Double Slit Experiment: Feynman's Lectures

[lavinia]

In the third book of Feymann's Lectures on Physics(section 1-8) he describes how "the Uncertainty Principle protects Quantum Mechanics." The experimental situation is a modified double slit experiment where the two slits are put on rollers in an attempt to detect which slit an electron passes through. The idea is that the change in the direction of momentum of the slits on rollers after collision with an electron will be opposite for the two slits.

He argues that if one could measure this change in momentum then uncertainty principle would make it impossible to exactly locate the position of the slits and this will blur out the interference pattern.

- Why can't the apparatus be reset each time for each electron? In this way we do not care where the slits are after the electron passes through - only the momentum.

Is it that the apparatus can not be reset because this would mean knowing where the slits are and knowing that their momentum is zero at the same time?But then doesn't the mere existence of an interference pattern mean that we know both?

 

[jfizzix]

In the quantum mechanical description, if there is literally any piece of information from any kind of measuring device that could determine which slit the particle went through, then the interference would disappear. From a quantum standpoint, this is because measurement is a physical interaction between the measurement device and the thing being measured. The electron can only exhibit perfect wave-like behavior when it is isolated from interaction. When a measurement device interacts with the electron, the joint wavefunction of electron-plus-device becomes entangled, and the individual state of the electron loses coherence.

If you put the slits on rollers, they could conceivably experience recoil due to the electrons changing paths within them, and you could conceivably reset the slits each time. But because the electrons interact with the slits in the first place, you will not see any interference (or at least the visibility of interference will go down, as your ability to tell which path goes up).

 

[lavinia]

If you put the slits on rollers, they could conceivably experience recoil due to the electrons changing paths within them, and you could conceivably reset the slits each time. But because the electrons interact with the slits in the first place, you will not see any interference (or at least the visibility of interference will go down, as your ability to tell which path goes up).

Thanks for your response. My daughter and I are still confused. Here is the question as it came up when we were reading Feynmann's book together.

One can tell from the interference pattern- apparatus not on rollers- exactly where the two slits are located. This supposedly means that one has absolutely no knowledge of their momentum. But in fact you know that they are not moving and if they were there would be no interference pattern.

Since one gets an interference pattern by shooting one electron at a time one should be able to reset the apparatus-on-rollers each time after each electron is shot through and still get the interference pattern. But before resetting we can measure the change in momentum of the slits and tell which slit the electron passed through.

- A related example discussed by Leonard Susskind in a Youtube video on the Special Theory of Relativity: The decay of the positronium atom.

When positronium decays the positron/electron pair almost always becomes two photons whose tracks can be recorded in a cloud chamber. If one follows the tracks back to their point of intersection one gets the exact place where the positronium atom was at the instant of decay (Before decay Susskind says that quantum mechanics says that one can not exactly locate the positronium atom) The Uncertainty Principle then tells us that at the instant of decay we have absolutely no information about the momentum of the positronium atom. But in fact it seems that we do since the total momentum of the photons can be fairly accurately measured from the cloud chamber tracks.

 

[Demystifier]

In the third book of Feymann's Lectures on Physics(section 1-8) he describes how "the Uncertainty Principle protects Quantum Mechanics."

In this section, Feynman attempts to explain some properties of quantum mechanics (QM) without actually using QM. By "actually using QM", I mean using wave functions. Without wave functions, the uncertainty principle cannot really be understood. All explanations of uncertainty principle without wave functions (the most famous example of which is the Heisenberg microscope) are at best heuristic and should not be taken seriously. So I would suggest to skip all those heuristic "explanations" of QM and jump to real QM in terms of wave functions. Since you are a mathematician, such an approach should also be easier for you.

