# Destructive interference in the wavefunction, is this conceptually explainable?

• Xilor
In summary, the two waves can be recombined in all interpretations, but the outcome is always the same--a single photon travels down each path.
Xilor
Hello, destructive interference seems like an important part of quantum physics, but I'm finding it very hard to grasp it conceptually. For instance in the Elitzur–Vaidman bomb tester, destructive interference in the mirror is used to determine if one of the paths is blocked. What exactly is happening that causes every single photon to choose only one of the paths when the wave comes from both sides? Why is the mirror so significant that it can cause this behavior and how is it decided which of the paths the photons will always take?

Bumping as I'm still interested.

Xilor said:
What exactly is happening that causes every single photon to choose only one of the paths when the wave comes from both sides?
The answer is somewhat controversial, because it depends on the interpretation of quantum mechanics, which is not unique. Perhaps the most intuitive interpretation is the Bohmian one, according to which particle is a pointlike object the velocity of which depends on the value of wave function on the position of the particle. Thus, particle takes only one path, depending on the initial position of the particle.

Xilor said:
Why is the mirror so significant that it can cause this behavior and how is it decided which of the paths the photons will always take?
Actually the mirror is not very important, it only plays an auxiliary role by directing the wave beam into a desired direction. A more important peace of apparatus is the beam splitter, which splits the initial wave beam into two beams. In the Bohmian interpretation, the wave is splitted but the particle is not.

So the wave goes both ways but the particle only one in Bohmian Mechanics? Is it then possible to combine that 'empty' wave and the particles wave again in a way that affects the particle? That sounds like something that should either be possible in all interpretations, or in none.

And how exactly does the initial position matter? Does it have something to do with the lengths of the paths they take until hitting the final beamsplitter? (when I said mirror earlier on, I meant the half-silvered one at the point where the beams combine again)

Edit:

Also, how do they even make the beamsplitters precise enough to make exactly 0 photons appear on one side? Wouldn't even the smallest deviation from the 50% reflection on either of the splitters cause different results? Or does the same thing happen if the mirror reflects 25% or 75% of the photons?

Last edited:
Xilor said:
So the wave goes both ways but the particle only one in Bohmian Mechanics?
Yes.

Xilor said:
Is it then possible to combine that 'empty' wave and the particles wave again in a way that affects the particle?
Yes.

Xilor said:
That sounds like something that should either be possible in all interpretations, or in none.
Yes, the two beams can be recombined in all interpretations.

Xilor said:
And how exactly does the initial position matter? Does it have something to do with the lengths of the paths they take until hitting the final beamsplitter?
I'm not sure I understand that question, try to reformulate it.

Xilor said:
(when I said mirror earlier on, I meant the half-silvered one at the point where the beams combine again)
Fine, that's the same as beam splitter.

Xilor said:
Also, how do they even make the beamsplitter at the combination point precise enough to make exactly 0 photons appear on one side? Wouldn't even the smallest deviation from the 50% reflection cause different results? Or does the same thing happen if the mirror reflects 25% or 75% of the photons?
Well, in reality they have something like 50.1:49.9 beam splitter, which at the end means that the number of photons at one side is close to zero, but not exactly zero.

Thank you for your responses! It's starting to make some sense now.

Demystifier said:
I'm not sure I understand that question, try to reformulate it.

My question was about the following thing which you said:

Thus, particle takes only one path, depending on the initial position of the particle.

My question was what exactly the influence of this initial position is, what detail about the position ends up translating to the eventual choice of paths? My own guess in that question was that it might have something to do with the pathlengths towards the final beamsplitter, as that would mean one half of the wave would reach the beamsplitter quicker.

Yes, the two beams can be recombined in all interpretations.

So, wild idea here. What if you direct the empty wave and the particle carying wave towards a double split screen, but you separate the slits with some wall in between (only on the side where the photons arrive from) so that you know the empty wave always goes through the left, and the particlewave through the right slit. Since you said they can still interact, a interference pattern should still emerge, while you do have which slit information. Am I correct? Because that does sound strange since I've always heard that which slit information destroys the interference pattern.

Xilor said:
So, wild idea here. What if you direct the empty wave and the particle carying wave towards a double split screen, but you separate the slits with some wall in between (only on the side where the photons arrive from) so that you know the empty wave always goes through the left, and the particlewave through the right slit. Since you said they can still interact, a interference pattern should still emerge, while you do have which slit information. Am I correct? Because that does sound strange since I've always heard that which slit information destroys the interference pattern.

