Exploring Heisenberg's Uncertainty Principle: Intuition & Explanations

[kith]

fuesiker, there's nothing wrong with questioning postulates, as long as the discussion remains serious. Especially the collapse postulate has stirred discussions since the CI took form and there are many interpretations which do not require it.

Frederik has proven in numerous threads that he is not a crackpot but a person with very good understanding of basic QM who has not stopped to ask questions. I find the discussion here quite stimulating, so it would be a pity if the thread was ruined by personal issues.

I'll be writing more about the physics when I find the time.

 

[Fredrik]

Thank you kith.

 

[atyy]

I agree with Kith's sentiments in #141 too.

I have been dreaming about corn flakes that taste like fish for the last few hours now

http://www.chowandchatter.com/2010/01/guest-post-fish-floss.html

 

[fuesiker]

No, it's a matter of definition (of "measurement"). So far, neither you nor anyone else have presented a valid argument for the definition that makes the above "plain wrong" should be preferred over the alternative.


This isn't about mathematical reasoning or physics. You just don't understand the meaning of the words "has something to do with".

Two of your statements are "plain wrong" regardless of what measurement is defined as. The HUP is not dependent on measurement, nor on preparation, no matter what you define the former or the latter. Now again dodge this like you do all the statements I make that prove what you're saying is at odds with QM theory.

Also, I think I understand very well that when a variable "has something to do" with a certain value, then that value depends on that variable. You know this is true, but you just want to dodge based on what I can conclude from your behavior on this thread.

 

[fuesiker]

i agree with kith's sentiments in #141 too.

I have been dreaming about corn flakes that taste like fish for the last few hours now

http://www.chowandchatter.com/2010/01/guest-post-fish-floss.html

lol.

 

[fuesiker]

What you said right now is an outrageous lie. You're lying about having refuted things I've said. If you think I have lied about anything, it's because you have reading comprehension issues and lack the honesty to admit your mistakes even to yourself. Your counterarguments are usually not wrong, but are always irrelevant.

When I throw things back at you, it's because you're falsely accusing me of exactly the things you're actually doing yourself. It's really bizarre. I haven't seen anyone behave this way since the last time I had an argument with someone who believes that there are people who can talk to ghosts.

That was actually me and there *are* people who talk to ghosts (just kidding).

You accuse me of reading comprehension issues and I think I only proved that I understand things quite fast by understanding the general meaning of the paper you posted (now of course you will deny that). But keep your false accusations going. I will stick, as usual, to the physics. My counterarguments "are usually not wrong", lol. They so far have been all correct, the physics ones, and you're the only one on here who has made axiomatically false statements (with conviction) about quantum mechanics, and I *did* refute them (read the many posts where I explained to you in excruciating detail why I am at odds with your statements and why they are wrong, those being that HUP "has something to do" with the wavefunction or measurement, which is plain wrong), but you still accuse me of refuting things you didn't say, when everyone on here sees that you did say them. Please tell me how my rebuttals to your above statements were irrelevant. They explicitly and directly address your statements, which again (I have to keep repeating myself with you) were that HUP has something to do (aka depends on as I explained earlier what "has something to do with" means mathematically, it means dependence) with preparation and/or measurement, notions that are false no matter what you define these two.

Please, be mature and honest, and don't just write back asking the same of me. Come up with your own stuff. Or ask Ballentine for some advice.

 

[Fredrik]

Two of your statements are "plain wrong" regardless of what measurement is defined as. The HUP is not dependent on measurement, nor on preparation, no matter what you define the former or the latter. Now again dodge this like you do all the statements I make that prove what you're saying is at odds with QM theory.

Also, I think I understand very well that when a variable "has something to do" with a certain value, then that value depends on that variable. You know this is true, but you just want to dodge based on what I can conclude from your behavior on this thread.

"Depends on" is your choice of words, not mine. My statements of the form "X has something to do with Y" are only wrong if one accepts your definition of "has something to do with", which everyone can see is at odds with the English language.

We were talking about a theorem, not a variable. Would you also say that the theorem

For each $x\in\mathbb R$, we have $x^2\geq 0$.​

has nothing to do with real numbers? By your definitions and logic, it must not have anything to do with real numbers, since that would mean that it depends on what number we're talking about.

