Uncertainty principle and momentum

[atyy]

No you can't, because if it is too close, the particle can come from ANY part of the slit! You now have an ambiguous origin on how to calculate your classical trajectory!

You have no way of assigning a unique momentum to that spot on the screen anymore!

I see. And as the width of the slit gets smaller, the state at the slit is closer and closer to a position eigenstate, then even for any finite distance, the momentum will become more and more accurate. So is your basic argument that if we do know the particle is in an eigenstate of one observable, we can make simultaneous accurate measurements of non-commuting observables on it, because we can measure one observable without disturbing the eigenstate, leaving the same state available for a measurement of the non-commuting observable?

 

[DevilsAvocado]

What am I missing in this screen/momentum scenario...??

Take two identical cars, traveling 100 km on a test track, ending in a rock wall. Car #1 makes the distance on exactly 60 min, and gets completely demolished when crashing into the rock wall; average speed 100 km/h.

Car #2 also makes the distance on exactly 60 min, and gets a tiny dent when bumping into the wall; average speed 100 km/h.

What happened?

Car #1 went 100 km/h from start to finish. Car #2 went 110 km/h for 54 min, hit the brakes and then went 10 km/h for 6 min.

So how on Earth can a smash into a screen tell us anything about simultaneous position and momentum?? (unless you check the 'recoil' of the screen)

I don't get it...

 

[Jilang]

What am I missing in this screen/momentum scenario...??
...

I don't get it...

Are you perhaps mixing up your components of momentum and position? The transverse momentum and the transverse position are the non-commuting variables. Px and Y will commute so no issue.

 

[ZapperZ]

What am I missing in this screen/momentum scenario...??

Take two identical cars, traveling 100 km on a test track, ending in a rock wall. Car #1 makes the distance on exactly 60 min, and gets completely demolished when crashing into the rock wall; average speed 100 km/h.

Car #2 also makes the distance on exactly 60 min, and gets a tiny dent when bumping into the wall; average speed 100 km/h.

What happened?

Car #1 went 100 km/h from start to finish. Car #2 went 110 km/h for 54 min, hit the brakes and then went 10 km/h for 6 min.

So how on Earth can a smash into a screen tell us anything about simultaneous position and momentum?? (unless you check the 'recoil' of the screen)

I don't get it...

OK, I didn't realize this is that difficult to understand, considering that this is almost first-year physics.

v is known, since you can prepare the particles to be of a certain speed/momentum, or in the case of light, this is c.

The time taken for the particle to go from the slit to the screen is

t = d/v_y = d/(v cosΘ)

For most cases, the slit is very small, and the screen is very far, so cosΘ ≈ 1 is a good approximation.

In that same time, the particle has a transverse x-component movement, which means that it has a transverse velocity/momentum, i.e.

v_x = x/t = xv/d.

Since v and d are know, this means that knowing x, i.e. where it landed on the screen transversely, give you the velocity, and thus, the momentum.

I've done this for a non-relativistic case, since this gives a clearer illustration how we detect momentum in practically ALL cases. Note that you cannot detect "collision", i.e. you don't measure it. You measure where the colliding particle landed at detectors, and via reconstruction, you arrive at the momentum. This is how we measure energy and momentum of particles in high energy physics. We DETECT where the particles are when they hit the detector. All we see are locations of these particles at the detector. We do NOT see a "collision"!

Zz.

 

[DevilsAvocado]

OK, I didn't realize this is that difficult to understand, considering that this is almost first-year physics.

OMG, c, why didn't I think of that... (maybe because my Volvo has a hard time keeping the pace in winter time < 100 km/h)… I get it (please don't laugh).

It's quite amazing that classical equations hold so nice. Who needs path integrals?

 

[DevilsAvocado]

Zz, one more question: What happen if we introduce another slit, between the first one and the screen, and make sure the openings are large enough not to produce single-slit interference.

Will the "shared opening", between Slit #1 & #2, create results like if it were one tight single-slit (= interference)?

(I'm a little bit tired today, so here comes probably more "free amusement"... )

 

[atyy]

@ZapperZ, can you derive or give a reference that the classical equations of motion hold for assigning a trajectory in the near field?

