Why doesn't the electron "wave" collapse at the double-slit?

[francis20520]

TL;DR Summary

Why doesn't the wave collapse at the double-slit???

This is dumb question, so please bear with me.

In the double-slit experiment where they fire a single electron at time, as you can see the electron gun fires a single electron.

Now the electron travels as a wave.

Now my question is, why doesn't the wave collapse when the wave encounters the double-slit?

The double slit is a sheet of metal (I think) with 2 slits in it.

So, the electron "wave" encounters this sheet of metal.

Why doesn't the wave "collapse" right then and there??

Why does the wave collapse only at the detector screen?

Because the detector screen is also a solid piece of plate just like the double-slit plate.

It looks like the electron "knows" the difference between a solid steel plate (i.e. the double-slit) and the detector.

Or, am I getting this all wrong?

 

[Vanadium 50]

You seem to be thinking that an electron is "really" a tiny billiard ball that QM somehow makes "wavy". That's not what QM says, so this is unlikely a path to greater understanding.

 

[francis20520]

I am aware of superposition. So, the electron "wave" is the probability wave function.
So, that means the "wave" gives likelihood of the electron being detected at certain points.

But what exactly is "detection"?

Why doesn't the probability wave collapse at the double-slit plate?

What's not there in the double-slit plate that is in the "detector" plate that collapse this "wave"??

 

[EPR]

You cannot know both position and momentum of something as small as an electron at the same time. Nature forbids it.

Why is it so? Ask a philosopher.

 

[vanhees71]

Forget about collapse. It's not clearly defined what it is and it is not part of the mathematical formalism of quantum mechanics.

All you need to know is that an electron in the non-relativistic approximation can be described by a wave function $\psi(t,\vec{x})$ which obeys the Schrödinger equation,
$$\mathrm{i} \hbar \partial_t \psi(t,\vec{x})=\hat{H} \psi(t,\vec{x})$$
and that
$$P(t,\vec{x})=|\psi(t,\vec{x})|^2$$
is the probability distribution for the position of the electron when looking for it at time $t$.

The slit is included in the Hamilton operator which describes the interaction between the electron and the material making up the slit, and that's how the slit indeed has an influence on the electron and on the behavior of its wave function.

In practice it's of course impossible to really write down the Hamiltinian in this case in all microscopic detail, but it's not necessary, because as an effective description you can say that the material is just absorbing any electron hitting it and only electrons get behind it if they go through one of the slits.

Then the math is very similar to the same calculation you do for this diffraction problem for electromagnetic waves. That's no surprise, because at the end you have to solve the same Helmholtz equation as in the electromagnetic case, and what comes out after some vector calculus is Huygens's principle, i.e., you get the wave behind the slits by superposition of spherical waves origining from each point in the openings, where the wave can be assumed to be an undisturbed free wave entering the slits. If the source of the electrons is very far from the screen you can assume a plane wave there and if you also observe the electrons behind the slits very far from the slits what finally results is that the interference pattern behind the screen is described as the Fourier transform of the slit openings (Fraunhofer refraction).

The only important difference between the picture for classical electromagnetic waves (light) and that for electrons (particles) is the meaning of the wave function: In the quantum case it's just the said position-probability distribution. You cannot think of the electron as a classical point particle, described by a specified position and momentum (or velocity) at any time nor as described as a classical field described by the Schrödinger wave function. All you know about the electron are the probabilities to find it at the detection screen given the setup with the double slits, and these probabilities you can compare with the experiment by letting go very many electrons through the slit, because indeed, each electron will just be registered as a point on the screen, not a smeared distribution described by the wave function, when (wrongly) interpreting it as a classical field. Only with very many electrons you'll get the predicted smooth probability distribution.

For any single electron you cannot predict where it will hit the screen, not even in principle, and it has nothing to do with some missing knowledge. It's just that electrons are quantum objects which cannot be described as a classical particle nor as a classical wave, and you have to accept the probabilistic nature of quantum objects as a fundamental fact that has been figured out 100 years ago by the founding fathers of quantum theory, in fact in three different formulations: matrix mechanics (Born, Jordan, Heisenberg 1925), wave mechanics (Schrödinger 1926), and "transformation theory" or operator formalism (Dirac 1926), but all these three formulations are nothing else than one and the same theory, quantum mechanics.

 

[Nugatory]

This is dumb question, so please bear with me.

