- #211
atyy
Science Advisor
- 15,168
- 3,374
It is uncontested that you can measure both at the same time if you don't care about accuracy.
Uncontested by you perhaps.atyy said:It is uncontested that you can measure both at the same time if you don't care about accuracy.
Fredrik said:Uncontested by you perhaps.
Fredrik said:I haven't looked at the Raymer article yet, but I also feel that if any kind of limit is supposed to be a part of the definition of this momentum measurement, it's L→0, not L→∞. The reason is that what we're measuring is more like an average momentum than the "momentum right now". The position measurement is performed on a particle with a wavefunction that has had some time to spread out. To claim that we have really performed a simultaneous measurement, we should measure the momentum when the particle is in the same state as when we measure the position, but the momentum measurement involves two different times, and the wavefunction is spreading out over time. So it seems that we are closer to a "true" simultaneous measurement when L is small.
I'll have to agree with this. Confidence levels and confidence intervals is what the SD is all about anyway.atyy said:It is uncontested that you can measure both at the same time if you don't care about accuracy.
But theory does not predict the outcomes of single measurements (single data points) anyway. It only predicts the ensemble properties.Fredrik said:It also seems to me that this is precisely the type of "inference" that measuring devices do when they test the accuracy of the theory's predictions, so how can anyone not call it a measurement?
Can you describe his method or quote the relevant part of the article?atyy said:I also suspect that Raymer's momentum "measurement" isn't a true momentum measurement. If it were, we would expect the state to collapse into a momentum eigenstate before the "measurement", since the "measurement" at infinite time is supposed to reflect momentum at finite t. Raymer says "This mapping of the momentum distribution into position for large L is analogous to far-field diffraction in optics". My guess is that it isn't a true momentum measurement, because it uses some knowledge of the state. At one extreme, if one knows the state of particle, one can get both position and momentum distributions with no measurement and no collapse at all.
I would describe it as a series of approximate position measurements that together can be considered a single momentum measurement. Do we want to call this a simultaneous measurement of both? Maybe. It seems to be a matter of semantics, and taste. The argument in favor of calling it a simultaneous measurement is of course that by the end of it, we have obtained a value of each of the position components and each of the momentum components. The argument against it would be that we only need one of the liquid drops to obtain the values of the position components, but we need several to obtain the values of the momentum components.atyy said:...what else could the tracks in a cloud chamber be but simultaneous position and momentum measurements?
Fredrik said:However, since all momentum measurements seem to involve at least two approximate position measurements (or at least a preparation of a state with sharply defined position, followed by a position measurement), I don't think there can exist a meaningful definition of what it would mean to measure the momentum of a particle in an arbitrary state. This is probably as good as it gets. Momentum measurements of the type suggested by von Neumann's projection axiom don't exist.
atyy said:Here's a free article that describes the same thing as Raymer: http://tf.nist.gov/general/pdf/1283.pdf .
It talks about the position and momentum "shadows" of an initial state. The shadow of position occurs at a different time from the shadow of momentum. So Ballentine is wrong because he is not talking about canonically conjugate position and momentum.
SpectraCat said:[EDIT] The more I think about this .. the more wrong it seems. The whole discussion of eigenstates as "the only possible results" of a "measurement" clearly has some kernel of truth to it, but it seems like a drastic over-simplification. On the other hand, it seems like any oversimplification must not matter very much, given the long, strong history of agreement between QM theory and experiment. I am getting more confused by the minute here.
I agree that one of the things we're discussing is if what Ballentine is describing is a momentum measurement, but I consider the issue of what new state is prepared by the measurement trivial. Either the particle is absorbed by the detector and no new state is prepared, or the particle makes it all the way through the detector and escapes on the other side. In that case, the wavefunction is sharply peaked at the location of the detector, and is close to zero outside of it.Fra said:what seems to be of debate is wether the example in the Ballentine notes Fredrik posted where one is using an inference to "measure" momentum can qualify as a measurement, and thus wether one can define at least loosely speaking (until proper full analyis is made) some "effective state" that derives from the mixed measurements + inference?
I don't understand what this means.atyy said:I think Ballentine's claim that the distribution of position values at large L corresponds to the momentum distrubution at small L is true.
Fredrik said:I don't understand what this means.
Fredrik said:The rest of what you said seems to be based on the assumption that the new state is going to be spread out all over the region between 0 and y+δy. I don't see a reason to think that.
Fredrik said:Ballentine's argument relies on his claim that [itex]\delta p_y[/itex] can be made arbitrarily small by making L large. This claim appears to be false. (It's correct if we leave out the L→0 statement from the definition of momentum measurement, but it's false if we include it).
atyy said:I think Ballentine's claim that the distribution of position values at large L corresponds to the momentum distrubution at small L is true.
