Photons and Heisenberg's Uncertainty Principle

[Meatlofe]

To summarize, my current understanding of how Heisenberg's uncertainty principal works suggests that there would be a contradiction (somewhere down the line) with any way that it applies to (or doesn't apply to) photons, due to the fact that they must always travel the speed of light.

I understand that photons have a poorly define “position” parameter, but interaction sites do not, so we can use the position parameter of them in place of that of the photons. Let’s say we have 2 photons emitted from a location that we have measured to an arbitrarily high level of precision (henceforth "precisely measured") that interact somehow at a separate, precisely measured position. My question is, will both photons always take precisely the same time to get there, or will Heisenberg’s uncertainty principal alter the position of the photon forwards or backwards (in the direction of travel). If it ever alters the position of the particle (after a “measurement”) in the direction of travel, then that implies that the photon’s average speed was not equal to c, thereby allowing for FTL communication. If however it does not cause alteration in the direction of travel, then this property can be exploited to find the exact position and momentum of a massive particle (using a system too complex for this question, but one I'll get into in another post if it ends up being necessary). Below is an example of how the start and end point could be measured so precisely (and the rough foundation of the complex system).

We have 2 electrons which we have precisely measured the position of. After this measurement is taken, both of them instantaneously* drop from excited level X to ground level Y and emit a gamma photon (this is t=0). This is repeated until 2 photons collide and pair production occurs forming electron-positron pair. The produced particles then travel some distance, where they hit detectors which precisely measure their position and the times of these measurements. Because we know the energy and momentum going in as well as the mass of the particles, we can calculate the velocity of the particles, which we can then use with the time difference between measuring to work backwards to find the point of origin of the particles. This also happens to be the place where the photons interacted. (the other being where they were emitted from)

In this scenario, would the collision site always be at precisely equidistant from the 2 emitters? Or would it sometimes be closer to one than the other due to the position being uncertain? My gut suggests that in this specific case, because a measurement was last made to the position of the light, it will retain this information and therefore it will always happen at the halfway point, however if we had last measured the velocity of the light, it would then randomize the position. The problem arises though, what counts as a measurement of momentum? If you know the start point, end point, mass and time, you can calculate velocity, so does that count?I should also mention that there are substitutes for the 2 initial electrons to get the 2 gamma photons started at the same time from a known location, they're just the simplest way to explain. Examples would include: Splitting a laser with a semi-transparent mirror into 2 pinholes, each with the width of wavelength of the photon. Fission reaction. Annihilation. Etc. I could be wrong but I don't think any of these would make a difference.

*Yes, I know that "instantaneously" is likely unfeasible with this method, however by using some fancy graphs you can compare the distribution of when you'd expect it to be emitted with your actual results to see if the distribution lines up, thereby working around the fact that "instantaneously" is unobtainable. Also, some of the other methods of initiation may allow for “simultaneously” or better yet “at precisely measurable starting times”, both of these can be used as substitutes if you use a bunch of math to compensate.

 

[PeroK]

As a general point of information, the Heisenberg Uncertainty Principle (HUP) is a statistical law. It says that:

$\sigma_p \sigma_x \ge \frac{\hbar}{2}$

Where $\sigma$ represents the standard deviation of a set of measurements on identically prepared systems.

The HUP doesn't say anything directly about the precision with which any particular measurement of a particle's position or momentum can be made.

 

[Meatlofe]

As a general point of information, the Heisenberg Uncertainty Principle (HUP) is a statistical law. It says that:

$\sigma_p \sigma_x \ge \frac{\hbar}{2}$

Where $\sigma$ represents the standard deviation of a set of measurements on identically prepared systems.

The HUP doesn't say anything directly about the precision with which any particular measurement of a particle's position or momentum can be made.