That's actually not such a dumb question - you've found the problem with most non-mathematical "explanations" that use the the collapse idea.

One way you could think about it: If you're using a collapse interpretation (but do remember what @vanhees said above about collapse not being part of the mathematical formalism of QM) then it is the position measurement that collapses the wave function. The screen behind the slits is a position measuring device - a dot appears on the screen and we know the electron was there. The two slits are not a position measuring device - they don't determine the position of the electron as it passes through the barrier. No position measurement, no collapse.

If the electron were absorbed by the barrier instead of passing through the slit (classically and misleadingly we would say that it "missed both slits and hit the barrier") that would collapse the wave function. In principle we could look for the tiny point of impact and say that the electron was detected at that point - and that's a measurement.

 

[vanhees71]

Well, you can consider the slits as "position-preparation device". Using the macroscopic effective description as the slits being such that the material is absorbing electrons except the ones going through the slits and if you subscribe to a collapse interpretation you can say the slits "collapse the wave function" such that only electrons which move at some time through the openings are considered, i.e., at the moment the electrons enter the slits they are localized in these slits and the wave function has support only at these slits.

As I said before, the collapse postulate however a somewhat problematic idea since it is not part of the formal description of the theory and just added as an ad-hoc assumption to the theory without any mathematical precision. It is very problematic in the relativistic context (causality problems) and thus I would not use it at all. There's also no necessity to use it in any way, because everything can be described without a collapse (minimal statistical interpretation).

 

[atyy]

Or, am I getting this all wrong?

No you are getting it right. In quantum mechanics, whether a measurement or observation occurs is subjective (Schroedinger's cat). When discussing the double slit, we "measure" or "observe" only at the screen. However, one could probably also use a continuous measurement formalism that collapses the wave function at every point in time - but because different measurements are carried put at different times, the collapse is different at different times. I haven't seen the continuous measurement formalism applied to the double slit, but there is a discussion of related issues for the cloud chamber in https://arxiv.org/abs/1209.2665 [Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929) by Rodolfo Figari, Alessandro Teta].

There is also a related discussion in http://de.arxiv.org/abs/2010.07575 [The time distribution of quantum events by Danijel Jurman, Hrvoje Nikolic].

 

[atyy]

There's also no necessity to use it in any way, because everything can be described without a collapse (minimal statistical interpretation).

This is misleading, since we do not know whether the OP is using "collapse" in the sense you use it. Collapse can be said to be part of the minimal statistical interpretation.

 

[PeterDonis]

why doesn't the wave collapse when the wave encounters the double-slit?

It does--if the electron doesn't go through the slits but hits the screen with the slits in it somewhere else. In the idealized experiments discussed in textbooks, possibilities like this are ignored, but in real experiments they would happen. If your electron source gave some indication every time it emitted a single electron, then you would find that there were some runs where you got that indication from the source but no bright dot ever showed up on the detector screen; that would be a run where the electron didn't make it through the slits at all but hit the first screen (the one with the slits in it) somewhere else.

 

[PeterDonis]

Forget about collapse. It's not clearly defined what it is and it is not part of the mathematical formalism of quantum mechanics.

Please bear in mind that in this forum (as opposed to the QM interpretations forum), the term "collapse" means "the projection postulate", which is Rule 7 in the 7 basic rules described in this Insights article (which is linked to from the guidelines for this forum):

https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/

Any discussion in which the term "collapse" is used with any other meaning, or in which there is an argument about "whether collapse is needed or not", belongs in the QM interpretations forum, not here.

 

[PeterDonis]

Everyone, please bear in mind that QM interpretation discussions belong in the interpretations forum, not here. Discussion in this forum should be using the 7 basic rules (linked to in my previous post just now), and no interpretation beyond that. The OP question can be, and should be in this thread, addressed purely in those terms.

 

[francis20520]

It does--if the electron doesn't go through the slits but hits the screen with the slits in it somewhere else. In the idealized experiments discussed in textbooks, possibilities like this are ignored, but in real experiments they would happen. If your electron source gave some indication every time it emitted a single electron, then you would find that there were some runs where you got that indication from the source but no bright dot ever showed up on the detector screen; that would be a run where the electron didn't make it through the slits at all but hit the first screen (the one with the slits in it) somewhere else.

I really like this answer. Mostly because it's put in non-technical detail.

Just out of curiosity, so when the electron is emitted by the source there is actually no electron but a probability wave.

