What Intuitive Insights Explain Heisenberg's Uncertainty Principle?

[Fredrik]

I think by detector you mean the resolution of the detectors at the wall.

Yes, I meant one of the little boxes to the right in the figure in Ballentine's article.

But IMO, the entire slit setup is part of the "detector", simply because in this "generalized" "measurement" where we also try to infer momentum, the inference depends on the entire setup, inlucing L. So I think in the case where we try to as you say, define or generalized some kind of inference of p_y in parallell to infering y, the entire setup is the "detector" IMO.

I disagree. A measuring device (an idealized one) only interacts with the system during the actual measurement, and the measurement is performed on the last state the system was in before the interaction with the measuring device began. In this case, we're clearly performing the measurement on the state that was prepared by the slit, so it can't be considered part of the momentum measuring device. The momentum measuring device consists of the wall of detectors and any computer or whatever that calculates and displays the momentum that we're going to call "the result". The coordinates and size of the slit will of course be a part of that calculation, but those are just numbers typed manually into the computer. Those numbers are part of the measuring device, but the slit isn't physically a part of it.

Why would the uncertainy of the inference increase with L? It seems to be the other way around? Holding
\delta y fixed, and increasing L, decreases \delta \theta and thus the error?

You're talking about the the contribution to the total error that's caused by the inaccuracy of the y measurement. I was talking about a different contribution to the total error. I started explaining it here, but I realized that my explanation (an elaboration of what I said in my previous posts) was wrong. I've been talking about how to define a momentum measurement on a state with a sharply defined position, but now that I think about it again, I'm not sure that even makes sense.

What we need here is a definition of a "momentum measurement" on the state the particle is in immediately before it's detected, and the only argument I can think of against Ballentine's method being the only correct one is that classically, it would measure the average momentum of the journey from the slit to the detector. However, classically, there's no difference between "momentum" and "average momentum" when the particle is free, as it is here. I don't see a reason to think this is different in the quantum world, so I no longer have a reason to think we're measuring "the wrong thing", and that means I can no longer argue for a second contribution to the total error that comes from "measuring the wrong thing". (That was the contribution I said would grow with L).

Also; I'm not thinking in terms of wavefunctions here. I'm thinking in terms of information state;

Huh? What's an information state? Are you even talking about quantum mechanics?

 

[BruceW]

I don't think anyone believes quantum theory is fine as it is :)

 

[Fredrik]

I don't know the derivation, but I believe what those papers say is this. Let's say the transverse wave function at the slit is u(x). If we measure its transverse position accurately, we expect it to be distributed as |u(x)|²; if we measure its transverse momentum accurately, we expect it to be distributed as |v(p)|², where v is the Fourier transform of u. If you measure the transverse position at large L, and for each measured position x_L you take the corresponding sinθ_L, where tanθ=x_L/L, then sinθ_L is distributed like |v(p)|².

OK, thanks. If anyone knows a derivation (or a reason to think this is wrong), I'd be interested in seeing it. (I haven't tried to really think about this myself).

This is the same procedure Ballentine uses to get the momentum. So I believe that his momentum distribution is an accurate reflection of the momentum at an earlier time.

I still don't understand the significance of this. If we replace the wall of detectors with a photographic plate and make L large, how does it help us to know that the image we're looking at is the momentum distribution of the initial state (the state that was prepared by the slit)?

I know that I've been talking about how to define a momentum measurement on that initial state (sorry if that has caused confusion), but what we really need to know is how to define a momentum measurement on the state immediately before detection. I mean, we're performing the position measurement on that state, so if we're going to be talking about simultaneous measurements, the momentum measurement had better be on that state too.

 

[atyy]

I still don't understand the significance of this. If we replace the wall of detectors with a photographic plate and make L large, how does it help us to know that the image we're looking at is the momentum distribution of the initial state (the state that was prepared by the slit)?

I know that I've been talking about how to define a momentum measurement on that initial state (sorry if that has caused confusion), but what we really need to know is how to define a momentum measurement on the state immediately before detection. I mean, we're performing the position measurement on that state, so if we're going to be talking about simultaneous measurements, the momentum measurement had better be on that state too.

Ballentine's procedure gives the position distribution of the state just before detection. It also gives the momentum distribution of the initial state (just after the slit), which is not the momentum distribution of the state just before detection. So he does not have simultaneous accurate measurement of both position and momentum.

 

[Fredrik]

Ballentine's procedure gives the position distribution of the state just before detection. It also gives the momentum distribution of the initial state (just after the slit), which is not the momentum distribution of the state just before detection. So he does not have simultaneous accurate measurement of both position and momentum.

Aha. You're saying that because of what you described in the post before the one I'm quoting now, the position distribution (which we are measuring) is the same function as the momentum distribution of the initial state, and that this means that we're performing the momentum measurement on the wrong state.

That promotes the issue of how to prove (or disprove) that claim to the main issue right now.

 

[Fredrik]

it's an extrinsic theory; depending on a classical observer context.

Isn't this a problem with all theories, i.e. all sets of statements that satisfy some kind of falsifiability?