 

[SpectraCat]

SpectraCat, of course you can do interferometry on single photons, even on one photon with itself. You can pass a photon (or even an electron, or with the proper cooling method, a whole car - this is way too difficult obviously, but can be done) and make that object (photon or car) interfere with itself. We're all waves, my friend.

What makes you think any of that was unclear to me prior to your post? I asked you to describe the design of an interferometer that could provide phase data about single photons so as to constitute a momentum measurement that does not also involve position measurement. You claimed in an earlier post that interferometry was capable of measuring momentum of individual photons without measuring their positions. I cannot see how that is possible, so I asked you to explain it.

 

[fuesiker]

Ok, then please describe in detail how you would construct an interferometer for *single photons*

This is what made me think it was unclear to you. You obviously (from your request) seem to not be familiar with how interferometry on a single photon works. This makes me conclude that it is not familiar to you to see single-photon interference akin to many-photon interference, hence my explanation.

This is too detailed to explain on here. Please refer to Loudon's Quantum Theory of Light.

 

[Naty1]

This WAS a fascinating discussion...before all the accusations started.

Instead, could one or more of the principals in the discussion maybe take a crack at stating points of mutual agreement??

for example post # 121 was very helpful:

Originally Posted by fuesiker
But this is not just the case with momentum, its the case with every observable and kith mentioned this in an earlier post. For example, if you want to measure the frequency (or energy) of a photon, you will not get a sharp frequency, but a spread of frequencies. This is because the photon interacts with your CCD camera (or SPAD, whatever) for a finite period of time dt, and we come back to HUP where dt*dE >= h (times some constant), and this means there will always be uncertainty in anything you measure...

from atyy: I agree

 

[fuesiker]

I asked you to describe the design of an interferometer that could provide phase data about single photons so as to constitute a momentum measurement that does not also involve position measurement. You claimed in an earlier post that interferometry was capable of measuring momentum of individual photons without measuring their positions. I cannot see how that is possible, so I asked you to explain it.

OK I decided to answer this quickly, but next time ask in a more polite way ;). Moreover, I think this is something many were keen about here because Fredrik's propaganda machine had wrongly (yet again) made people on here think you can't measure momentum unless you measure position (I don't have the time or the motivation to dig this up in his many posts, but he did say something in that color). I may add that I heard drug addicts at the local train station here make more sense than that. I hope the below experimental description (which is way too simplified in terms of technical, yet not conceptual, matters) will put an end to this myth of having to measure position to measure momentum.

As I mentioned before, a photon is a wave and a particle. Now, you want to measure a wave phenomenon, so you better not know anything about the particle behavior of the photon. You can do this using either a double-slit experiment or the Mach-Zehnder interferometer. Let's take the latter, because it is easier, in my opinion. I think this is done in experimental quantum optics courses as a lab task too.

Now, your Mach-Zehnder interferometer has two arms of lengths $l_1$ and $l_2$. The whole essence of interference is the fact that you cannot tell which path the photon takes, whether the first arm or the second arm. You can think of it in a naive way as the photon takes both at the same time (wave nature of the photon). Now you don't know what the wavelength $\lambda$ of your photon is, but we assume that you are sure only one photon is going through the interferometer at any given time (this is experimentally laborious, but it is done). When $l_1=l_2$, then the photon interferes contructively with itself, and you hear a click at a detector that you have placed at the output of the interferometer. Now don't fool yourself into thinking this is measuring the position of the photon by the detector. We are inferring nothing from the measurement on the detector other that a photon has hit it. We make no further use of this knowledge, nor do we need to. Now as you vary $l_2$ (in some non-random process, such as a linear ramp), while keeping $l_1$ fixed, you start going away from constructive interference. Then at a certain value of $l_2$, you don't hear any clicks on the detector anymore -> destructive interference (of course, you are sending similarly-prepared photons each time you make a measurement). This allows you to determine the momentum $k=\frac{2\pi}{\lambda}$ of the photon you have up to machine precision, without knowing anything about its position. To see this more clearly, intentionally make your detector one huge pixel whereby anywhere the photon hits it, it gives you a click, yet you can have this pixel occupy a huge spatial extent such that the uncertainty in your position can be considered, for all practical purposes, infinite. I call such an extension "the Fredik-propaganda negator factor".