For ones missing the slits, I suppose that this happens when the highest point in the probability wave is not in the vicinity of the slit when the "wave" is at the plate??

Is this correct??

 

[vanhees71]

The electron emitted by the source is an electron. It is described by a mathematical formalism called quantum theory, in which the probability distribution for finding the electron is given by the squared modulus of a wave function, which obeys Schrödinger's equation. The probability for being absorbed at the "first screen" is given by this probability distribution, and all these electrons are not further considered. All you observe at the "second screen" are those electrons going through the slits. You can take the first screen as a device for "preparation of the electrons" to be observed on the second screen.

 

[PeterDonis]

Moderator's note: A number of posts which were off topic, since they did not use the meaning of "collapse" that is appropriate for this forum, per my previous posts, have been deleted. Any further arguments or speculations about what "collapse" means in this thread, or any further attempts to discuss what particular QM interpretations say about "collapse", will receive a warning and a thread ban.

 

[PeterDonis]

so when the electron is emitted by the source there is actually no electron but a probability wave

No. The electron is an electron. We describe it in the math using a wave function. Don't confuse the model with the thing that the model is describing. Discussions about whether the electron "really is" the probability wave or not belong in the interpretations forum, not here.

For ones missing the slits, I suppose that this happens when the highest point in the probability wave is not in the vicinity of the slit when the "wave" is at the plate??

No. The wave function is the same for every electron that comes out of the source. That wave function has a more or less constant probability of reaching any particular position on the first screen (the one with the slits in it). But the first screen doesn't give any indication when an electron hits it, so the only relevant probability at the first screen is the probability of the electron going through the slits vs. not. You could, in the idealized setup we are discussing, measure this probability by counting how many times an electron is emitted from the source but never shows up on the second screen (the detector screen), vs. how many times it does show up on the second screen.

Another way to view this is the way @vanhees71 describes it in post #14: the first screen serves as a preparation device. It filters out all the electrons from the source that don't pass through one of the slits, so the only ones that get measured at the second screen are the ones that do pass through the slits.

Either way of viewing it is fine; they basically end up the same way in the math, at least as far as predicting what will be observed at the second screen.

 

[vanhees71]

Moderator's note: A number of posts which were off topic, since they did not use the meaning of "collapse" that is appropriate for this forum, per my previous posts, have been deleted. Any further arguments or speculations about what "collapse" means in this thread, or any further attempts to discuss what particular QM interpretations say about "collapse", will receive a warning and a thread ban.

Which meaning of "collapse" is appropriate for this forum in your opinion?

To my understanding we had an agreement about the minimal interpretation of quantum theory, and there's no place for a collapse. Just deleting postings is nonsense, because then you cannot follow the arguments for the one or the other point of view anymore. So the entire thread should go to the interpretation subforum, where a discussion about "collapse" is "allowed".

 

[vanhees71]

No. The wave function is the same for every electron that comes out of the source. That wave function has a more or less constant probability of reaching any particular position on the first screen (the one with the slits in it).

Here one has to be careful. It depends on the way you "prepare the electrons" or, in a more operationally formulated way, on how the electron source is constructed and also how far from the slits you put it. If you take a very narrow beam of electrons just aiming it to go through one of the slits, you'd not get a two-slit interference pattern (maybe a single-slit interference pattern, if the beam is wide compared to the width of the single slit).

The same holds for the observation screen: Even if you have the preparation such that both slits are well-covered by the incoming wide beam, if you put it too close to the slits, you'll not see two-slit interference patterns but then you can still follow through which slit each electron came and you get the corresponding pattern of two spots (i.e., an incoherent addition of the single-slit interference patterns). Only if the observation screen is far enough away from the slit where the partial waves from each slit overlap, you see the two-slit interference pattern.

All this you can calculate by using the Green's function (propagator) of the electron's Schrödinger equation including the slits.

 

[PeterDonis]

Which meaning of "collapse" is appropriate for this forum in your opinion?

It isn't a matter of my or anyone's opinion. It's what's stated in the guidelines for this forum. I stated explicitly what those guidelines say about the term "collapse" in post #11.

To my understanding we had an agreement about the minimal interpretation of quantum theory, and there's no place for a collapse.

We have been over and over this, and you should know better than to try to rehash it here. I am now banning you from further posts in this thread, in accordance with forum policy and with the warning I gave in post #15.