 

[Demystifier]

When positronium decays the positron/electron pair almost always becomes two photons whose tracks can be recorded in a cloud chamber. If one follows the tracks back to their point of intersection one gets the exact place where the positronium atom was at the instant of decay (Before decay Susskind says that quantum mechanics says that one can not exactly locate the positronium atom) The Uncertainty Principle then tells us that at the instant of decay we have absolutely no information about the momentum of the positronium atom. But in fact it seems that we do since the total momentum of the photons can be fairly accurately measured from the cloud chamber tracks.

The measured tracks are not mathematical lines. They are physical lines, i.e. lines with a finite width. This means that neither position nor momentum is measured with perfect accuracy. Instead, both position and momentum are measured with some inaccuracy, and product of these inaccuracies is of the order of the Planck constant.

 

[vanhees71]

Thanks for your response. My daughter and I are still confused. Here is the question as it came up when we were reading Feynmann's book together.

One can tell from the interference pattern- apparatus not on rollers- exactly where the two slits are located. This supposedly means that one has absolutely no knowledge of their momentum. But in fact you know that they are not moving and if they were there would be no interference pattern.

Since one gets an interference pattern by shooting one electron at a time one should be able to reset the apparatus-on-rollers each time after each electron is shot through and still get the interference pattern. But before resetting we can measure the change in momentum of the slits and tell which slit the electron passed through.

- A related example discussed by Leonard Susskind in a Youtube video on the Special Theory of Relativity: The decay of the positronium atom.

When positronium decays the positron/electron pair almost always becomes two photons whose tracks can be recorded in a cloud chamber. If one follows the tracks back to their point of intersection one gets the exact place where the positronium atom was at the instant of decay (Before decay Susskind says that quantum mechanics says that one can not exactly locate the positronium atom) The Uncertainty Principle then tells us that at the instant of decay we have absolutely no information about the momentum of the positronium atom. But in fact it seems that we do since the total momentum of the photons can be fairly accurately measured from the cloud chamber tracks.

Your confusion is due to not distinguishinging clearly the macroscopic setup of the expreiment and the measured (quantum object, say and electron as an example).

The two slits are classical objects and can, for this experiment, be described with an overwhelming accuracy by good old classical mechanics. The two slits are at rest and have a well-defined distance (and also a well-defined width). This definiteness of position and momentum of a macroscopic object is meant to be accurate on macroscopic scales. The Heisenberg (quantum) uncertainty in position and momentum of the screen making up the double slit are way smaller than the accuracy you can determine these quantities, but that's irrelevant for the here performed experiment with a single electron (or better said with a lot of equally prepared single electrons).

For the single electron, as a microscopic object, it's much more likely that quantum effects play a role, and in this case of the double-slit experiment that's indeed the case. What you have to do is to prepare the electrons in the ensemble with quite some accuracy in momentum. According to the Heisenberg uncertainty relation the wave function describing such an electron must be pretty broad in spatial extension (in fact the momentum wave function and the position wave function are Fourier transforms of each other and thus if the momenum-wave function is very well localiced in momentum space the position-wave function must be pretty broad in position space, which is the math behind the uncertainty relation expressed in terms of wave mechanics). This means, when the electron goes through the slits it's impossible to know through which slit it came, because the position is rather indetermined because of the preparation with a quite well determined momentum. That's why for a large ensemble of such prepared electrons you see interference effects in the pattern on the detector behind the screen.

As Feynman beautifully stresses that's how electrons behave, according to many very accurate observations and experiments done with them. There's no simpler way to describe their behavior than quantum mechanics, and there's no scientifically sound explanation of "why it behaves like this". According to quantum theory you can principally know only probabilities (or probability distributions) due to preparation procedures (which define the quantum mechanical state of the system in terms of wave functions) about the outcome of measurements.

Note that this point of view is only a special formulation of non-relativistic quantum theory, called "wave mechanics". Later you'll learn an even more abstract formalism, the representation free formulation, with vectors in Hilbert space and operators acting on them, defining a set of rules to calculate probabilities, for future measurements on the quantum system, and these probabilities are the only thing we can now after knowing how the quantum system has been prepared. So far there's no hint that this physical theory is incomplete, and thus we must acknowledge that according to quantum physics nature behaves indeterministically in stark contrast to the deterministic world view of classical physics.