 

[atyy]

@ZapperZ, a further objection to the particle having a trajectory even after the second position measurement in the far field is that if the slit has finite width, then the initial position is still uncertain - in which case what definite trajectory is there?

If the slit is narrowed such that the initial position is certain, then the near field assignment of momentum using classical equations will become "accurate", while the distribution of "accurate" momenta will certainly be inaccurate. In this case, it would be peculiar to say that the classical equations hold in the near field.

So either way, whether the initial position is accurate or not, I don't see how to assign a trajectory to the particle. I do agree that in the far field the initial momentum distribution can be accurately calculated from the position measurements, and that the classical equations are a quick and dirty way to do it. However, I don't see how they are conceptually correct within quantum mechanics. An analogy might be Rutherford's derivation of the correct formula for quantum scattering from the inverse squared potential using classical equations, which although it gave the correct result is certainly conceptually incorrect.

 

[atyy]

http://arxiv.org/abs/quant-ph/0609185v3 seems to have a discussion of whether a trajectory can be assigned in the single slit experiment with far field determination of the momentum on p21-22. Apparently, Quadt argued that the assignment of a trajectory is not possible without Kochen-Specker contradictions. Unfortunately I don't have access to Quadt's paper.

 

[curious bishal]

What you said here made very little sense. A "calculation" is a mathematical operation. How could an electron "escapes" while one is calculating? Furthermore, in a calculation, you can make your detection as accurately as you want! You have no instrument resolution to deal with such as in an actual experiment!

The rest of what you wrote is equally puzzling.

Zz.

If you find out the data and calculate its position and says that the electron is there, the electron is not now, there , it had reached somewhere else...

 

[ZapperZ]

If you find out the data and calculate its position and says that the electron is there, the electron is not now, there , it had reached somewhere else...

So? I can still show my avatar in a paper and when I claim that this is the band that crosses the Fermi surface, people accept it. They didn't have to ask me WHEN this was taken just for them to know these electrons are no longer there!

You are making an issue out of nothing.

Zz.

 

[ZapperZ]

@ZapperZ, a further objection to the particle having a trajectory even after the second position measurement in the far field is that if the slit has finite width, then the initial position is still uncertain - in which case what definite trajectory is there?

If the slit is narrowed such that the initial position is certain, then the near field assignment of momentum using classical equations will become "accurate", while the distribution of "accurate" momenta will certainly be inaccurate. In this case, it would be peculiar to say that the classical equations hold in the near field.

So either way, whether the initial position is accurate or not, I don't see how to assign a trajectory to the particle. I do agree that in the far field the initial momentum distribution can be accurately calculated from the position measurements, and that the classical equations are a quick and dirty way to do it. However, I don't see how they are conceptually correct within quantum mechanics. An analogy might be Rutherford's derivation of the correct formula for quantum scattering from the inverse squared potential using classical equations, which although it gave the correct result is certainly conceptually incorrect.

Sigh.

Look at a typical diffraction experiment. Do you think it matters here from which part of the slit the particle came from? Unlike the case where the screen is at a distance comparable with the slit width, the location of the spots on the screen are large, it is orders of magnitude larger than the slit width!

You know, this is why even theorist should learn how to do experiments. It appears that the sense of SCALE here is missing.

I recommend you look at a review of angle-resolved photoemission spectroscopy techniques and how momentum and energy of the photoelectrons are determined JUST from the image they make on the screen. As a matter of fact, look up mass spectroscopy techniques as well while you are at it!

Zz.

 

[atyy]

Sigh.

Look at a typical diffraction experiment. Do you think it matters here from which part of the slit the particle came from? Unlike the case where the screen is at a distance comparable with the slit width, the location of the spots on the screen are large, it is orders of magnitude larger than the slit width!

You know, this is why even theorist should learn how to do experiments. It appears that the sense of SCALE here is missing.

Zz.

So do you retract the claim that position and momentum can be simultaneously measured with arbitrary accuracy?

 

[atyy]

I recommend you look at a review of angle-resolved photoemission spectroscopy techniques and how momentum and energy of the photoelectrons are determined JUST from the image they make on the screen. As a matter of fact, look up mass spectroscopy techniques as well while you are at it!

