focussing only on correlations which exclude counterfactual artefacts, perhaps some related thinking is appropriate here. I.e., we "measure" p_y only by "correlating" various other raw data.

In this context, Ballentine's treatment of measurement and apparatus is section 9.2 of his textbook increases in importance. His emphasis there is that measurement of an "object" by an "apparatus" really consists only arranging an interaction such as to produce correlations between object initial state and apparatus final state.

I.e., correlations are what's critically important in QM, not correlata.

I don't see what it is about section 9.2 that helps us understand the "measurement" in the single slit experiment. I think of section 9.2 as a sophisticated version of the Schrödinger's cat argument. (If microscopic systems can be in superpositions, then the linearity of the Schrödinger equation implies that macroscopic systems can be too). Section 9.1 seems more relevant to the problem at hand. There he describes how a Stern-Gerlach magnet enables us to measure spin component operators. They do this by creating a correlation between spin states and momentum states, so that we can infer a value of the spin component by..."observing the deflection of the particle".

But we do this by detecting the particle either in the "upper" location or the "lower" location. This is a position measurement. So we infer both momentum and spin from that position measurement.

I don't see anything in Ballentine's book that suggests that what's going on in the single-slit experiment isn't a momentum measurement. I mean, everyone thinks of Stern-Gerlach apparatuses as measuring spin components, not as measuring positions from which we can "infer" spin component values. So why shouldn't the detection in the S-G experiment also be considered a momentum measurement?

The only thing I've seen that looks like an argument against that, is what atyy has been saying, but unfortunately I haven't had the time to examine the details of that argument yet.

 

[kith]

I think even the statistical interpretation runs into problems if an individual particle has exact position and momentum.

Think about the ground-state hydrogen atom. If the electron actually had exact position, then the hydrogen atom would have an electric dipole moment.
If we imagine a statistical ensemble of hydrogen atoms, each would have a different, non-zero electric dipole moment. So the state representing this ensemble would be a superposition of different dipole moment eigenstates.

This isn't interpretation dependent, since the ground state can always be expanded in the position base. This yields \left|E_1\right\rangle = \int d\vec{r} \,\psi(\vec{r}) \,\left|\vec{r}\right\rangle which is a superposition of dipole moment eigenstates also in the Copenhagen Interpretation. So in principle, measuring the dipole moment of this state should yield non-zero values. Or am I overlooking something?

 

[atyy]

The only thing I've seen that looks like an argument against that, is what atyy has been saying, but unfortunately I haven't had the time to examine the details of that argument yet.

Another point to consider is - if the position distribution at large L "reflects" the momentum distribution of an earlier time, and is thus not an "accurate simultaneous position and momentum measurement", is it then a "momentum measurement" of the earlier time?

Naively, the answer should be "no", since if it were, we would expect it to collapse into a momentum eigenstate at the earlier time. But what is the formal way of justifying the "no"? My guess is that it's because it requires knowledge of initial p to convert the later position distribution into the earlier momentum distribution. Prior knowledge should disqualify some forms of knowledge acquisition from being measurements, since in an extreme case, if we know the initial state, we know all future momentum and position distributions without having to do any measurements at all.

Another way out, which I think doesn't work, is that this a generalized measurement, not a projective measurement (I don't think a collapse to a momentum eigenstate at an earlier time plus unitary evolution is equivalent to a collapse to a position eigenstate at a later time).

 

[BruceW]

This isn't interpretation dependent, since the ground state can always be expanded in the position base. This yields \left|E_1\right\rangle = \int d\vec{r} \,\psi(\vec{r}) \,\left|\vec{r}\right\rangle which is a superposition of dipole moment eigenstates also in the Copenhagen Interpretation. So in principle, measuring the dipole moment of this state should yield non-zero values. Or am I overlooking something?

I think you're right. I guess the ground-state hydrogen atom does create an electric field. Which I guess is obvious, since it is the electric dipole moment that allows most of the transitions to higher-states...

I was totally wrong about all this. I suppose the statistical interpretation does allow for particles to have exact position and momentum.

I was probably confused because my recommended text-book says this:
"If the charge were localised at this point, the atom would possesses an electric dipole moment, which would in principle seem to be measureable: e.g., by observing its effect on a nearby test charge. Experimentally, the dipole moment is found to be zero, so the charge appears to be associated with the spherically symmetric wave function rather than being localised on the particle, which contradicts our starting assumption. Of course, a full DBB calculation of the situation appropriate to such a measurement must produce results identical to those of quantum mechanics, as it does in any other situation, but the ontological implications remain."

 

[Demystifier]

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).

Here you (as well as Ballentine) fail to see a difference between INTERPRETATION of measurement and measurement itself. Yes, this experiment is INTERPRETED as a measurement of momentum, but this is exactly why this NOT a true measurement of momentum. For more details see
https://www.physicsforums.com/showpost.php?p=3552334&postcount=26

 

[atyy]

Here you (as well as Ballentine) fail to see a difference between INTERPRETATION of measurement and measurement itself. Yes, this experiment is INTERPRETED as a measurement of momentum, but this is exactly why this NOT a true measurement of momentum. For more details see
https://www.physicsforums.com/showpost.php?p=3552334&postcount=26

I also think there has to be something like that. The "collapse" (sorry, I have to use naive textbook terminology, I will try to learn BM some day;) is to a position eigenstate at late time although it is a "measurement" of an early time momentum. Thus it cannot be a projective measurement of early time momentum. I wonder whether "generalized measurements" would help with this?

 

[Demystifier]

I also think there has to be something like that. The "collapse" (sorry, I have to use naive textbook terminology, I will try to learn BM some day;) is to a position eigenstate at late time although it is a "measurement" of an early time momentum. Thus it cannot be a projective measurement of early time momentum. I wonder whether "generalized measurements" would help with this?

Generalized measurements would not help. Let me explain.

Instead of naive collapse terminology, let us use the more precise decoherence terminology. (You don't need to know about BM to understand decoherence, but you do need to know about decoherence to understand BM.) In both projective and generalized measurements, there is decoherence involved. The decoherence happens at late time only, and it happens in the (approximate) position basis.

 

[Demystifier]

I was probably confused because my recommended text-book says this:
"If the charge were localised at this point, the atom would possesses an electric dipole moment, which would in principle seem to be measureable: e.g., by observing its effect on a nearby test charge. Experimentally, the dipole moment is found to be zero, so the charge appears to be associated with the spherically symmetric wave function rather than being localised on the particle, which contradicts our starting assumption. Of course, a full DBB calculation of the situation appropriate to such a measurement must produce results identical to those of quantum mechanics, as it does in any other situation, but the ontological implications remain."

Can you say which textbook says this?

 

[BruceW]

Can you say which textbook says this?

"Quantum Mechanics" by Alastair I. M. Rae (5th edition). The book is generally pretty good (as an introduction to QM). But in this case, maybe he made a mistake? He said the dipole moment is zero, but clearly if a measurement was made, the dipole moment would be some nonzero value.

I guess he meant to say the statistical average of the dipole moment over many separate experiments is equal to zero. (Instead of just 'the dipole moment is found to be zero'). So either he got it wrong, or just didn't explain it very well. He wrote this paragraph near the end of his book, in a short chapter on the conceptual problems on QM, so I don't think he was trying to give a rigorous explanation.

 

[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