I'd agree with the first part, however I'd suggest the latter to be missing the point. It is a fundamental law, as far as I can tell it is an emergent property in all systems, usually propagated by the observer effect or wave-particle duality (but this changes a great deal depending on which interpretation of physics you're using), which means that in effect, it can be treated as a force such that when you measure one variable precisely, then the other variable will instantly change (slightly out of context but straight from the wiki): "A nonzero function and its [position and momentum conjugate variables] cannot both be sharply localized"

 

[PeroK]

I'd agree with the first part, however I'd disagree on the latter. It is a fundamental law, as far as I can tell it is an emergent property in all systems, usually propagated by the observer effect or wave-particle duality (but this changes a great deal depending on which interpretation of physics you're using), which means that in effect, it can be treated as a force such that when you measure one variable precisely, then the other variable will instantly change (slightly out of context but straight from the wiki): "A nonzero function and its [position and momentum conjugate variables] cannot both be sharply localized"

Nonetheless, you could measure the position of a particle with high precision (say the result was $x_1$), then at some later time measuring its new position with high precision ($x_2$, say). Assuming the time between the measurements was know to a high precision, $t$, and the mass of the particle, and you reasoned that the momentum of the particle was $m(x_2 -x_1)/t$, then there's not a lot Heisenberg can say about that.

In addition to the statistical matter, QM deals with particle states and wavefunctions and not with inferred classical trajetories. It is not, in other words, classical mechanics with an Uncertainty Principle.

 

[weirdoguy]

wave-particle duality

Wave-particle duality is not part of modern quantum physics. It's part of so called "old quantum theory" which died around 1924-25. Besides, and don't get me wrong, this is not a matter of agreeing or disagreeing. What PeroK stated is a fact about HUP - it does not state anything about precision of single measurement. Just look at the derviation of it.

 

[Meatlofe]

Nonetheless, you could measure the position of a particle with high precision (say the result was $x_1$), then at some later time measuring its new position with high precision ($x_2$, say). Assuming the time between the measurements was know to a high precision, $t$, and the mass of the particle, and you reasoned that the momentum of the particle was $m(x_2 -x_1)/t$, then there's not a lot Heisenberg can say about that.

In addition to the statistical matter, QM deals with particle states and wavefunctions and not with inferred classical trajetories. It is not, in other words, classical mechanics with an Uncertainty Principle.

Yeah, there isn't much that Heisenberg can say about us calculating the past velocity of it, however by taking the measurement of the position $x_2$, we have re-randomized the momentum of it, so we may know how its momentum and position was in the past, that does not mean that we know it anymore.
As to the second part, there's a decent chance that my logic fails somewhere due to that, but currently I'm still unable to see where, how, or in what way. That is why I included a precise experiment, so that even if the theory of knowledge I was using turned out to be full of holes, I could still use the quantitative data of the (expected) results to get my point across, and to better comprehend other people's answers.

 

[ZapperZ]

I'd agree with the first part, however I'd disagree on the latter. It is a fundamental law, as far as I can tell it is an emergent property in all systems, usually propagated by the observer effect or wave-particle duality (but this changes a great deal depending on which interpretation of physics you're using), which means that in effect, it can be treated as a force such that when you measure one variable precisely, then the other variable will instantly change (slightly out of context but straight from the wiki): "A nonzero function and its [position and momentum conjugate variables] cannot both be sharply localized"

The HUP is NOT a "fundamental law". If you look at the postulates of QM, nowhere in there is there any mention of the HUP.

Instead, the HUP is a consequence of how we define operators and observables. It follows directly from the idea that two observable operators may not commute with each other, i.e. AB ≠ BA. If I have another pair of observables such as CD = DC, then the quantities that I measure associated with those two observables will NOT have any kind of a HUP relationship. So this doesn't happen all the time for every single quantity that we measure! It is not "fundamental".

Zz.

 

[PeroK]

Yeah, there isn't much that Heisenberg can say about us calculating the past velocity of it, however by taking the measurement of the position $x_2$, we have re-randomized the momentum of it, so we may know how its momentum and position was in the past, that does not mean that we know it anymore.
As to the second part, there's a decent chance that my logic fails somewhere due to that, but currently I'm still unable to see where, how, or in what way. That is why I included a precise experiment, so that even if the theory of knowledge I was using turned out to be full of holes, I could still use the quantitative data of the (expected) results to get my point across, and to better comprehend other people's answers.

