Dual slit with controlled electron trajectories

[Buckethead]

TL;DR Summary

dual slit outcome if pseudorandomly generated positions of electrons are used

In the traditional single electron duel slit experiment, I assume a cathode emits electrons in an unfocused direction spreading across the dual slits like a flashlight beam, but one electron at a time. Electrons however can be finely focused and controlled using magnetic or electric fields (think cathode ray tubes). Imagine a single electron gun with an aiming field controlled by a pseudo-random number generator designed such that the numbers allow for an even wash of the electron stream as in the opening statement.

When run, and if the experiment is unobserved except for the effect on the destination screen (no detectors at the slits), will there be interference? What if the pseudo-random generator (which can determine the trajectory of the electron after the fact) were replaced by a more random source such as a natural noise source, the noise of which was not recorded? Thanks for reading.

 

[anuttarasammyak]

Say electrons go z direction with distance L and hit slits standing y direction with interval of d, electrons have uncertainty of y -momentum
$$\triangle p_y = \frac{\hbar}{d}$$`
error of ejection angle can be
$$\frac{\triangle p_y}{p_z}=\frac{\hbar}{dp_z} << d/L$$
by preparing large enough $p_z$, high speed electrons wrt size of slits. In this case electrons behave like bullets so if you fire electron machine gun to one of two slits randomly there are two separate damaged marks on the screen after two slits.

 

[Buckethead]

Sadly I'm math illiterate, but OK it should be 2 slits on the screen and not an interference pattern. How does this setup (flooding the slits evenly using a random number generator) and a standard single electron unguided stream setup that does interfere differ? In both cases the slits get flooded randomly.

 

[anuttarasammyak]

Your setting seems to be as Fig.1.1 described in
https://www.feynmanlectures.caltech.edu/III_01.html
Try reading it and you will get clearer idea on your problem.

 

[Buckethead]

Thanks for the link. And yes the setup is similar, with the exception that I take the electron gun experiment one step further. Instead of the direction of the electrons being left to chance I am directing the x,y position of the electron using a field. The field is controlled (in one setup) by a natural noise source such as thermal noise meaning the electron will "wash" over the two slits the same way as in the original electron gun experiment. It seems to me in both cases the electrons arrive at the slits in an entirely random way and as such in both setups an interference pattern should show on the screen.

If that is the case, then what happens if the voltage on the noise source is simultaneously measured so that at a later time, the exact trajectory of the electrons can be calculated?

It may be that this experiment is impossible as the accuracy of the focusing field may not be sufficient, and a quantum "wobble" on the electron may be greater than the distance between slits making it impossible to tell which slit the electron goes through even if the predicted x,y coordinates are recorded.

 

[PeterDonis]

I assume a cathode emits electrons in an unfocused direction spreading across the dual slits like a flashlight beam, but one electron at a time.

This is more or less correct.

When run, and if the experiment is unobserved except for the effect on the destination screen (no detectors at the slits), will there be interference?

No. The source that is doing the focusing "knows" that it is doing the focusing (i.e., produces a focused electron wave function) even if nothing is recorded that humans can detect afterwards. See below.

What if the pseudo-random generator (which can determine the trajectory of the electron after the fact) were replaced by a more random source such as a natural noise source, the noise of which was not recorded?

Doesn't matter. If the source is focusing, it's focusing.

It seems to me in both cases the electrons arrive at the slits in an entirely random way

That's not the key point. The key point is that you have focused the beam so that it is only wide enough to cover one slit, instead of being wide enough to cover both slits. That means each electron's wave function only has a non-negligible value in the vicinity of one slit, so you get one-slit behavior at the detector. Even if you don't know which slit each electron's wave function is focused on, even if nothing is recorded that let's you know that afterwards, the source is still producing an electron wave function that is only wide enough to cover one slit, and that's the key fact.

 

[Buckethead]

No. The source that is doing the focusing "knows" that it is doing the focusing (i.e., produces a focused electron wave function) even if nothing is recorded that humans can detect afterwards. See below.

It seems from this that the electron when emitted is already a probability wave and that by focusing it, you are no longer messing with an electron but you are instead messing with the probability distribution of the wave function. At first look it just seems like you would be dealing with an electron that is either random or..um..random which would give the same result. So it seems we are not dealing with electrons even before they are emitted. The emission itself must be just a probability wave.

 

[PeterDonis]

by focusing it, you are no longer messing with an electron but you are instead messing with the probability distribution of the wave function.