 

[lavinia]

The measured tracks are not mathematical lines. They are physical lines, i.e. lines with a finite width. This means that neither position nor momentum is measured with perfect accuracy. Instead, both position and momentum are measured with some inaccuracy, and product of these inaccuracies is of the order of the Planck constant.

Sorry Demystifyer - I didn't explain the question very well. Our idea was that at the instant when the positronium atom decays its wave function collapses to a point just as in the instant when an electron is detected on the detector screen in the double slit expriment. So it is located at an exact point for that instant. While it is true that one can not exactly measure that point from the tracks, it is also true that one can not exactly determine the point of detection on the detector screen. So we thought that here is a point with a collapsed wave function where the momentum is reasonably well determined. So maybe when it decays the wave function does not collapse and something more complicated happens.

 

[bhobba]

In this section, Feynman attempts to explain some properties of quantum mechanics (QM) without actually using QM. By "actually using QM", I mean using wave functions.

In fact the uncertainty principle (along with the principle of superposition) is the explanation for the double slit:
https://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

To the OP - you have hit on one of the issues in learning physics. At the start its often a morass as you try to develop an intuitive understanding of what's going on. You need to unlearn stuff as you proceed and gain a more nuanced understanding. Its not the ideal way to learn, but even Feynman, great educator he was, understood it really not possible to do it any other way. He would have preferred to give the whole truth, and nothing but the truth, from the start but knew it couldn't really be done that way. So as you proceed you need to relearn and unlearn things. Its not really a big issue - it will happen pretty much subconsciously. But sometimes when discussing issues of principle it comes back to bite you and that's when you need to be aware of it - expecially here where we have some really advanced posters. You get the whole truth and nothing but the truth here - warts and all, no sugar coating.

I really like the paper I linked above, but just to indicate even it has issues from an even more advanced level see:
http://arxiv.org/abs/1009.2408

And to make matters even worse we have some really advanced regular posters here who are professors of physics - and they even have issues with the above paper. Physics is a never ending journey.

Just as an aside it is really bad with QFT. At the start to develop intuition you get some downright untruths. People who only know QFT at that level for some reason get very 'jumpy' when told that and you have long thread after thread that goes nowhere because they don't want to unlearn what they have been told. You even get posters that claim what they are being told by experts can't be correct and simply refuse to accept it - its a very very interesting phenomena.

Thanks
Bill

 

[bhobba]

So maybe when it decays the wave function does not collapse and something more complicated happens.

Indeed it does.

First collapse isn't really part of the QM formalism - only of some interpretations. This is part of the more advanced understanding I mentioned above.

If you want to pursue it here are the books to get:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

With your background in math they should be understandable. But only if you want to pursue it.

Thanks
Bill

 

[jfizzix]

Since one gets an interference pattern by shooting one electron at a time one should be able to reset the apparatus-on-rollers each time after each electron is shot through and still get the interference pattern. But before resetting we can measure the change in momentum of the slits and tell which slit the electron passed through.

The only way for the slits to change momentum is for the electron to have interacted with them, which inevitably disturbs the electron as well. The fringe pattern will be different between whether or not the slits are on rollers. When the interaction is strong enough that you can reliably tell which slit the electron went through, the interaction will also be strong enough to destroy the interference pattern. Incidentally, a very weak interaction would be one where the momentum imparted

- A related example discussed by Leonard Susskind in a Youtube video on the Special Theory of Relativity: The decay of the positronium atom.

When positronium decays the positron/electron pair almost always becomes two photons whose tracks can be recorded in a cloud chamber. If one follows the tracks back to their point of intersection one gets the exact place where the positronium atom was at the instant of decay (Before decay Susskind says that quantum mechanics says that one can not exactly locate the positronium atom) The Uncertainty Principle then tells us that at the instant of decay we have absolutely no information about the momentum of the positronium atom. But in fact it seems that we do since the total momentum of the photons can be fairly accurately measured from the cloud chamber tracks.