As you can see, this is pretty much the same procedure for interfering two photons, or two laser beams. That's why I thought you did not know that a single photon can interfere with itself, and I still think you could not have actually believed this when you did not think it could be done experimentally, because then obviously you when you think that is not possible you also have not seen it in nature (otherwise that would be your experiment), and hence, it does not make sense to say you do know a single photon interferes with itself when you believe it is impossible to see this effect.

But I really suggest Loudon's book. It's my favorite quantum optics book.

 

[Fra]

What a long thread.

Now that we are convinced of this, we can look at the properties of the Fourier transform. If a function K is the Fourier transform of a function F, then the sharper F is, the broader is K.

The uncertainty principle has nothing to do with observation. It does not arise due to observation. It is the nature of things that there are non-commuting observables (time and energy, momentum and position) that cannot be determined simultaneously to arbitrary precision.

I personally think this is the best answer to the OT.

The rest of the discussion seems to be about howto get from single datapoints to knowledge, and howto distinguish information from single measurements from information from an acquired history of measurments (also called preparation). This tends to be very interpretation dependent as always.

I in particular like the perspective Fredrik added that you can INFER momentum from position (and vice versa). But at least in my picture, such an inference does not constitute a measurement, it constitutes an internal process in the observing system. What would however be extremelt interesting (but also off topic) is to discuss if there is such a thing as primary and secondary observables, which are linked by such inferences. This reminds me of The principle of relative locality http://arxiv.org/abs/1101.0931 where they argue that spacetime is the result of an inference made from local momentum measurements.

One can take such a thing more seriously, and question wether this "inference" is internal processes or not. After all, NO local observer, can possibly make measurements of remote points. EVERYTHING is ultimately a local inference (and extrapolation) fro locally acquired information.

/Fredrik

 

[Fredrik]

Instead, could one or more of the principals in the discussion maybe take a crack at stating points of mutual agreement??

The thing is, we agree about essentially everything except the meaning of the words "has something to do with" and whether it makes sense to define a "measurement" as what a measuring device does in an experiment that tests the accuracy of the theory's predictions. (The alternative is to define it in purely mathematical terms as a projection onto an eigenspace). Feusiker has a very strange interpretation of "has something to do with" and thinks anyone who uses the word measurement in the former sense is a complete idiot who doesn't know anything about physics and deserves to be insulted and ridiculed at every possible opportunity. (I think I've been extremely polite to him under the circumstances). He hasn't offered any arguments to support his position, but instead insists that he's right because he has a PhD and because (according to him) Cohen-Tannoudji and Sakurai would both agree with him and Ballentine is an idiot.

We obviously also disagree about whether he has successfully refuted the things I said. He has repeatedly made arguments that are essentially correct, but don't even come close to addressing the things I actually said.

 

[fuesiker]

The thing is, we agree about essentially everything except the meaning of the words "has something to do with" and whether it makes sense to define a "measurement" as what a measuring device does in an experiment that tests the accuracy of the theory's predictions. (The alternative is to define it in purely mathematical terms as a projection onto an eigenspace). Feusiker has a very strange interpretation of "has something to do with" and thinks anyone who uses the word measurement in the former sense is a complete idiot who doesn't know anything about physics and deserves to be insulted and ridiculed at every possible opportunity. (I think I've been extremely polite to him under the circumstances). He hasn't offered any arguments to support his position, but instead insists that he's right because he has a PhD and because (according to him) Cohen-Tannoudji and Sakurai would both agree with him and Ballentine is an idiot.

Fredrik, I was aggressive with you as much as you've been with me. And no, we don't agree on everything. Cohen-Tannoudji and Sakurai (if the latter was alive) would agree with me unless their books are wrong. Theirs are my quantum mechanics bibles. Anyway, read the post I made, it is 2 before your post that I am quoting here. Here's a way to measure the momentum of a single photon without knowing anything about its position, something you claimed in earlier posts (and do not deny this) cannot be done. And I hope you won't say this method is crap or not right. I am certain it is done experimentally in basic quantum optics lab work, and I do not feel I need to go searching for the proof. Moreover, if you truly understand quantum mechanics, my description will make perfect sense to you.

Moreover, despite what you may think, I have respect for you. In the end, you do seem to love physics, though yourself more.

 

[Fredrik]

Fredrik, I was aggressive with you as much as you've been with me.