 

[PeterDonis]

It depends on the way you "prepare the electrons" or, in a more operationally formulated way, on how the electron source is constructed and also how far from the slits you put it.

Yes, this is a fair point. I was assuming that the source was configured in such a way, and placed far enough from the slits, that my statement would be true, since that seems to be the most relevant configuration for this thread. But of course it is possible to configure and place the source such that my statement is not true or even close to true.

 

[skyw33]

It does--if the electron doesn't go through the slits but hits the screen with the slits in it somewhere else. In the idealized experiments discussed in textbooks, possibilities like this are ignored, but in real experiments they would happen. If your electron source gave some indication every time it emitted a single electron, then you would find that there were some runs where you got that indication from the source but no bright dot ever showed up on the detector screen; that would be a run where the electron didn't make it through the slits at all but hit the first screen (the one with the slits in it) somewhere else.

The interesting realization about this explanation for me is that if an election doesn't go through the slits, then that means that the election hit somewhere else on the screen with the slits in it. When this happens, there is obviously no detection of an electron on the detector screen. Therefore, can we conclude that there is no wave traveling through the slits? If so, can we say that the screen with the slits collapses the wave function in the same way that the detector collapses the wave function?

 

[PeterDonis]

When this happens, there is obviously no detection of an electron on the detector screen. Therefore, can we conclude that there is no wave traveling through the slits?

Yes.

If so, can we say that the screen with the slits collapses the wave function in the same way that the detector collapses the wave function?

For the cases where the electron hits the screen instead of going through the slits, yes. But this case is trivial in the usual experimental setup, because "collapsing the wave function" does not help you to make any further predictions, since the electron is not detected when it hits the screen.

 

[skyw33]

Thank you. I guess my point is that people talk about the "act of measurement" collapsing the wave function. But is it really an act of measurement in addition to any interaction with matter?

 

[PeterDonis]

people talk about the "act of measurement" collapsing the wave function.

"People talk" is not a good reference. Actual QM textbooks will carefully explain what "collapse of the wave function" does and does not mean as far as using the theory to make predictions. For a brief summary of what they say, please see post #11 in this thread.

 

[Vanadium 50]

people talk

People say lots of things. You can't really trust them.

 

[skyw33]

Sorry, all threads eventually lead to "interpretation". I'm sure you are tired of that.

 

[PeterDonis]

Sorry, all threads eventually lead to "interpretation".

The concept of "collapse" that is described in post #11 and the references there is independent of any interpretation. It is part of the basic machinery of QM that is used to make predictions. But that concept makes no claims at all about what "really happens", or whether collapse is a "real process" or not. Claims of that sort are interpretation dependent, yes.

 

[Demystifier]

The double slit is a sheet of metal (I think) with 2 slits in it.

So, the electron "wave" encounters this sheet of metal.

Why doesn't the wave "collapse" right then and there??

When the electron hits the metal, it gets absorbed and hence "collapsed". But when it hits one of the slits, it goes through it hence does not get absorbed. The electron wave is an extended object that does both; a part of the wave hits the metal and another part of the wave hits the slits. This means that there is a finite probability that the electron will get absorbed, but there is also a finite probability that it will go through the slits. In practice this means that, if you have many electrons, some will be absorbed (collapsed), and some will not. Those that will not are the ones that physicists are usually interested in, because only those will later create the interference pattern at the detector screen. If the detector screen itself had slits in it, some of the electrons would go through them as well.

 

[Mister T]

TL;DR Summary: Why doesn't the wave collapse at the double-slit???

Now my question is, why doesn't the wave collapse when the wave encounters the double-slit?

The double slit is a sheet of metal (I think) with 2 slits in it.

So, the electron "wave" encounters this sheet of metal.

Why doesn't the wave "collapse" right then and there??

Why does the wave collapse only at the detector screen?

Collapse of the wave function is part of some of the interpretations of quantum mechanics, most famously the Copenhagen Interpretation. I think what you're really asking is why is the electron not detected as a particle at the double-slit screen, but is instead detected at the detector screen.

The answer is simply that in the usual setup you are observing what happens at the detector screen, not at the double-slit screen. So you observe the electron as a particle at the detector screen. But if instead you place a detector at the double-slit screen, you will indeed observe the electron there as a particle. If such a setup allows you to detect which slit the electron goes through, then you no longer get the typical pattern of interference fringes on the detector screen.