In quantum mechanics, although the uncertainty principle forbids a photon from having a well-defined position and momentum, it does allow pairs of photons to have an unlimited strength of correlation. Indeed, there are experiments that show that the uncertinties in the sum of the momenta of the photon pair, and in the difference if their positions can both be arbitrarily small.

That is, with position $x$ and momentum $p$, and standard deviation $\sigma$,

The uncertainty principle tells us:
$\sigma_{x}\sigma_{p}\geq\frac{\hbar}{2}$
but the uncertainty principle also tells us:
$\sigma_{(x_{1}-x_{2})}\sigma_{(p_{1}+p_{2})}\geq 0.$

If we try to deduce the statistics of the positronium atom from that of the photon pair, one can use conservation of momentum to say the positronium atom can have an arbitrarily well-defined momentum, but the position of the positronium atom is best estimated at the mean (half the sum) of the positions of the photon pair, and those uncertainties cannot both be arbitrarily small. In fact:

$\sigma_{(\frac{x_{1}+x_{2}}{2})}\sigma_{(p_{1}+p_{2})}\geq \frac{\hbar}{2} .$

As one can see, the uncertainties in the estimations are bounded by the same limits as the uncertainties in directly measuring the thing of interest.

 

[Demystifier]

So we thought that here is a point with a collapsed wave function where the momentum is reasonably well determined.

When you determine position, the wave function indeed collapses to a very narrow wave function. But it is precisely this narrowing of the wave function which makes the momentum badly undetermined. Even if the momentum was well determined before the measurement of position, after the measurement of position the momentum is no longer well determined. It is this very act of measurement that changes the properties of the system. Indeed, if you measure the momentum after the measurement of position, from the result of position-measurement you will not be able to predict well the result of the momentum-measurement.

 

[Eye_in_the_Sky]

In the third book of Feymann's Lectures on Physics(section 1-8) he describes how "the Uncertainty Principle protects Quantum Mechanics." The experimental situation is a modified double slit experiment where the two slits are put on rollers in an attempt to detect which slit an electron passes through. The idea is that the change in the direction of momentum of the slits on rollers after collision with an electron will be opposite for the two slits.

He argues that if one could measure this change in momentum then uncertainty principle would make it impossible to exactly locate the position of the slits and this will blur out the interference pattern.

Here is a heuristic argument.
_____

Let

λ = wavelength of incoming plane-wave ,

d = slit separation ,

L = distance from slits to screen .

Let "<" mean "less than by roughly an order of magnitude".
_____

Then:

The momentum p (of the quantum) associated with the incoming plane-wave is

p = h / λ .

The momentum imparted to the slit assembly will be in one of two opposite directions and is on the order of

p (d / L) .

Thus, the uncertainty Δp associated with a measurement of this impulse must be small enough to satisfy

Δp < p (d / L) .

But, simultaneously, in order to preserve the interference pattern, the uncertainty Δx in the position of the slits must satisfy

Δx < λ (L / d) ,

since λ (L / d) is on the order of magnitude of the fringe separation.

Therefore, the product of the uncertainties would have to satisfy

Δx Δp << λ p = h .

- Why can't the apparatus be reset each time for each electron? In this way we do not care where the slits are after the electron passes through - only the momentum.

Is it that the apparatus can not be reset because this would mean knowing where the slits are and knowing that their momentum is zero at the same time?But then doesn't the mere existence of an interference pattern mean that we know both?

The presence of an interference pattern implies

Δx < λ (L / d) .

In that case, by the uncertainty principle,

Δp > p (d / L) .