This is easily the most absurd statement you have made so far.

Since you have made me spend almost the entire day defending myself, I haven't yet had time to get up to speed about momentum measurements with interferometers. I will as soon as I can.

 

[Naty1]

Is the following agreed upon:

Atyy:

The uncertainty relation is defined as the non-commutation of position and momentum operators. A state with definite momentum is an eigenstate of momentum, and state with definite position is an eigenstate of position. The commutation relation prevents an eigenstate of momentum from being an eigenstate of position, so there is no state with definite momentum and position, and so it cannot be prepared.

And Fra's post endorsing this:
Originally Posted by fuesiker

Now that we are convinced of this, we can look at the properties of the Fourier transform. If a function K is the Fourier transform of a function F, then the sharper F is, the broader is K.

The uncertainty principle has nothing to do with observation. It does not arise due to observation. It is the nature of things that there are non-commuting observables (time and energy, momentum and position) that cannot be determined simultaneously to arbitrary precision.

These two,atyy and Fuesiker, don't conflict, right?

I thought I'd check Wikipedia for comparsion with all the discussion above; it was a bit discouraging (This is the first line) :

The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus...

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

 

[fuesiker]

Is the following agreed upon:

Atyy:


And Fra's post endorsing this:
Originally Posted by fuesiker



These two,atyy and Fuesiker, don't conflict, right?

Right! Any two observables that do not commute cannot have common eigenstates.

 

[Fredrik]

Regarding atyy's statement: I disagree with the words "defined as", but the commutator is certainly involved in the derivation. The rest of it is OK.

Regarding Fra's Feusiker quote: I disagree with the words "has nothing to do with", but I agree with everything else in that quote. Edit: Uh wait, I didn't look at it closely enough. Give me a minute to think.

OK, I have thought about it. I have some issues with the statement that the two non-commuting observables can't be determined simultaneously. I interpret "determined" as "measured", and I consider a measurement to be what a measuring device does in an experiment that tests the accuracy of the theory's predictions. If experimentalists consider a detection of a particle, followed by a calculation of a momentum from the coordinates of the detection event, a valid way to measure momentum, then position and momentum can be measured simultaneously. In fact, we would be measuring momentum by measuring position.

Also, "time" isn't an observable.

 

[atyy]

Naty1, the interpretation of quantum mechanics has no consensus, but I think we aren't discussing that. We are discussing things within the simple-minded textbook interpretation of say Cohen-Tannoudji, Sakurai, or Chuang and Nielsen.

The statements at issue:

1. Non-commutativity of position and momentum means that sharp measurements (projective measurements) of position and momentum cannot be simultaneous.

2. A sharp momentum measurement is the same as two sharp position measurements plus timing.

3. After simultaneous sharp measurements of position and momentum, the state collapses to a position eigenstate.

In my current understanding (remember I am a biologist, and I claim no authority in physics - nor biology for that matter): 1 is true, if we also add that nothing about the initial position and momentum of the particle is known (ie. for arbitrary initial states). 2 is not true, again for arbitrary initial states, since if successive position measurements are used to approach an accurate momentum measurement, the position measurements must not be sharp. 3 is not true since it directly contradicts the projection postulate. I believe the Ballentine, Park and Margenau argument that 1 is not true only works for certain initial states and if in using 2 to measure momentum the second sharp position measurement is taken at infinite time, which is why I have added the qualification "arbitrary states" (see also the quote from Bell in post #89).

 

[fuesiker]

Regarding atyy's statement: I disagree with the words "defined as", but the commutator is certainly involved in the derivation. The rest of it is OK.

Regarding Fra's Feusiker quote: I disagree with the words "has nothing to do with", but I agree with everything else in that quote.

It indeed has nothing to do with observation in the sense that you do not need to observe for the HUP to hold. You keep dodging this. What effect does observation have to do with the HUP? Please explain it to me. What does observation have to do with the relation

$\langle (\Delta x)^2\rangle \langle(\Delta p)^2\rangle \geq \frac{\hbar^2}{4}$

Perhaps you mistakenly think it is because of the expectation values involved which "have to do something with wavefunctions" as you put it? Well, the thing is, these expectation values are there whether you observe or not. They can take on any value, whether one you can measure, or one you cannot (expectation values of the momentum operator with respect to a position wavefunction). In such a sense, the HUP has nothing to do with observation. Your observation, if you make it and you don't have to for the HUP to hold, will only alter $\Delta x$ and $\Delta p$, but not the HUP, and I believe you understand this. But it is perfectly correct to say the HUP has nothing to do with observation, since, as I discuss above, the expectation values in the HUP need not be those resulting from an observation. I insist you kindly address this post before claiming again that it is wrong to say the HUP has nothing to do with observation. It indeed has nothing to do with it.