 

[atyy]

Aha. You're saying that because of what you described in the post before the one I'm quoting now, the position distribution (which we are measuring) is the same function as the momentum distribution of the initial state, and that this means that we're performing the momentum measurement on the wrong state.

That promotes the issue of how to prove (or disprove) that claim to the main issue right now.

Yes, that's what I'm thinking.

 

[Fra]

I disagree. A measuring device (an idealized one) only interacts with the system during the actual measurement, and the measurement is performed on the last state the system was in before the interaction with the measuring device began. In this case, we're clearly performing the measurement on the state that was prepared by the slit, so it can't be considered part of the momentum measuring device. The momentum measuring device consists of the wall of detectors and any computer or whatever that calculates and displays the momentum that we're going to call "the result". The coordinates and size of the slit will of course be a part of that calculation, but those are just numbers typed manually into the computer. Those numbers are part of the measuring device, but the slit isn't physically a part of it.

This is where I think we either disagree or aren't trying to do the same thing. Normally I agree with you, ie. if all we are doing is measuring position at the plate. Then I agree.

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. And in THIS case, since as you acknowledge below, we really have an "average" throughout the construct, then this has to be respected by the y measurement as well, other wise we are IMO not inferring y and p_y from the same information - thus the comparasion of uncertainties make no sense at all.

I've been talking about how to define a momentum measurement on a state with a sharply defined position, but now that I think about it again, I'm not sure that even makes sense.

Mmm ok. Then we were trying to accomplish different things. I don't think this makes sense either. I mean, sure we could come up with some type of calculation of dy and dp, but in the way you seek it I think it would not correspond to the same information state (see below).

Huh? What's an information state? Are you even talking about quantum mechanics?

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

I just mean that wavefunction gives a part classical flavour. I think more in terms of an abstracte state vector (which is of course suppsedly encoding the same info as the wavefunction) but interpreted differently from ballentines stat int.

The interpretation is that, instead of a thinking of the state vectors as encoding information about statistical ensemble, realized as an infinity of identical prepared systems etc, I'm thinking of the observers state of information/knowledge about the system.

Technically this is not a properpy of the system, it's a property of hte STATE of the observing system. Only at equilibrium, does the state of the observers information about the system match at least in some sense the system. The point is that this interpretation allows understanding the concept of information state, even when no ensemble can be realized, or when the information that "should need to go into the ensemble" must be truncated simply because the observing system is NOT and infinite environment serving as informaiton sink, but rather a finite mass subsystem of the universe.

But it's not news that my interpretation of QM, is not at all like Ballentines statistical view.

What we need here is a definition of a "momentum measurement" on the state the particle is in immediately before it's detected, and the only argument I can think of against Ballentine's method being the only correct one is that classically, it would measure the average momentum of the journey from the slit to the detector. However, classically, there's no difference between "momentum" and "average momentum" when the particle is free, as it is here. I don't see a reason to think this is different in the quantum world, so I no longer have a reason to think we're measuring "the wrong thing", and that means I can no longer argue for a second contribution to the total error that comes from "measuring the wrong thing". (That was the contribution I said would grow with L).

This sounds like the objection I have too.

I phrased it differently ,but the objection is similar. What we do infer is the momentum "spread out" over the time from where the information used for inference originates. This is why this is also the "time stamp" for any y measurement we want to "associate" to the same inforation. Ie. this is why I ague for the extra uncertaint in y. It's not because the error at the screen is larger than the detector cell, but because we are force to add this error if we insist on associating it with the infered p_y average.

So instead of saying "we infer the wrong thing", I took that whatever we measured as the starting point, given ballentimes scheme, and then suggested that to make it a coherent inferent we need to adjust the y-inference as well and add the error.

When you look at what information, that is used for an inference. This becomes more clear. The only "time stamp" we have are parameterizations of how the information set evolves. An intrinsic comparasion must work on the same information set (corresponding to the generalisation of conjugate at same time).

Atyy's points has been similar althjough I didn't read all the quoted papers, except the way you put it depends on your interpretation.

/Fredrik

 

[Fra]

Isn't this a problem with all theories, i.e. all sets of statements that satisfy some kind of falsifiability?

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.

This is also why these old scheme works perfectly fine for subsystems, but not for cosmological scale theories and also presents problems for understanding unification of interactions. The cosmologicla theory issues here (where the ensembles and subsystem idealizaion obviouls does break down) becomes hot topic in understanding unification if you think that any subsystem acts rational upong what information it does have about it's environemtn. Then it's clear that the "inference" that ultimately results in "theories" here are an essential part of a proper "theory scaling", which is essentialy IMO at the core of unification.

/Fredrik

 

[Fra]

Example: what is the important trait of life? It's not just that fact that we are mortal. No, the magic lies in the variation/adaption/reproduction learning from mistakes etc. What one needs to assert, is not that the theory can be wrong, but that the theory comese with a framework that allows progress THROUGH falsification in a rational way.

Biology has accomplished this, and I think so has phyislca laws, it's just that we humans scientists just haven't understood it "that way" YET ;)

This is the difference between seeing a theory as a "static description" or an "interaction tool", where the latter obviously means we rather have "evolving expectations".

/Fredrik

 

[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.