Double slit probability question

[entropy1]

and you will find an interference pattern

Are you sure? The screens are physically separated! (boxed)

 

[Nugatory]

The point I was trying to make at #8, is that screen A is first-in-line because it is closer, so the 'particle' (wavefunction) has to 'decide' to impact there or travel some further to screen B. In effect, both screens are hit 50% of the time. So, what makes screen A decide to let 50% of the electrons go? Then screen B obviously decides to keep 100% of the ones let through by screen A, while it as an exactly identical screen. (I realize screen A and B are physically separated - I hope you understand what I mean)

This is no different than when there is one screen and the particle "decides to land" somewhere on the left-hand side of the screen instead of the right. One way of seeing this is to compare your two-screen setup with one in which there is only one screen, but we've angled it so that it is no longer parallel to the barrier and one side is closer than the other. Another way of seeing this is to consider what happens if in your setup we completely remove either screen: the rate at which dots appear on the remaining screen and the pattern on that screen doesn't change.

For every point in space we calculate the probability of a particle leaving a dot if we place a screen at that point. We do that by summing the amplitudes contributed by every available path to that point; the presence or absence of other screens at other points only affects that calculation if they happen to block some paths so we don't get a contribution from them.

 

[DrClaude]

Are you sure? The screens are physically separated! (boxed)

Then I misunderstood the picture in post #13. If there is a divider, then there is no interference.

So, what makes screen A decide to let 50% of the electrons go? Then screen B obviously decides to keep 100% of the ones let through by screen A, while it is an exactly identical screen. (I realize screen A and B are physically separated - I hope you understand what I mean)

Then it is simply a question of probability. Right after the slit, there is a 50% probability of the electron being on either side. The electron is in a superposition of being in the A "box" and the B "box." Once a detection is made, that superposition goes away.

If the distance of the two screens from the slits is so different that not detecting the event at A after a while tells you that the electron is flying towards B before hitting the screen, then you can see A as always making a detection, even if no electron is observed.

 

[Nugatory]

Are you sure? The screens are physically separated! (boxed)

By "physically separated" do you mean that there is no path from slit A to screen B, or from slit B to screen A because the middle dashed line in your picture is a solid barrier? If so, there will be no interference pattern on either screen because for any point on either screen there is only one path available. But that's not how you drew your picture.

 

[entropy1]

By "physically separated" do you mean that there is no path from slit A to screen B, or from slit B to screen A because the middle dashed line in your picture is a solid barrier?

Yes.

But that's not how you drew your picture.

No, I realize my mistake.

 

[entropy1]

So, there are four possibilities:

• The electron did not impact either screen. In that case, the electron hit something else in the box or something outside the box, or:
• The electron impacted screen A, or:
• The electron impacted screen B.

But:

• The electron never impacts both screen A and screen B.

It is this mutual exclusiveness that puzzles me.

You could assert that the electron travels through slit A or through slit B, but the wavefunction travels through both. So what is mutually exluding the impacts on screen A and B? You can't say that the electron has a 50% chance of being on either side, for then it could be detected with a 25% chance on both sides simultaneously, right?

I hope I posed a clear question/issue.

 

[Nugatory]

So, there are four possibilities:

• The electron did not impact either screen. In that case, the electron hit something else in the box or something outside the box, or:
• The electron impacted screen A, or:
• The electron impacted screen B.

But:

• The electron never impacts both[/I screen A and screen B.
  

It is this mutual exclusiveness that puzzles me.

And when I toss a coin, it might come up heads or it might come up tails but it never comes up both heads and tails. That's the same sort of mutual exclusiveness, for the same reason. Why should the one be any more puzzling than the other?

You could assert that the electron travels through slit A or through slit B, but the wavefunction travels through both.

Yes, you could assert that. But there's nothing in the theory of quantum mechanics that says that has to be what's going on. So if thinking about it that way seems to imply absurdities like a 25% chance of one particle making two dots... Then don't think about it that way.

 

[entropy1]

And when I toss a coin, it might come up heads or it might come up tails but it never comes up both heads and tails. That's the same sort of mutual exclusiveness, for the same reason. Why should the one be any more puzzling than the other?

I have trouble imagining how the coin toss fits in this picture. Where is my setup equivalent with a coin toss?

So if thinking about it that way seems to imply absurdities like a 25% chance of one particle making two dots... Then don't think about it that way

That is not my line of reasoning. I was referring here to possible consequences of the reasoning of DrClaude.

 

[Karolus]

Excuse me if I intrude, but I do not understand what the problem is. In the experiment proposed an electron hits the screen A is passed through the hole A and if it hits the screen B, passes through the hole B. Once the experiment performed, you can count how many electrons there are on the screen A and the screen B, and the two numbers are almost equal, without interference. The electron that starts from the source has an equal chance of passing through the slit 1 or 2. Where is the problem? (Maybe I will not understand it, but the question I'm curious ...)
The different distance to which the screens are placed behind the slits, does not affect the probability of the electron passing through the hole 1 or 2
Sorry for the intrusion, I would not introduce "entropy" ...