This means the uncertainty in the velocity of the slit assembly, Δv=Δp/m, will then have a lower bound so small that (over the time scales of the experiment) the position of the slits is practically fixed (relative to the scale set by the fringe separation).

 

[Tollendal]

Dear Lavinia,

Some errors survive a long time after refuted. The double slits experiment is one of those. Prof. Aephraim Steinberg (Toronto, Canada) since 2011 demonstrated something that negates the traditional conclusions.

"By applying a modern measurement technique to the historic double-slit experiment, we were able to observe the average particle trajectories undergoing wave-like interference, which is the first observation of its kind. This result should contribute to the ongoing debate over the various interpretations of quantum theory," said Steinberg. "It shows that long-neglected questions about the different types of measurement possible in quantum mechanics can finally be addressed in the lab, and weak measurements such as the sort we use in this work may prove crucial in studying all sorts of new phenomena.

"But mostly, we are all just thrilled to be able to see, in some sense, what a photon does as it goes through an interferometer, something all of our textbooks and professors had always told us was impossible."

Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer, Science 3 June 2011: Vol. 332 no. 6034 pp. 1170-1173 DOI: 10.1126/science.1202218

 

[vanhees71]

When you determine position, the wave function indeed collapses to a very narrow wave function. But it is precisely this narrowing of the wave function which makes the momentum badly undetermined. Even if the momentum was well determined before the measurement of position, after the measurement of position the momentum is no longer well determined. It is this very act of measurement that changes the properties of the system. Indeed, if you measure the momentum after the measurement of position, from the result of position-measurement you will not be able to predict well the result of the momentum-measurement.

Well, in the usual setup the electron's position is measured by letting it hit a screen (photoplate or nowadays CCD). Then it's absorbed by the material and it doesn't make much sense anymore to associate a wave function with it, because it now has become part of a many-body system. Your collapse is an illusion, in this case even in a very drastic way!

Collapse is an effective way to talk about socalled ideal projection measurements a la von Neumann, but it's not a well founded description in terms of quantum theory.

 

[vanhees71]

Dear Lavinia,

Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer, Science 3 June 2011: Vol. 332 no. 6034 pp. 1170-1173 DOI: 10.1126/science.1202218

This doesn't make much sense to me. How can single photons be on trajectories. You cannot even define a position observable for them. This must mean something different than trajectories in the usual sense. The sentence is also a contradiction in adjecto, because it talks about "average trajectories", but this can be only measured by making measurements on an ensemble of photons. So I guess they talk about detection events of photons. I cannot access the paper since I'm on travel.

 

[Demystifier]

Well, in the usual setup the electron's position is measured by letting it hit a screen (photoplate or nowadays CCD). Then it's absorbed by the material and it doesn't make much sense anymore to associate a wave function with it, because it now has become part of a many-body system. Your collapse is an illusion, in this case even in a very drastic way!

Collapse is an effective way to talk about socalled ideal projection measurements a la von Neumann, but it's not a well founded description in terms of quantum theory.

That's all correct, but irrelevant in the present context where the issue is to understand the position-momentum uncertainty relations. In that context, the ideal projection measurement is a suitable idealization.

Or to quote Karl Popper:
"Science may be described as the art of systematic over-simplification."

 

[bhobba]

This doesn't make much sense to me. How can single photons be on trajectories.

Its the typical misunderstanding of weak measurements.

I have lost count of the number of what is obviously 'silly' claims that rely on misinterpreting weak measurements.

Thanks
Bill

 

[vanhees71]

That's all correct, but irrelevant in the present context where the issue is to understand the position-momentum uncertainty relations. In that context, the ideal projection measurement is a suitable idealization.

Or to quote Karl Popper:
"Science may be described as the art of systematic over-simplification."

The position-momentum uncertainty relation is well understood without any need of over-simplification. It's just the fact that the Fourier transform of a sharply peaked position-wave function leads to a broad postition-wave function which is its Fourier transform (and vice versa). No need for over-simplifications, let alone philosophers, who tend to make simple math complicated to sound more profound than they are ;-).