 

[Fredrik]

It indeed has nothing to do with observation in the sense that you do not need to observe for the HUP to hold.

Of course. It's a theorem. (That's the reason why I can't make myself call it a "principle", no matter how standard that term is). But it doesn't make much sense to say that it has nothing to do with observation when of the things it tells us is how the results of measurements (i.e. observations) will be statistically distributed around the expectation values.

 

[fuesiker]

Of course. It's a theorem. (That's the reason why I can't make myself call it a "principle", no matter how standard that term is). But it doesn't make much sense to say that it has nothing to do with observation when of the things it tells us is how the results of measurements (i.e. observations) will be statistically distributed around the expectation values.

Yes, but this is better stated as observation has something to do with the Heisenberg Uncertainty Principle rather than the Heisenberg Uncertainty Principle has something to do with observation. It is the nature of things, embedded in the HUP, that makes observation the way it is, and not the other way around. Moreover, like I mentioned, even if you're not "looking", the HUP is there. In this sense alone, the HUP has nothing to do with observation, because, like I said repeatedly (which you keep ignoring), the HUP is there whether you observe or not.

 

[jtbell]

This thread is temporarily closed pending moderation.

 

[jtbell]

This thread is now open for discussion again. My apologies for the delay. It took me a long time to read and digest the whole thread, because I had not been following it closely, and it sort of "blew up" rapidly.

I'd like to issue a general reminder that it's OK to disagree with other people, but PF's policy is that you need to at least "play nice" while doing so. Personal attacks and other ways of "personalizing" an argument only serve to degrade the tone of discussion and detract from facts and logical arguments. (Click the "Rules" link at the top of any page and note the section Guidelines on Langauge and Attitude.)

This is an interesting subject, and I'd hate to see the discussion dragged downhill again.

 

[jtbell]

Much of this thread has revolved around a thought experiment in which one sends a particle through a single slit, and then observes its transverse position as it hits a screen further on. This also allows you to infer the particle's transverse momentum by reconstructing its path from the slit. The question is, how is this consistent with the Heisenberg uncertainty principle?

Here's a variation of the experiment that might shed some light on it. Imagine the screen as a collection of small pixel-detectors. Remove one of the pixels to create a small hole in the screen. If the particle passes through the hole, we know its transverse position immediately after the screen, just as if it had hit a detector there. But its momentum immediately after the screen is uncertain, because it has once again undergone diffraction.

It seems to me that in this case the HUP applies to the "future" momentum of the particle and not to the "past" momentum of the particle which we inferred from the position measurement, and that inferring the "past" momentum doesn't qualify as a "measurement" in the sense of the HUP and the common statement that we cannot "measure" two non-commuting variables simultaneously.

 

[atyy]

Much of this thread has revolved around a thought experiment in which one sends a particle through a single slit, and then observes its transverse position as it hits a screen further on. This also allows you to infer the particle's transverse momentum by reconstructing its path from the slit. The question is, how is this consistent with the Heisenberg uncertainty principle?

Here's a variation of the experiment that might shed some light on it. Imagine the screen as a collection of small pixel-detectors. Remove one of the pixels to create a small hole in the screen. If the particle passes through the hole, we know its transverse position immediately after the screen, just as if it had hit a detector there. But its momentum immediately after the screen is uncertain, because it has once again undergone diffraction.

It seems to me that in this case the HUP applies to the "future" momentum of the particle and not to the "past" momentum of the particle which we inferred from the position measurement, and that inferring the "past" momentum doesn't qualify as a "measurement" in the sense of the HUP and the common statement that we cannot "measure" two non-commuting variables simultaneously.

But why should the "past momentum" be disqualified? Is it because the past momentum cannot be measured for arbitrary initial states, ie. for certain initial states, if one has some knowledge of the initial state, then one can perform "single meaurements" that yield a distribution of (p,q) values that when marginalized give the correct p and q distributions of non-simultaneous measurements? (I'm not sure that's what's really happening, but I think that is Park and Margenau's definition of simultaneous precise measurement, as well as Busch et al's.)