So these experiments are simultaneous measurements of the position and momentum of single electrons with arbitrary accuracy?

 

[ZapperZ]

So do you retract the claim that position and momentum can be simultaneously measured with arbitrary accuracy?

No, because this "accuracy" in a single measurement depends on the INSTRUMENT accuracy, not on the HUP! The "arbitrary" accuracy means that the accuracy of a momentum measurement and the accuracy of the position measurement are NOT COUPLED as in the HUP!

Zz.

 

[ZapperZ]

So these experiments are simultaneous measurements of the position and momentum of single electrons with arbitrary accuracy?

The accuracy is never part of the determination of the energy and momentum. The accuracy do the CCD screen is.

Why haven't you try to look this up yourself?

Zz.

 

[atyy]

No, because this "accuracy" in a single measurement depends on the INSTRUMENT accuracy, not on the HUP! The "arbitrary" accuracy means that the accuracy of a momentum measurement and the accuracy of the position measurement are NOT COUPLED as in the HUP!

Zz.

But you do retract the claim that the particle simultaneous had a unique position and a unique momentum - as implied by the claim that it had a trajectory?

 

[ZapperZ]

But you do retract the claim that the particle simultaneous had a unique position and a unique momentum - as implied by the claim that it had a trajectory?

Huh? Where did I even implied this?

Look at my original description of the diffraction experiment. I can determine both the position of the particle since it passed through the slit, and when that particle landed and make a spot at the screen, I can THEN determine its transverse momentum. The accuracy of each of this measurement depends on the width of the slit for the position and the size of the spot on the screen for the momentum.

Isn't this what I wrote earlier, or have we forgotten that?

I feel like I'm going around in circles!

Zz.

 

[atyy]

Huh? Where did I even implied this?

Look at my original description of the diffraction experiment. I can determine both the position of the particle since it passed through the slit, and when that particle landed and make a spot at the screen, I can THEN determine its transverse momentum. The accuracy of each of this measurement depends on the width of the slit for the position and the size of the spot on the screen for the momentum.

Isn't this what I wrote earlier, or have we forgotten that?

I feel like I'm going around in circles!

Zz.

Back in post #8 you said that single particles have trajectories, by which I thought you meant that a single particle has a trajectory.

 

[ZapperZ]

Back in post #8 you said that single particles have trajectories, by which I thought you meant that a single particle has a trajectory.

And?

Zz.

 

[DevilsAvocado]

The accuracy of each of this measurement depends on the width of the slit for the position and the size of the spot on the screen for the momentum.

Just to make it clear for everyone: If the width of the slit gets too narrow (for better position), we will get interference, hence the wider spot on the screen will result in increased momentum uncertainty.

And I guess, that in any case, we are not even close to \hbar / 2 ...

Right?

 

[atyy]

And?

Zz.

A particle with a trajectory has at each instant of time a simultaneously well-defined position and momentum.

 

[atyy]

Just to make it clear for everyone: If the width of the slit gets too narrow (for better position), we will get interference, hence the wider spot on the screen will result in increased momentum uncertainty.

And I guess, that in any case, we are not even close to \hbar / 2 ...

Right?

If one calculates everything quantum mechanically with no trajectories, the far field position distribution is indeed related to the momentum distribution one would have obtained from a momentum measurement on the initial state (ie. the state after the slit, of course the momentum of the state before the slit is disturbed by the slit). It is also true that the classical equation for assigning the momentum gives a result that is increasingly close to the fully quantum answer, and I'm not sure, but it seems reasonable to imagine that the answers by both methods are identical in the far field limit. So that is not in question. The question is are we then allowed to say that the particle had a trajectory governed by the classical equations? Or is this coincidence, just as Rutherford obtained the correct quantum result for inverse squared scattering by classical calculations? Do we say then that Rutherford scattering means the particles had trajectories for inverse square potentials? Or do we say that that's a coincidence, since the classical calculations fail to reproduce the quantum ones for other potentials?