 

[entropy1]

Excuse me if I intrude, but I do not understand what the problem is. In the experiment proposed an electron hits the screen A is passed through the hole A and if it hits the screen B, passes through the hole B. Once the experiment performed, you can count how many electrons there are on the screen A and the screen B, and the two numbers are almost equal, without interference. The electron that starts from the source has an equal chance of passing through the slit 1 or 2. Where is the problem? (Maybe I will not understand it, but the question I'm curious ...)
The different distance to which the screens are placed behind the slits, does not affect the probability of the electron passing through the hole 1 or 2

The electron goes through both slits. Why would it go through one slit?

 

[Nugatory]

I have trouble imagining how the coin toss fits in this picture. Where is my setup equivalent with a coin toss?

Your setup has a set of mutually exclusive outcomes (particle detected at A, particle detected at B, no detection) and a probability associated with each outcome. The coin toss also has a set of mutually exclusive outcomes (heads, tails) and a probability associated with each outcome.

In both cases we have a model that starts with a set of mutually exclusive outcomes. In both cases we might reasonably ask whether that model accurately describes the universe we live, and we answer that question by doing experiments. If there is a difference, it is that we're much quicker to accept that the results of a coin toss are mutually exclusive because "it's obvious" when we pick up a coin, look at it, and think back on a lifetime of experience with coins and other objects.

If you look at the mathematical formalism of quantum mechanics without layering any interpretational questions on top of it, you'll see a theory for predicting the results of measurements. Measurement outcomes are more or less by definition mutually exclusive; a theory that didn't work that way might be logical and internally consistent, but it wouldn't match the observed behavior of the universe that we live in so we wouldn't bother much with it.

 

[ueit]

So, there are four possibilities:

• The electron did not impact either screen. In that case, the electron hit something else in the box or something outside the box, or:
• The electron impacted screen A, or:
• The electron impacted screen B.

But:

• The electron never impacts both screen A and screen B.

It is this mutual exclusiveness that puzzles me.

You could assert that the electron travels through slit A or through slit B, but the wavefunction travels through both. So what is mutually exluding the impacts on screen A and B? You can't say that the electron has a 50% chance of being on either side, for then it could be detected with a 25% chance on both sides simultaneously, right?

I hope I posed a clear question/issue.

An electron is a point particle. No electron has been detected in more than one place. If the electron is in one place it cannot be in another one.

The wavefunction does not "travel". It is a mathematical object used to compute detection probabilities.

So, the electron arrives to either screen A or B as a point particle and is detected as such.

Andrei

 

[Karolus]

The electron goes through both slits. Why would it go through one slit?

then suppose to fire a single electron. This electron, in your opinion, is broken into two halves - electrons, and each half hits the screen?
I do not think so. If you fire only one electron, the trace of this electron in the screen A, or B, but not in both!

 

[zonde]

The electron goes through both slits.

Interesting interpretation. What would be prediction of such a model when two beams from two separate but coherent sources meet at the screen? Will there be interference?

 

[entropy1]

then suppose to fire a single electron. This electron, in your opinion, is broken into two halves - electrons, and each half hits the screen?
I do not think so. If you fire only one electron, the trace of this electron in the screen A, or B, but not in both!

It has to be possible to travel through both slits for a single electron, otherwise I can't explain the possibility of a (possible) interference pattern.

It seems to me that only at detecting the electron it "has travelled" a certain path, right?

 

[entropy1]

Interesting interpretation. What would be prediction of such a model when two beams from two separate but coherent sources meet at the screen? Will there be interference?

I don't understand the question very well, but when firing two electrons at once - I don't know what happens.

 

[zonde]

I don't understand the question very well, but when firing two electrons at once - I don't know what happens.

Two coherent sources fire electrons at whatever time they fire them (firing times are not synchronized). Two beams are arranged so that they end up on the screen like coming from two slits of the double slit. Do you expect to see the interference pattern given your model (the electron goes through both slits)?

 

[Nugatory]

t has to be possible to travel through both slits for a single electron, otherwise I can't explain the possibility of a (possible) interference pattern.
It seems to me that only at detecting the electron it "has travelled" a certain path, right?

Not right. The electron doesn't have any path or position at all except when it is interacting with something else, and that only happens at the source and at the screen. There is no justification for saying the that electron ever was at or passed through any point in the space between source and screen unless you actually put something (a detector? A random air molecule?. ...) for the electron to interact with at that point. Quantum mechanics tells us how to calculate the probability of finding the electron at that point if we do have something there, but it says nothing about what is happening in between.

Of course our lifetime of experience with classical objects leaves us with an almost irresistible temptation to assume that the electron that left a dot on the screen must surely have traveled from through space from the source to the screen. But that's not a necessary part of the theory, and if you can resist the temptation to make that assumption then your problem will go away.