I am not a supporter of Bohm so if I'm incorrect please someone correct me, but what you in fact have in this interpretation are two waves, and the knowledge that one of them is real and the other is empty is not available (that is, you don't know which of the waves is empty and which is in fact a particle).

The interference pattern will be destroyed, yes, and you will only find out which wave is the empty one and which is the particle after you detect it beyond the two slits.

Xilor said:
My question was what exactly the influence of this initial position is, what detail about the position ends up translating to the eventual choice of paths? My own guess in that question was that it might have something to do with the pathlengths towards the final beamsplitter, as that would mean one half of the wave would reach the beamsplitter quicker.
The answer is much easier to draw than to tell in words, but let me try. In such a situation, two different possible paths never cross. So, for example, if the initial position of the particle is at at left side of the beam, then after the splitting the particle will end up in the left beam.

Xilor said:
So, wild idea here. What if you direct the empty wave and the particle carying wave towards a double split screen, but you separate the slits with some wall in between (only on the side where the photons arrive from) so that you know the empty wave always goes through the left, and the particlewave through the right slit. Since you said they can still interact, a interference pattern should still emerge, while you do have which slit information. Am I correct? Because that does sound strange since I've always heard that which slit information destroys the interference pattern.
Well, if something caries the which-slit information, then it eats up the information about the phase of the wave, which means that the wave decoheres and the interference pattern is destroyed. In some cases that destruction of information is reversible (i.e., information about the phase can be restored), but in most cases it is not.

However, your separation of the slits is something different. While it prevents interference as long as the separation is present, it does not carry any which-slit information. In other words, the mere fact that there is a wall does not enable you to tell which slit the particle passed through. In particular, I don't see how did you conclude that the empty wave will be the left one, and not the right one. Indeed, if the wall is no longer present at some larger distance from the slits, the interference can be seen there.

Demystifier said:
The answer is much easier to draw than to tell in words, but let me try. In such a situation, two different possible paths never cross. So, for example, if the initial position of the particle is at at left side of the beam, then after the splitting the particle will end up in the left beam.

I'm afraid I don't understand.. What do you mean with the left side of the beam? The photons are the beam themselves, so this would suggest to me that there's not only particles on the left side, but also on the right side. Following the logic, some photons would also have to end up traveling the other path, creating a stream of photons on both sides of the final beamsplitter.

Well, if something caries the which-slit information, then it eats up the information about the phase of the wave, which means that the wave decoheres and the interference pattern is destroyed. In some cases that destruction of information is reversible (i.e., information about the phase can be restored), but in most cases it is not.

However, your separation of the slits is something different. While it prevents interference as long as the separation is present, it does not carry any which-slit information. In other words, the mere fact that there is a wall does not enable you to tell which slit the particle passed through. In particular, I don't see how did you conclude that the empty wave will be the left one, and not the right one. Indeed, if the wall is no longer present at some larger distance from the slits, the interference can be seen there.

Let me draw it out, I might just be misunderstanding what which path information actually is, but it seems to make sense to me.

First the destructive interference setup as I understand it:

http://i.imgur.com/wVTvK.jpg

Now, add the double split setup with a separating wall. Seemingly this altered setup should either not give an interference pattern, or it should retroactively change what way the photons went, or should give an interference pattern while the path is known with a very high degree of certainty (there's the not completely 0 photons thing, and I suppose photons can tunnel through the wall).

http://i.imgur.com/RDuM5.jpg

In the last situation you drew, there will be an interference pattern, yes. I really don't know how to explain it in Bohmian terms so I will leave it to someone who might, but yes, in that case, the wall doesn't matter at all, and you will observe the interference pattern on the detector.

So there's an interference pattern while you can say through which slit the photon went?

What if you go even further and add the delayed choice experiment at the end instead of just a regular detector screen. Am I correct in saying that if the choice were made to detect which slit information (After the photon would've passed through the slits), that it's always going to be the 'right' slit, while if the choice is made to detect interference that there's going to be interference?
I would really love to see the results of an actual experiment like that because it sounds so mind boggling.

Xilor said:
So there's an interference pattern while you can say through which slit the photon went?

No, not at all. The way you drew it, you cannot say through which slit the photon went until the detector. Even though you drew one of the waves as being photon-containing and the other as being empty, you do not know that. You don't have access to the information of which wave is empty and which wave is not. You didn't put any detectors anywhere that would contain which-wave information. If you had, then yes, the interference pattern would show up, but the way you drew it, you do not have any means of saying 'photon beam' or 'empty wave' before the detector screen.