Fredrik said:I don't understand what this means.
What figure 2c seems to indicate is that in a double-slit experiment, we won't get the typical "both slits open" interference pattern if the particles are moving too fast. Making L small should have the same effect. In either case, the wavefunction won't have spread out enough in the y direction by the time its peaks reach the screen.atyy said:Try http://tf.nist.gov/general/pdf/1283.pdf, figure 2. In the text on the left column of p25, they say: "Figure 2c shows the results predicted by theory for atoms with a wide range of propagation times. In the extreme Fresnel regime, we recognize the spacelike shadow of the two slits. With increasing td, the wavepackets start to overlap and interfere until, for large td, we arrive at the Fraunhofer regime in which the diffraction pattern embodies the momentum-like shadow of the state."
...
So the position distribution on the screen at large L (Fraunhofer regime) corresponds to the momentum distribution of the initial state, whereas the position distribution on the scrren at small L (Fresnel regime) corresponds to the position distribution of the initial state.
Only if it was known to be in a state with a sharply defined position earlier. Maybe not even then. This is still a matter of some debate in this thread. Ballentine's thought experiment is just a particle going through a single slit, and then reaching a wall of detectors. This wall of detectors could be a photographic plate. Those details aren't important here.dx said:The momentum of an electron can be measured just by letting it fall on a photographic plate, and so we know both the position and momentum.
I'm not sure I understand what you're saying. What do you mean by "spread of the information set"? If you're talking about the width of the wavefunction after the detection, how could it be larger than the detector?Fra said:And the spread of the y above, IS the spread of the information set we do use for the inference.
See my answer to dx above. Does this help you understand what I mean at least?Fra said:Not sure what you mean. It seems to me that ballentine is right on that point.
So we know that it's a momentum eigenstate, and just need to find out which one that is? I think we would need to detect the particle twice to be able to calculate a momentum, and if we do, the first detection will change the state of the particle. I don't know if this should be called a "momentum measurement". (I'm thinking it probably shouldn't).dx said:What I had in mind was simply a state which is prepared with a definite momentum, which is then measured by the photographic plate. So when the particle falls on the plate, we know its position and also its momentum because we have measured ('prepared') the momentum before.
Yes, I think this is what we are discussing, and I was proposing something in the direction.Fredrik said:I think we have to define what we mean by a "momentum measurement" in this situation
I think by detector you mean the resolution of the detectors at the wall.Fredrik said:I'm not sure I understand what you're saying. What do you mean by "spread of the information set"? If you're talking about the width of the wavefunction after the detection, how could it be larger than the detector?
Fredrik said:So I want to use a definition that implies that the value of py that's inferred from the y measurement is only an approximate measurement, and that the inaccuracy of the y measurement isn't the only thing that contributes to the total error. There's also a contribution that depends on L (and goes to zero when L goes to zero) that must be added to the contribution from the inaccuracy in the y measurement.
Since the error depends on L, it should grow at least linearly with L. So I want to define a "momentum measurement with minimum error L" as a y measurement at x coordinate L, followed by a calculation of py. Maybe that should be "with minimum error kL", where k is some number, but right now I don't know what number that would be, so I'm just setting it to 1.
Fredrik said:What figure 2c seems to indicate is that in a double-slit experiment, we won't get the typical "both slits open" interference pattern if the particles are moving too fast. Making L small should have the same effect. In either case, the wavefunction won't have spread out enough in the y direction by the time its peaks reach the screen.
I see that the pattern will depend on the initial wavefunction (and therefore on its Fourier transform), and that it will "look like" the wavefunction itself when L is small. I don't see how the pattern will "correspond to the momentum distribution of the initial state" when L is large. Do you mean that it will actually "look like" the Fourier transform of the wavefunction?
I still don't see how to interpret your statement in the first quote above. Did Ballentine even say something like that? Which one of his statements have you translated into what you're saying now?
I also don't see what this implies about the single-slit experiment.
I'm not saying that you're wrong, only that I don't understand what you're thinking.
Yes, I meant one of the little boxes to the right in the figure in Ballentine's article.Fra said:I think by detector you mean the resolution of the detectors at the wall.
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.Fra said: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.
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.Fra said:Why would the uncertainy of the inference increase with L? It seems to be the other way around? Holding
[itex]\delta y[/itex] fixed, and increasing L, decreases [itex]\delta \theta[/itex] and thus the error?
Huh? What's an information state? Are you even talking about quantum mechanics?Fra said:Also; I'm not thinking in terms of wavefunctions here. I'm thinking in terms of information state;
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).atyy said: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 xL you take the corresponding sinθL, where tanθ=xL/L, then sinθL is distributed like |v(p)|2.
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)?atyy said: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.
Fredrik said: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.
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.atyy said: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.