 

[entropy1]

Not right. The electron doesn't have any path or position at all except when it is interacting with something else, and that only happens at the source and at the screen. There is no justification for saying the that electron ever was at or passed through any point in the space between source and screen unless you actually put something (a detector? A random air molecule?. ...) for the electron to interact with at that point. Quantum mechanics tells us how to calculate the probability of finding the electron at that point if we do have something there, but it says nothing about what is happening in between.

So the electron (when 'nowhere') has a probability to be detected somewhere, but not a probability to 'be' somewhere?

Of course our lifetime of experience with classical objects leaves us with an almost irresistible temptation to assume that the electron that left a dot on the screen must surely have traveled from through space from the source to the screen. But that's not a necessary part of the theory, and if you can resist the temptation to make that assumption then your problem will go away.

How do we explain the presence/absence of an interference pattern when we cannot say: "it went through one slit" or: "it went though both slits"?

 

[entropy1]

Two coherent sources fire electrons at whatever time they fire them (firing times are not synchronized). Two beams are arranged so that they end up on the screen like coming from two slits of the double slit. Do you expect to see the interference pattern given your model (the electron goes through both slits)?

If I understand correctly, yes, I would think so, in the general case, but not in my setup.

 

[ueit]

Not right. The electron doesn't have any path or position at all except when it is interacting with something else, and that only happens at the source and at the screen.

The electron does interact with the slits. We know that for a fact because its momentum is changed. If the electron only interacts at the source and screen the slits should have no influence on the observed pattern which is not the case.

There is no justification for saying the that electron ever was at or passed through any point in the space between source and screen unless you actually put something (a detector? A random air molecule?. ...) for the electron to interact with at that point.

As I said before, there is something between the source and the screen, the slits.

There is also the point that the electron always interacts with all charged particles in the universe, as a result of it being charged and it always interacts with all massive particles in the universe as a result of it being massive.

An electron never interacts like a bullet, bumping into objects that happen to be in its path, that is a crude approximation.

 

[zonde]

If I understand correctly, yes, I would think so, in the general case, but not in my setup.

Well, but then we would have to extend your model and claim that electron is fired by both sources, right?

 

[Ostrados]

It has to be possible to travel through both slits for a single electron, otherwise I can't explain the possibility of a (possible) interference pattern.

It seems to me that only at detecting the electron it "has travelled" a certain path, right?

In you setup the wave function will not go in both slits it will collapse because the which path info is now revealed due to splitting the paths using a barrier (ex: if particle hits screen A then it came from slit A). Which is equivalent to measurment at slits in normal double slit experiement.

Note: which path info is enough to destroy the interferance even with no direct measurment (see delayed choice experiement).

No puzzle here .. I hope it is clear for you now.

 

[entropy1]

Well, but then we would have to extend your model and claim that electron is fired by both sources, right?

As far as I understand it, you suggest a setup with two sources. In a setup with two slits and one screen, I expect to see interference. In my setup, with two slits and two screens in two separated compartments, there would be two (separated) bulbs (maxima) and no interference pattern. What difference it makes to use two sources instead of one, I can't assess, but it seems to complicate the matter by the possibility of having two particles entering the boxes simultaneously; I don't know if these particles (wavefunctions) blend into one.

 

[mike1000]

Do the slits increase the number of possible paths? The narrower the slit the more undefined the momentum becomes, which I think would translate into an increase in the number of possible paths after the electron passes through the slit.

In other words, the number of possible paths is a function of 1)the number of slits and 2)how wide is each slit. Can I then jump to the conclusion that as the number of possible paths increases we see interference patterns because the electron can (and will) travel any of the possible paths.

 

[DrClaude]

Do the slits increase the number of possible paths? The narrower the slit the more undefined the momentum becomes, which I think would translate into an increase in the number of possible paths after the electron passes through the slit.

In other words, the number of possible paths is a function of 1)the number of slits and 2)how wide is each slit. Can I then jump to the conclusion that as the number of possible paths increases we see interference patterns because the electron can (and will) travel any of the possible paths.

The number of possible paths is always infinite.

 

[entropy1]

The number of possible paths is always infinite.

So what decides which one is actualized?

 

[DrClaude]

So what decides which one is actualized?

We don't know. As far as we understand, the (quantum) world is truly random.

 

[mike1000]

The number of possible paths is always infinite.

Well that kind of takes the uncertainty in momentum, caused by the electron passing through the narrow slit, out of the equation doesn't it? Let me rephrase it this way, passing through the slit causes the uncertainty in momentum which makes certain paths more probable.

 

[DrClaude]

Well that kind of takes the uncertainty in momentum, caused by the electron passing through the narrow slit, out of the equation doesn't it? Let me rephrase it this way, passing through the slit causes the uncertainty in momentum which makes certain paths more probable.

This is where the Feynman path formulation becomes very valuable. What the width of the slits do is change the relative interference of the different paths.