What if you go even further and add the delayed choice experiment at the end instead of just a regular detector screen. Am I correct in saying that if the choice were made to detect which slit information (After the photon would've passed through the slits), that it's always going to be the 'right' slit, while if the choice is made to detect interference that there's going to be interference?
I would really love to see the results of an actual experiment like that because it sounds so mind boggling.

I will take a raincheck on answering that because I haven't read about the delayed choice experiment yet so I cannot say :P

JamesOrland said:
No, not at all. The way you drew it, you cannot say through which slit the photon went until the detector. Even though you drew one of the waves as being photon-containing and the other as being empty, you do not know that. You don't have access to the information of which wave is empty and which wave is not. You didn't put any detectors anywhere that would contain which-wave information. If you had, then yes, the interference pattern would show up, but the way you drew it, you do not have any means of saying 'photon beam' or 'empty wave' before the detector screen.

Could you explain why this is the case? Because it does seem like there is information. You know from previous experiments that the particle always went into a certain direction, so why can't you say that you know the particle had gone in that direction?
We didn't actually measure the which path information in any way that would collapse the wavefunction, but to require a wavefunction collapse for a wavefunction collapse seems weird. It would seem to reduce the statements about how knowledge of the path affects things to circular logic like: 'If you collapse the wavefunction by measuring the particles path, then the wavefunction is collapsed'.

JamesOrland said:
The way you drew it, you cannot say through which slit the photon went until the detector. Even though you drew one of the waves as being photon-containing and the other as being empty, you do not know that. You don't have access to the information of which wave is empty and which wave is not. You didn't put any detectors anywhere that would contain which-wave information. If you had, then yes, the interference pattern would show up, but the way you drew it, you do not have any means of saying 'photon beam' or 'empty wave' before the detector screen.

This is plain wrong. The setup drawn is a typical Mach-Zehnder interferometer. For such an interferometer the probability distribution at the final output ports is a function of the phase difference along the two paths. If a single coherent beam is used at the entrance of the interferometer, the remaining phase difference is entirely a function of geometry. In this case this is the path length difference between the two paths. As one can measure this difference, one also knows the probability distribution at the exit port. It is not necessary to actually measure it.

Cthugha said:
This is plain wrong. The setup drawn is a typical Mach-Zehnder interferometer. For such an interferometer the probability distribution at the final output ports is a function of the phase difference along the two paths. If a single coherent beam is used at the entrance of the interferometer, the remaining phase difference is entirely a function of geometry. In this case this is the path length difference between the two paths. As one can measure this difference, one also knows the probability distribution at the exit port. It is not necessary to actually measure it.

Xilor said:
Could you explain why this is the case? Because it does seem like there is information. You know from previous experiments that the particle always went into a certain direction, so why can't you say that you know the particle had gone in that direction?
We didn't actually measure the which path information in any way that would collapse the wavefunction, but to require a wavefunction collapse for a wavefunction collapse seems weird. It would seem to reduce the statements about how knowledge of the path affects things to circular logic like: 'If you collapse the wavefunction by measuring the particles path, then the wavefunction is collapsed'.

I'm quite very sorry. I was looking at the experiment you drew and thinking of another, different experiment (one that contained another photon-gun). Please disregard my previous explanations!

I am not sure what would happen in the case of the last drawing, but I believe it would be equivalent to the case where there is a single photon-gun shooting at a single-slit wall.

Xilor said:
The photons are the beam themselves ...
No, by beam I meant the wave beam, and in the Bohmian interpretation the wave is not the photons.

JamesOrland said:
I am not sure what would happen in the case of the last drawing, but I believe it would be equivalent to the case where there is a single photon-gun shooting at a single-slit wall.

To be honest I also misinterpreted the drawing twice before I understood what it is intended to show. I agree with you. At least if the intentionof the setup is that only one of the two slits is illuminated by the photon beam. If the photon beam illuminates both slits, it gets complicated.

Cthugha said:
To be honest I also misinterpreted the drawing twice before I understood what it is intended to show. I agree with you. At least if the intentionof the setup is that only one of the two slits is illuminated by the photon beam. If the photon beam illuminates both slits, it gets complicated.

Yes, that's the intention. Only one slit being hit by the photon beam, and the other slit by the empty wave.
So, how is it explained then? Do the photons not interact with the wave? Is that combineable with what demystifier said earlier about photons being able to interact with the empty wave?

Xilor said:
Is that combineable with what demystifier said earlier about photons being able to interact with the empty wave?
I don't think I said that. The empty wave does not influence the particle, as long as it is empty. However, an empty wave now may became a part of non-empty wave later, in which case it can influence the particle later.