 

[Fredrik]

I think it's a little bit misleading to say that we're "inferring the past momentum", because the state before the measurement was a state with a sharply localized position, not momentum. The particle didn't have a well-defined momentum before this inference.

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? These are the answers I can think of right now:

a) Because if we do, the common statement that we can't simultaneously measure position and momentum wouldn't be true.

b) Because if we do, von Neumann's axiom that all measurements project the state onto an eigenspace of the measured observable would contradict itself.

c) Because if we do, we would have to distinguish between the terms "measurement" and "state preparation".​

Those are, in my opinion, very bad reasons to not use the term "measurement" for what measuring devices do when they test the accuracy of the theory's predictions.

 

[atyy]

I think it's a little bit misleading to say that we're "inferring the past momentum", because the state before the measurement was a state with a sharply localized position, not momentum. The particle didn't have a well-defined momentum before this inference.

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? These are the answers I can think of right now:

a) Because if we do, the common statement that we can't simultaneously measure position and momentum wouldn't be true.

b) Because if we do, von Neumann's axiom that all measurements project the state onto an eigenspace of the measured observable would contradict itself.

c) Because if we do, we would have to distinguish between the terms "measurement" and "state preparation".​

Those are, in my opinion, very bad reasons to not use the term "measurement" for what measuring devices do when they test the accuracy of the theory's predictions.

Could you clarify what your procedure is (is it to measure two positions sharply, and divide by the time between measurements), and how you use it to measure the momentum of a state with sharply defined momentum?

 

[Fredrik]

Could you clarify what your procedure is (is it to measure two positions sharply, and divide by the time between measurements), and how you use it to measure the momentum of a state with sharply defined momentum?

I'm certainly no expert in experimental methods, so I don't have a general definition that I'm satisfied with yet. For the moment, I don't have anything better than what I said in #91, and I don't find that definition satisfactory at all. (It might be OK as a definition of what it means to measure the momentum of a particle that has a sharply defined position, but I would like to find a procedure that we can use to test the theory's predictions about momentum regardless of what state the particle is in).

I don't see how two position measurements would constitute a momentum measurement. The first measurement prepares a new state that can be very different from the one we want to measure.

 

[Fredrik]

Yes, but this is better stated as observation has something to do with the Heisenberg Uncertainty Principle rather than the Heisenberg Uncertainty Principle has something to do with observation.

There's no difference between those statements. "has something to do with" is a reflexive relation.

Moreover, like I mentioned, even if you're not "looking", the HUP is there. In this sense alone, the HUP has nothing to do with observation, because, like I said repeatedly (which you keep ignoring), the HUP is there whether you observe or not.

You say that I "keep ignoring it", but you're quoting a post where I'm not only acknowledging it but also agreeing with it.

Now, your Mach-Zehnder interferometer has two arms of lengths l1 and l2. The whole essence of interference is the fact that you cannot tell which path the photon takes, whether the first arm or the second arm. You can think of it in a naive way as the photon takes both at the same time (wave nature of the photon). Now you don't know what the wavelength λ of your photon is, but we assume that you are sure only one photon is going through the interferometer at any given time (this is experimentally laborious, but it is done). When l1=l2, then the photon interferes contructively with itself, and you hear a click at a detector that you have placed at the output of the interferometer. Now don't fool yourself into thinking this is measuring the position of the photon by the detector. We are inferring nothing from the measurement on the detector other that a photon has hit it. We make no further use of this knowledge, nor do we need to. Now as you vary l2 (in some non-random process, such as a linear ramp), while keeping l1 fixed, you start going away from constructive interference. Then at a certain value of l2, you don't hear any clicks on the detector anymore -> destructive interference (of course, you are sending similarly-prepared photons each time you make a measurement). This allows you to determine the momentum k=2πλ of the photon you have up to machine precision, without knowing anything about its position. To see this more clearly, intentionally make your detector one huge pixel whereby anywhere the photon hits it, it gives you a click, yet you can have this pixel occupy a huge spatial extent such that the uncertainty in your position can be considered, for all practical purposes, infinite. I call such an extension "the Fredik-propaganda negator factor".