 

[Demystifier]

The position-momentum uncertainty relation is well understood without any need of over-simplification. It's just the fact that the Fourier transform of a sharply peaked position-wave function leads to a broad postition-wave function which is its Fourier transform (and vice versa). No need for over-simplifications, let alone philosophers, who tend to make simple math complicated to sound more profound than they are ;-).

Again, it depends on the context, or more precisely on the person who is supposed to understand that. It seems to me that the thread starter in this case would not find your explanation satisfying. I always try to adjust my explanation to the person who asks a question, not to some abstract idea of what a perfect explanation should look like.

 

[vanhees71]

That doesn't matter. If you want to understand quantum theory you have to learn very basic functional analysis and generalized functions (aka distributions) anyway, and in my experience you make it more difficult, particularly for beginners, if you are not strict on that point. Again: The wave functions that represent (pure) states in non-relativistic single-particle quantum mechanics are the square integrable functions in the sense of the Hilbert space $L^2(\mathbb{R}^3)$ and nothing else.

Generalized eigenfunctions for continuous spectral values of a self-adjoint operator are not square integrable but distributions (in the sense of generalized functions). Any socalled "simplification" of this very basic principle of quantum theory leads to confusion, because it leads to the ill-defined use of distributions, and I don't want to suggest to teach the rigorous math to physicists starting to learn quantum theory but to use the usual care in hand-waving physicists use to get to results without too much rigor but also without too much confusion. You to find the right balance. To claim that $\delta$ distributions or plane-wave sulutions to the Schrödinger equation represent pure states in quantum theory, however, is clearly beyond what should be allowed to any textbook writer in quantum theory, because it leads to sever misunderstandings of quantum theory, and not only to some mathematical finesse, but also in the physical meaning of the theory!

 

[lavinia]

When you determine position, the wave function indeed collapses to a very narrow wave function. But it is precisely this narrowing of the wave function which makes the momentum badly undetermined. Even if the momentum was well determined before the measurement of position, after the measurement of position the momentum is no longer well determined. It is this very act of measurement that changes the properties of the system. Indeed, if you measure the momentum after the measurement of position, from the result of position-measurement you will not be able to predict well the result of the momentum-measurement.

OK. Susskind uses the decay of positronium to illustrate conservation of energy and momentum in relativity. Starting in center of mass coordinates he deduces that the two photons must have opposite trajectories (zero momentum) and total energy very slightly less than the sum of the mass of the positron plus the mass of the electron. At the time of decay we thought that the center of mass would be the actual decay point.

 

[Demystifier]

he deduces that the two photons must have opposite trajectories (zero momentum)

This is classical reasoning, while photons obey laws which are not classical. In quantum mechanics, if trajectories are not directly measured, then the trajectories do not even exist (standard interpretation) or exist but do not always obey momentum conservation (Bohmian interpretation).

 

[ShayanJ]

This is classical reasoning, while photons obey laws which are not classical. In quantum mechanics, if trajectories are not directly measured, then the trajectories do not even exist (standard interpretation) or exist but do not always obey momentum conservation (Bohmian interpretation).

What about this?

1. If a particle gets through the slit, then I can say that my knowledge of the position of the particle at the slit has an uncertainty equal to the width of the slit. Thus, the width of the slit Delta(y) is the uncertainty in the position of the particle when it passes through the slit. 2. The y-component of the momentum of the particle can be found by looking at how far the particle drifts along the y-direction when it hits the screen. This makes the explicit assumption that no external forces acts on the particle at and after it passes through the slit, so that it’s momentum remains constant from the slit to the screen (which is a reasonable assumption). Let’s say the particle drifts from the center, straight-through line and hits the screen at a distance Y. If it takes the particle a time T to reach the screen (which we can assume to be a constant if screen distance from the slit is much larger than the width of the slit (i.e. L >> Delta(y)), then the y-component of the momentum is p_y prop. Y/T. Now, there is a measurement uncertainty here in determining where exactly the particle hits the detector. This measurement uncertainty depends on the resolution of the detector, how fine is the “mesh”, etc. But this is NOT the “uncertainty” that is meant in the HUP. We haven’t gotten to the uncertainty of the momentum YET. All we have is a measurement of the y-component of the momentum of the particle.