Okay, I'm threading in dangerous territories here again by trying to discuss the dBB interpretation which I do not fully understand myself, so correct me if I make any wrong moves.

What I understood from the theory was that once you know whether a wave is empty or not (as you do in the case of the experiment shown), it no longer has any power to influence the experiment. That is, throughout the whole experiment, there is an empty wave and a photon wave (which one of those two 50/50 waves it is, we do not know), but after the second beam-splitter, we know for a fact that the vertical wave is "empty" and the horizontal one is not, and so the vertical one ceases to exist for all practical purposes.

I need to read more on this idea. I myself don't think it's a model of reality, but I think it can be used as a good approximation to the experimental results, or at least better than the CI's assumptions.

Xilor said:
Yes, that's the intention. Only one slit being hit by the photon beam, and the other slit by the empty wave.
So, how is it explained then? Do the photons not interact with the wave? Is that combineable with what demystifier said earlier about photons being able to interact with the empty wave?
I don't think this works. If there is zero probability of particle going some path then there is no wave either.
In this case in the output of beam-splitter where two waves undergo destructive interference there is no wave. But even if we assume that there are two waves of equal amplitude but in opposite phases they effect will average out to no effect whenever they will interact with other wave (because of opposite phase their effect is always opposite).

Btw to make your setup more realistic you can replace double slit with third beam-splitter so that you have something like two chained Mach-Zehnder interferometers. Analysis basically will be the same.

zonde said:
Btw to make your setup more realistic you can replace double slit with third beam-splitter so that you have something like two chained Mach-Zehnder interferometers. Analysis basically will be the same.

I think, though, he was trying to just understand what the effect of said 'empty wave,' if any, would be felt by the photon beam, so the experiment designed is just simple enough to allow that. And according to what you said, then, it appears my conjecture would be right, and the empty wave does not interfere at all with the photon-beam.

Cthugha said:
To be honest I also misinterpreted the drawing twice before I understood what it is intended to show. I agree with you. At least if the intentionof the setup is that only one of the two slits is illuminated by the photon beam. If the photon beam illuminates both slits, it gets complicated.

The diagram is two separate experiments, I think.

Experiment 1 (i.e. the mach-zehnder)

Once the path is known (either by phase difference or detector or other means) then the other path (labelled as "empty wave") is left empty. i.e. nothing has traveled that path after the second/last beam splitter... i.e. after the experiment is over

Experiment 2 (i.e the one with the wall)

The path to the right/south of the wall will create interference (if it's illuminating both slits). Else it will be single slit, single photon pattern.

The path to the left/north of the wall will showing nothing because nothing has traveled that path.

Now one could argue that you have which-way as well as interference. however that argument is not valid since you have two separate experiments.

Last edited:

## 1. What is destructive interference in the wavefunction?

Destructive interference in the wavefunction is a phenomenon in quantum mechanics where two or more waves combine to produce a resultant wave with a smaller amplitude than the individual waves. This results in a cancellation of the wave amplitude at certain points, leading to a decrease in the intensity of the wave at those points.

## 2. How is destructive interference conceptually explainable?

Destructive interference can be explained using the principle of superposition, which states that when two or more waves interact, the resulting wave is the algebraic sum of the individual waves. In the case of destructive interference, the waves have opposite phases, causing them to cancel out and reduce the overall amplitude of the wave.

## 3. What causes destructive interference in the wavefunction?

Destructive interference is caused by the interaction of two or more waves with opposite phases. This means that when the crests of one wave align with the troughs of another wave, they cancel out and result in a decrease in the overall amplitude of the wave.

## 4. How does destructive interference affect the wavefunction?

Destructive interference affects the wavefunction by reducing the amplitude of the wave at certain points. This can lead to a decrease in the intensity of the wave, and in some cases, complete cancellation of the wave at specific locations.

## 5. What are some real-life examples of destructive interference in the wavefunction?

Destructive interference in the wavefunction can be observed in various natural phenomena such as sound waves, light waves, and water waves. For example, in noise-canceling headphones, destructive interference is used to cancel out unwanted background noise by producing a wave with an opposite phase to the noise wave, resulting in cancellation. Another example is the colorful patterns produced by soap bubbles, where destructive interference of light waves creates regions of dark colors.

Replies
32
Views
3K
Replies
76
Views
5K
Replies
7
Views
1K
Replies
47
Views
5K
Replies
8
Views
2K
Replies
16
Views
2K
Replies
3
Views
936
Replies
26
Views
4K
Replies
9
Views
1K
Replies
81
Views
5K