That's where the statistics come in. The HUP says nothing about individual measurements. In the above example, the precise measurement $x_1$ would lead to a large variance in momentum. That doesn't say anything about the precision of a subsquent momentum measurement. But, it does mean that subsequent measurements of mometum on an ensemble of identically prepared particles would have a large variance. And having a large variance on a set of data says nothing about the precision of any individual item of data. Each measurement may be very precise, but the range of values measured (in different experiments) is large.

 

[Meatlofe]

That's where the statistics come in. The HUP says nothing about individual measurements. In the above example, the precise measurement $x_1$ would lead to a large variance in momentum. That doesn't say anything about the precision of a subsquent momentum measurement. But, it does mean that subsequent measurements of mometum on an ensemble of identically prepared particles would have a large variance. And having a large variance on a set of data says nothing about the precision of any individual item of data. Each measurement may be very precise, but the range of values measured (in different experiments) is large.

Alright, that all makes sense. So in other words, what I've been calling "high uncertainty" would just mean "high variance". Everything should still work the same. If a photon starts from a well measured location, the variance of it's direction will be very high, however it will still travel the same speed in every case? And what if the velocity were precisely measured? The subsequent measurements of position then have a high variance, but would it change at all in the direction of the photon's travel? Because if so, the same problems occur, as the photon would be found to be traveling faster than c allowing for FTL... etc.

 

[PeroK]

Alright, that all makes sense. So in other words, what I've been calling "high uncertainty" would just mean "high variance". Everything should still work the same. If a photon starts from a well measured location, the variance of it's direction will be very high, however it will still travel the same speed in every case? And what if the velocity were precisely measured? The subsequent measurements of position then have a high variance, but would it change at all in the direction of the photon's travel? Because if so, the same problems occur, as the photon would be found to be traveling faster than c allowing for FTL... etc.

The photon does not have a position operator. See, for example:

https://www.physicsforums.com/threads/why-no-position-operator-for-photon.906932/

 

[ZapperZ]

Alright, that all makes sense. So in other words, what I've been calling "high uncertainty" would just mean "high variance". Everything should still work the same. If a photon starts from a well measured location, the variance of it's direction will be very high, however it will still travel the same speed in every case? And what if the velocity were precisely measured? The subsequent measurements of position then have a high variance, but would it change at all in the direction of the photon's travel? Because if so, the same problems occur, as the photon would be found to be traveling faster than c allowing for FTL... etc.

I think you still have a misunderstanding of what the HUP is about. I'll be tacky and link to my Insight article. Even if you don't read it, scroll down to the MinutePhysics video that essentially is saying the same thing.

https://www.physicsforums.com/insights/misconception-of-the-heisenberg-uncertainty-principle/

Zz.

 

[Meatlofe]

I think you still have a misunderstanding of what the HUP is about. I'll be tacky and link to my Insight article. Even if you don't read it, scroll down to the MinutePhysics video that essentially is saying the same thing.

https://www.physicsforums.com/insights/misconception-of-the-heisenberg-uncertainty-principle/

Zz.

Well, the video is from Veritasium, not minute physics (though I have seen all videos of both channels). From the video and what I can skim from the page (I will do a full reading of it later) it seems to give a better explanation of HUP, however it is still still dances around the question. The video suggests (if you listen closely) that because the light is confined in only 1 axis, it only spreads in that axis (which suggests that in the example I posted that if you substitued the starting mechanism for the split laser beam through pinholes/slits, the velocity would not be shifted forwards or backwards.) However the question still remains, if you measure the forwards/backwards velocity of the light, will the forwards/backwards position of it change?

 

[Meatlofe]

The photon does not have a position operator. See, for example:

https://www.physicsforums.com/threads/why-no-position-operator-for-photon.906932/

So yes, that is true, I mentioned it in my original post, however due to c being the speed of causality, we don't actually have to care if the theoretical "photon" or "waveform" has a "position" that is moving. What I actually care about is weather it can cause a measurable effect to a place that is outside the cone of influence from which it was emitted. Because of this we can get around the fact that it doesn't have a position operator by never measuring it's position, but by measuring the position of the point it was emitted from and the point it interacts with. This let's us model the photon as if it had a position operator, because we are never actually interacting with it, because it doesn't exist.