The electron is the wave function. More precisely, the wave function is the best description we have in QM of the electron, and as far as the math of QM is concerned, there is no "electron" other than the wave function. Any manipulation of an electron, or any other quantum object, during an experiment is a manipulation of the wave function. What else would it be?

 

[vanhees71]

Schrödinger's idea that the electron "is" the wave function leads to serious contradictions with experience, as particularly this famous double-slit example shows and to all empirical facts we have today.

Originally Schrödinger thought his wave function is just a classical field describing the electron as a dynamical field entity. Now, when you do the double-slit experiment with a single electron, you don't see a smeared distribution, including interference fringes, on the screen but a single spot.

That's why Born introduced the probability interpretation of the wave function and shortly thereafter the modern probabilistic interpretation of the general quantum state. Though there are a plethora of attempts to somehow abandon this irreducible randomness from our description of nature so far non of these ideas work, and quantum theory including its probabilistic interpretation passes all tests.

 

[Nugatory]

It seems to me in both cases the electrons arrive at the slits in an entirely random way and as such in both setups an interference pattern should show on the screen.

The interference patten is the result of each electron interfering with itself, not successive electrons interfering with one another. For any individual electron we calculate the probability of it landing at any given point on the screen by adding contributions from all physically possible paths between the source and that point.

When the electron source is uniformly illuminating both slits then paths through both slits are possible and contribute to the probability. We do the calculation and we find that that for some areas of the screen the paths through different slits add while in others they cancel; there are bands where the electron is likely to land alternating with bands where it is unlikely to land.

When the electron source sometimes illuminates one slit and sometimes the other, then for any given electron there are only paths through one slit. That's a different set of paths, so when we add them up of course we get a different result: in this case a spot of high probability behind whichever slit this particular electron was aimed at.

 

[vanhees71]

Maybe it helps to add that you just solve the wave equation (Helmholtz equation), and in 3D the solution of this kind of problems is given by Huygens's principle, as in (or even simpler, because it's only a scalar field) electrodynamics. It's clear that if you only partially "illuminate" the slits, you'll not get the same interference pattern as when you use something close to a plane wave illuminating both slits as a source. It also depends, where you observe the particles behind the slit. If you put your detector too close to the slits you don't see interference patterns, because the spherical wavelets behind the slits must sufficiently overlap at the place of detection to show the interference effects.

The math is the same as for any wave equation, i.e., as for classical wave fields, only the interpretation is different: The Schrödinger wave function is not a classical field but it's modulus squared is the probability ditribution for the particle's position.

 

[Buckethead]

Thanks everyone for contributing to this. It seems, in general, I always hear much more about the wave function during propagation and the resulting pattern on the screen, but nothing about the nature of the electron source and this discussion is shedding light on that.

That's not the key point. The key point is that you have focused the beam so that it is only wide enough to cover one slit, instead of being wide enough to cover both slits. That means each electron's wave function only has a non-negligible value in the vicinity of one slit, so you get one-slit behavior at the detector. Even if you don't know which slit each electron's wave function is focused on, even if nothing is recorded that let's you know that afterwards, the source is still producing an electron wave function that is only wide enough to cover one slit, and that's the key fact.

One thing I can't help but wonder about is the fundamental difference between an electron whose trajectory is determined by a random number generator, and random trajectories of electrons when emitted. Clearly there is a difference since they create different wave functions. The idea that electrons are not particles at all but instead are probability waves (regardless of the interpretation of what the probability wave actually is), is a compelling idea, since I can see how a naturally generated probability wave, when focused, will change and give different results.

So it seems there is really no such thing as emitting a single electron. One does not emit electrons, but only probability waves which, when detected, result in a single point of energy that only "looks" like a single electron.

I may not completely understand this, (who does?) but I am leaning more now toward seeing all particles (photons included) as either pure probabilities with no real "reality" to them or as probability waves with some kind of makup in the real world. I'm fine with either interpretation.

 

[PeterDonis]

One thing I can't help but wonder about is the fundamental difference between an electron whose trajectory is determined by a random number generator, and random trajectories of electrons when emitted. Clearly there is a difference since they create different wave functions.

Or, if you want to avoid possible misinterpretation (which, given the rest of your post, would be a good idea), these two different preparation procedures for electrons lead to different predictions for the results of experiments.