What you're describing here is a way to find out which momentum eigenstate your photons are in when you already know that they're all in the same momentum eigenstate. I would be more interested in a way to measure the momentum of any given photon (a photon that's in an arbitrary unknown state before the measurement).

Now don't fool yourself into thinking this is measuring the position of the photon by the detector. We are inferring nothing from the measurement on the detector other that a photon has hit it.

How is this not a position measurement? Is it just because we already know what the result will be? You have a detector at position 1 and a detector at position 2. The theory predicts that detector 1 will click 100% of the time that one photon is sent through. You test that prediction by sending photons through one at a time to see what percentage of times each detector clicks. What could you possibly use the term "position measurement" for if not the experiments you would use to test the theory's predictions about which detectors are going to click?

 

[Fredrik]

Just going to put this out there, in Griffiths book "Introduction to QM" I remember him mentioning that theoretically you could measure position AND momentum; however, I don't have the book on hand but will comment tomorrow on the page #.

I would be interested in seeing that page number reference. This discussion (or whatever I should call it) has made me curious about the terminology in standard textbooks.

 

[fuesiker]

There's no difference between those statements. "has something to do with" is a reflexive relation.

I'm tired of arguing with you about semantics. It seems to me you keep using semantics to cover your mistakes in physics.

What you're describing here is a way to find out which momentum eigenstate your photons are in when you already know that they're all in the same momentum eigenstate. I would be more interested in a way to measure the momentum of any given photon (a photon that's in an arbitrary unknown state before the measurement).

This is trivial. There's no reason to believe one cannot prepare the same photon over and over again. Moreover, you need this in order to find out in any measurement process what the state of your system on which you're measuring the observable is. Your measurement is a probability value, and over many measurements you can reconstruct your state.

How is this not a position measurement? Is it just because we already know what the result will be? You have a detector at position 1 and a detector at position 2. The theory predicts that detector 1 will click 100% of the time that one photon is sent through. You test that prediction by sending photons through one at a time to see what percentage of times each detector clicks. What could you possibly use the term "position measurement" for if not the experiments you would use to test the theory's predictions about which detectors are going to click?

As mentioned before, read my post again. I never said you have two detectors. Moreover, we do not know what the result will be a priori. Moreover, all you need is one detector, and the photon will always collapse onto it. All you know is that you have a photon that you can prepare over and over again (which is what is done in all experiments that do projective measurements to determine a state), and you then measure its momentum to machine precision without inferring anything from its position. Also, to counter your argument about measuring position (or your implicit belief that measuring position in this experiment "has something to do" with measuring momentum), I explained in the post how you can make the detector arbitrarily low in resolution, yet this will have no effect on your momentum measurement. You can make the position measurement uncertainty arbitrarily large, but your momentum measurement will always have machine resolution. So please tell me again, how are you inferring momentum from position here?

Think about one monochromatic wave of light instead of one photon, and you are passing it through a double-slit configuration. From the interference pattern, you will get a measurement of the momentum of that beam, without inferring anything about position. Now you can argue the pattern on the wall (where we see the interference pattern) is a position measurement, but that would be silly, because you cannot tell which photon on that wall is which photon before the slits.

 

[fuesiker]

I think it's a little bit misleading to say that we're "inferring the past momentum", because the state before the measurement was a state with a sharply localized position, not momentum. The particle didn't have a well-defined momentum before this inference.

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? These are the answers I can think of right now:

a) Because if we do, the common statement that we can't simultaneously measure position and momentum wouldn't be true.

b) Because if we do, von Neumann's axiom that all measurements project the state onto an eigenspace of the measured observable would contradict itself.

c) Because if we do, we would have to distinguish between the terms "measurement" and "state preparation".​

Those are, in my opinion, very bad reasons to not use the term "measurement" for what measuring devices do when they test the accuracy of the theory's predictions.

b) Because if we do, von Neumann's axiom that all measurements project the state onto an eigenspace of the measured observable would contradict itself.

This is what I mean that you ignore my rebuttals that you can't refute. I will say it here again, Fredrik.

