Uncertainty principle and momentum

[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?

 

[atyy]

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.

Just an analogy, I'm not defending the use of a trajectory as conceptually correct.

 

[Maui]

Just an analogy, I'm not defending the use of a trajectory as conceptually correct.

I am fairly certain that a trajectory can be derived for massive particles and afaik Zz specializes in high energy physics(this is based mostly on his references on Tevatron etc, sorry if this is incorrect) so maybe we are talking past each other.

 

[atyy]

I am fairly certain that a trajectory can be derived for massive particles and afaik Zz specializes in high energy physics(this is based mostly on his references on Tevatron etc, sorry if this is incorrect) so maybe we are talking past each other.

The particle "trajectories" are joint inaccurate measurements of position and momentum. You can measure position and momentum simultaneously, just not both accurately. For unbiased measurements (and maybe some other conditions), the uncertainty principle is the Arthurs-Kelly relation. There are other inequalities for joint or sequential measurements of non-commuting observables, eg. http://arxiv.org/abs/1304.2071 , http://arxiv.org/abs/1306.1565.

 

[Maui]

The particle "trajectories" are joint inaccurate measurements of position and momentum. You can measure position and momentum simultaneously, just not both accurately. For unbiased measurements (and maybe some other conditions), the uncertainty principle is the Arthurs-Kelly relation. There are other inequalities for joint or sequential measurements of non-commuting observables, eg. http://arxiv.org/abs/1304.2071 , http://arxiv.org/abs/1306.1565.

Sure, but massive particles behave semi-classically or almost classically because their momentum is very large and their wave behavior negligible(little x uncertainty due to lambda being very small). The uncertainty principle is not very useful for very massive particles is it?

 

[DevilsAvocado]

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

Of course, if the slit is too small, we will get single-slit interference, but I think the trick is to make it just small enough (not to get interference), and then utilize the distance between the slit and screen to "even out" uncertainty in position.

For instance; if we have a slit of 5 mm and 10 m between slit & screen, and fire photons (c) at the screen, maximal deviance in 'trajectories' will be 0.00125 mm, and at c this means a maximal 0.000004125 ns difference in travel time.

But... now I ask myself... what if we make the distance between slit & screen extremely large...?
Would that ever make a 'threat' to HUP??

How large must it be to get close to \hbar / 2 ... ?

 

[atyy]

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!

I still don't understand what you mean by a simultaneous accurate measurement of position and momentum are possible. Let's say we have the slit of width x, the state immediately after the slit is ψ_x, and this is the state on which you would like to have simultaneous accurate position and momentum information. By placing a screen in the far field you can accurately measure the momentum of ψ_x. If you wish to get better position information you must add a narrower slit of width y, and the state after the narrower slit is ψ_y, so that now the momentum distribution on the screen at large distance is that of ψ_y, which is an accurate measurement of momentum on the wrong state, since it is ψ_x whose momentum distribution we wanted to measure. So I don't see how changing position resolution does not change your momentum accuracy, since you don't have the same state any more on which to make your accurate momentum measurement. It is an accurate momentum measurement on the wrong state, which means it is inaccurate.

The usual way of demonstrating the "intrinsic" uncertainty principle, where position and momentum measurements are both accurate, is with separate, non-simultaneous measurements of position and momentum. In one set of experiments you measure position, and in a separate set of experiments you measure momentum. You don't measure position first then measure momentum. Since position and momentum are not measured simultaneously, they can each be arbitarily accurate.

 

[atyy]

Here perhaps is a derivation of ZapperZ's claim that the classical equations hold at near field. For a free particle or harmonic potential, the time evolution of the Wigner function is the same as the classical Liouville equation for a distribution in phase (p, x) space. The Wigner function in general is not a probability distribution because it can be negative. However, for a Gaussian wave function, the Wigner function is positive. Also the free time evolution preserves Gaussianity. Under these conditions, a particle does have simultaneously well defined position and momentum (really? If so indeed SURPRISE!). So if ZapperZ meant the initial wave function to be Gaussian, his claim may make sense.

As I understand it, the fact that the far field position distribution reflects the initial momentum distribution holds for arbitrary initial wave functions, even those with Wigner functions with negative bits.