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

It seems to me this reasoning also depends on assigning a trajectory to the particle.

 

[Demystifier]

What about this?

It also uses some classical reasoning, e.g. "The y-component of the momentum of the particle can be found by looking at how far the particle drifts along the y-direction".

 

[ShayanJ]

It also uses some classical reasoning, e.g. "The y-component of the momentum of the particle can be found by looking at how far the particle drifts along the y-direction".

Are there any conditions under which its safe to use such classical reasonings?
I discussed it in the discussion thread related to that insight post but atyy could only give the conditions in terms of phase space formulation of QM which I haven't studied about.

 

[Demystifier]

Are there any conditions under which its safe to use such classical reasonings?

You can use such classical reasoning as an approximation for wave packets which satisfy the following two conditions at once:
$$\Delta y(t) \ll \langle y(t)\rangle$$
$$\Delta p_y(t) \ll \langle p_y(t)\rangle$$

 

[ShayanJ]

You can use such classical reasoning as an approximation for wave packets which satisfy the following two conditions at once:
$$\Delta y(t) \ll \langle y(t)\rangle$$
$$\Delta p_y(t) \ll \langle p_y(t)\rangle$$

But the expectation value of position depends on the choice of origin!
Also, before the electron passes through the slit, $\langle p_y(t) \rangle=0$ and after that, we assume $\langle p_y(t) \rangle \approx \Delta p_y(t)$. So it seems to me that in the double slit experiment, the condition about momentum is not satisfied!

 

[Demystifier]

But the expectation value of position depends on the choice of origin!

Yes, but the position of the origin does not change with time, while my conditions depend on time. So if the conditions are satisfied for most of the time, then classical reasoning can be used for most of the time. Note that those are sufficient conditions, not necessary conditions. Sometimes classical reasoning can be used in other conditions, but it's not so easy to present a more general rule.

Also, before the electron passes through the slit, $\langle p_y(t) \rangle=0$ and after that, we assume $\langle p_y(t) \rangle \approx \Delta p_y(t)$. So it seems to me that in the double slit experiment, the condition about momentum is not satisfied!

So you cannot use classical reasoning in the double slit experiment. This is fully compatible with the fact that double slit experiment is a prime example of quantum behavior.

 

[lavinia]

I found this on the Wikipedia page on the Double Slit Experiment

"An experiment performed in 1987 [38][39] produced results that demonstrated that information could be obtained regarding which path a particle had taken without destroying the interference altogether. This showed the effect of measurements that disturbed the particles in transit to a lesser degree and thereby influenced the interference pattern only to a comparable extent. In other words, if one does not insist that the method used to determine which slit each photon passes through be completely reliable, one can still detect a (degraded) interference pattern.[40]"

 

[lavinia]

Thanks everyone for your responses. At this point, Rebecca and I understand your answers as follows: The double slit apparatus has a well determined location in space - not perfect but within a small $Δx$. So its momentum is fairly uncertain. If an electron hits the screen-on-rollers and changes the momentum by an amount less than the uncertainty in the momentum (or some fraction of it, not sure) one will not be able to determine which slit it passed through and the interference pattern will not be disturbed. If it hits the apparatus with large enough momentum to cause a detectible change in its momentum, the center of the interference pattern will be shifted and over many electrons the pattern will be blurred out.

- One wonders whether some of the electron bounce off of the apparatus and can be detected on a second screen on the opposite site of the two slits. Will this also produce an interference pattern?

- One wonders if the center of the interference patten can be exactly deduced.