The number operator $\hat{N}$ is that when you have a two-site lattice where you trap a number of atoms, where this site is described by the state $|\psi\rangle = |n_l,n_r\rangle$, then $\langle\psi|\hat{N}|\psi\rangle=(n_l,n_r)$. Experimentally, you can measure the number observable on this site. Let's say the initial state is $\psi(t=0)\rangle=|1,0\rangle$ and let's assume the lattice is bosonic. The eigenstates of $\hat{N}$ form a complete basis of the Hilbert space of this state. We time-evolve this state using a Bose-Hubbard model, with nonzero tunneling and interaction factors. The density at each cite will oscillate and the amplitude of the oscillations with time will get smaller, with an offset at $0.5$. If you want, I can refer you to a paper that experimentally deals with the above.

However, when you measure in the lab the number of atoms at any time $t$ on the state $|\psi(t)$, your device, which is classical, will give you non-negative integer numbers for $n_l$ and $n_r$. Now experimentally, no-one has gotten non-integer values for the above measurement, and a lot has been done on this. This shows how somehow during the measurement process, your state collapses to an eigenstate of the observable being measure. If it doesn't, then your state is a superposition of at least two distinct eigenstates of your system and would look something like $|\psi_m(t)\rangle=\frac{1}{\sqrt{2}}|1,0\rangle+\frac{1}{\sqrt{2}}|0,1\rangle$, (subscript $m$ to indicate state after measurement), which would literally mean that you are simultaneously counting one atom on the left site, zero atoms on the right site and zero atoms on the left site, one atom on the right site, which is ridiculous.

Now, in your above quote, you seem to imply that von Neumann's axiom is a "very bad reason" (exactly and 100% as you put it) to reject the notion that you can measure both momentum and position simultaneously to arbitrary precision (which, whether you acknowledge it or not, is contradictory to the HUP). To me, the von Neumann axiom is very intuitive considering the above explanation that I made. Moreover, this can be readily extended to any other observable.

 

[Cthugha]

This is trivial. There's no reason to believe one cannot prepare the same photon over and over again.

Correct. However, there is also no reason to believe that one can prepare the same photon over and over again.

Now you can argue the pattern on the wall (where we see the interference pattern) is a position measurement, but that would be silly, because you cannot tell which photon on that wall is which photon before the slits.

How is that silly? It is not required that you can tell which photon at the wall is which photon before the slit to perform a position measurement. The concept that every measurement is a position measurement after all, is widely accepted and also central to several interpretations of qm. It is also one of the reason why Bohmian mechanics works and is equivalent to standard qm as here particle positions are the central well-defined quantities of interest.

 

[Fredrik]

I'm tired of arguing with you about semantics. It seems to me you keep using semantics to cover your mistakes in physics.

I'm sure it seems that way to you, but if it hadn't been for your constant misreads and misinterpretations of plain English, you wouldn't have come to the incorrect conclusion that I'm making elementary physics mistakes. This entire discussion (which by the way is more insulting than anything I've seen in my seven years here) is a result of those mistakes.

It's certainly possible that some of the things I've said in this thread are wrong, especially those statements that start with "it seems to me...", "I believe...", or something like that. But your attacks on me have mostly been focused on two things: a) I used the word "measurement" to refer to what measuring devices do, instead of a projection operator. b) I interpreted the words "has something to do with" the way everyone but you does.

There's no reason to believe one cannot prepare the same photon over and over again.

I know.

Moreover, you need this in order to find out in any measurement process what the state of your system on which you're measuring the observable is. Your measurement is a probability value, and over many measurements you can reconstruct your state.

Agreed. If you want to find out what the state is before the measurement, you need to do something like this. I'm just saying that that's not what I'm interested in. You said earlier that interferometers can be used to measure momentum, but so far you have only described a way to determine what wavefunction is produced by a state preparation procedure that's already known to give us a momentum eigenstate. So I'm asking you now, do you know a way to use interferometry to measure the momentum of a single photon that's in an unknown state (not necessarily a momentum eigenstate) before the measurement?

I explained in the post how you can make the detector arbitrarily low in resolution, yet this will have no effect on your momentum measurement. You can make the position measurement uncertainty arbitrarily large, but your momentum measurement will always have machine resolution. So please tell me again, how are you inferring momentum from position here?

You can't make the detector arbitrarily large, because then it would cover the region of space where the other beam emerges. You need to keep the resolution high enough to distinguish between those two positions. As long as the setup can distinguish between two positions, it's a position measurement.