My understanding is that unless one is using some variant of de Broglie-Bohm theory, a particle does not have a trajectory. I'm trying to understand ZapperZ's claim that it does, after the second position measurement on the screen.

 

[ZapperZ]

A particle with a trajectory has at each instant of time a simultaneously well-defined position and momentum.

Note what I said. WHEN the particle makes a spot on the screen, I THEN calculate its momentum based on the location of the spot.

I claim that this WORKS, because many other techniques (ARPES, mass spectroscopy, energy spectroscopy, etc.) use the same technique, i.e. by using a signal on where a particle landed, one then calibrate that to arrive at another quantity. The position on the screen MEANS something.

And this, I believe, is where the whole issue is. You are somehow surprised by this fact, and I am also surprised by your puzzlement because it appears as if you are not aware of how many of the quantities that we have adopted are measured this way! One only has to look at ATLAS and CMS to know that they ALL use images of where such-and-such particles cause a signal in their detectors. They don't measure "momentum, energy, etc", they measure WHERE in the detector the signal came from! Then, via a proper model, either reconstruct the trajectory, or obtain the quantity they want, which includes energy AND momentum!

So, SURPRISE!

Zz.

 

[atyy]

Note what I said. WHEN the particle makes a spot on the screen, I THEN calculate its momentum based on the location of the spot.

I claim that this WORKS, because many other techniques (ARPES, mass spectroscopy, energy spectroscopy, etc.) use the same technique, i.e. by using a signal on where a particle landed, one then calibrate that to arrive at another quantity. The position on the screen MEANS something.

And this, I believe, is where the whole issue is. You are somehow surprised by this fact, and I am also surprised by your puzzlement because it appears as if you are not aware of how many of the quantities that we have adopted are measured this way! One only has to look at ATLAS and CMS to know that they ALL use images of where such-and-such particles cause a signal in their detectors. They don't measure "momentum, energy, etc", they measure WHERE in the detector the signal came from! Then, via a proper model, either reconstruct the trajectory, or obtain the quantity they want, which includes energy AND momentum!

So, SURPRISE!

Zz.

No surprise, that is well accepted - just as Rutherford obtained the correct quantum formula for inverse square scattering by classical equations - that works too - but both are ad hoc calculations that must be justified quantum mechanically. It's your unusual language that I was trying to understand. Good, so we agree simultaneous accurate measurements of position and momentum are not possible, and we do not assign a single particle a trajectory with a well-defined position and momentum at all times.

There is still one issue: you said the classical equations are valid even in the near field. Can you give the derivation for that, or a reference?

 

[DevilsAvocado]

The question is are we then allowed to say that the particle had a trajectory governed by the classical equations?

I don't know atyy, except that this approximation seems to work in practice, and my guess is that it's dependent on the distance between the slit and the screen (of course); large distance = approximation okay, tiny distance = approximation fails (i.e. when the distance between the slit and the screen equals the slit width, you're out in the blue, guessing...).

 

[Maui]

I don't know atyy, except that this approximation seems to work in practice, and my guess is that it's dependent on the distance between the slit and the screen (of course); large distance = approximation okay, tiny distance = approximation fails (i.e. when the distance between the slit and the screen equals the slit width, you're out in the blue, guessing...).

If I read this correct, it's not really an approximation but a post-factum derivation of a trajectory. Which also says something about the quantum world.

 

[Maui]

No surprise, that is well accepted - just as Rutherford obtained the correct quantum formula for inverse square scattering by classical equations - that works too - but both are ad hoc calculations that must be justified quantum mechanically.

Why? What if the particle is detected at a classically forbidden portion on the detector screen? What kind of trajectory would that be? Zz seems to be talking of accelerated massive particles and this would be an important distinction.

 

[DevilsAvocado]

If I read this correct, it's not really an approximation but a post-factum derivation of a trajectory. Which also says something about the quantum world.

Maybe safest to let Zz verify this...

For most cases, the slit is very small, and the screen is very far, so cosΘ ≈ 1 is a good approximation.

 

[Maui]

Maybe safest to let Zz verify this...

I thought small slit size guarantees interference(for non massive particles) and a trajectory would be impossible to derive?