You would be well advised to try to think of things in terms of results of possible experiments instead of trying to interpret them according to your personal beliefs about wave functions, particles, etc.

You would also be well advised to write down the actual math for each case you describe, instead of using vague ordinary language. There is a reason physicists use math instead of vague ordinary language to actually do physics.

The idea that electrons are not particles at all but instead are probability waves

...is an incorrect idea. Electrons are electrons. We describe them in our models sometimes as particles, sometimes as wave functions, and sometimes as other things (like quantum fields). Don't confuse the model with the thing it is modeling.

So it seems there is really no such thing as emitting a single electron.

Wrong. It is perfectly possible for an electron source to emit an electron in an eigenstate of particle number with eigenvalue $1$.

I may not completely understand this, (who does?)

Even if nobody completely understands this, there are a lot of people, including other posters besides you in this thread, who understand a lot more about it than you do.

I am leaning more now toward seeing all particles (photons included) as either pure probabilities with no real "reality" to them or as probability waves with some kind of makup in the real world. I'm fine with either interpretation.

Discussion of interpretations of QM belongs in the interpretations forum, not here.

There is no interpretation required to understand the actual math or the experimental predictions. And if you don't have a solid undertstanding of those, you will have a hard time understanding any of the standard QM interpretations, much less trying to make up your own, which is what you are trying to do here.

 

[Buckethead]

Or, if you want to avoid possible misinterpretation (which, given the rest of your post, would be a good idea), these two different preparation procedures for electrons lead to different predictions for the results of experiments.

You would be well advised to try to think of things in terms of results of possible experiments instead of trying to interpret them according to your personal beliefs about wave functions, particles, etc.

This is a good point and I can see why. There is no way to use common sense to figure out what the electron actually is when it acts like both a wave and a particle. If the math model works to describe what the outcome will be, then the math can be used to predict what else will happen under new and different circumstances and that advances science. However, I think it is valuable to spend time trying to imagine what this electron/wave function could be so that it could be described using words, if in fact that could be done (not saying it can).

...is an incorrect idea. Electrons are electrons. We describe them in our models sometimes as particles, sometimes as wave functions, and sometimes as other things (like quantum fields). Don't confuse the model with the thing it is modeling.

Again, a good point. The word electron is the label put on the model, not a description of an actual thing. I will keep that in mind.

Wrong. It is perfectly possible for an electron source to emit an electron in an eigenstate of particle number with eigenvalue 1.

Are you saying here the emitted particle has a definite vector? (Sorry, new word eigenstate). But isn't this determined only after detection in an experiment meaning one can't really know the nature of the particle before then?

Even if nobody completely understands this, there are a lot of people, including other posters besides you in this thread, who understand a lot more about it than you do.

Yes, of course this is true. I apologize if it seemed I was saying otherwise.

There is no interpretation required to understand the actual math or the experimental predictions. And if you don't have a solid undertstanding of those, you will have a hard time understanding any of the standard QM interpretations, much less trying to make up your own, which is what you are trying to do here.

I thought some scientists think of the probability wave as something that has a physical reality and others think of it as purely mathamatical in nature with no real physical form. At least that's how I have seen it expressed in popular literature.

 

[PeterDonis]

I think it is valuable to spend time trying to imagine what this electron/wave function could be so that it could be described using words

This is QM interpretation and belongs in the interpretation forum, not here (and you should not be trying to make up your own interpretation, you should be reading the extensive literature that already exists on QM interpretations).

Are you saying here the emitted particle has a definite vector?

A state vector which is an eigenvector of the particle number operator, yes.

(Sorry, new word eigenstate).

If you are not familiar with this term (or the other terms I used), you would be well advised to spend some time with a QM textbook learning the basics of the theory and terminology. I personally would recommend Ballentine, but personal preferences vary widely and you might get a lot of different recommendations from different people.

I thought some scientists think of the probability wave as something that has a physical reality and others think of it as purely mathamatical in nature with no real physical form.

This is a matter of interpretation and discussion of it belongs in the QM interpretations forum, not here.

 

[Buckethead]

I just finished reading with great interest the thread "Layman asks about Quantum "interaction" and it has really helped me to understand the nature of the dual slit experiment. Now that I clearly understand that there is no such thing at a transition from particle to wave and back again and so on but instead just a wave function that can change during an interaction (such as a detector at a slit) with no implication as to what is going on "physically" during the propagation of the electron, I can (hopefully) ask more intelligent questions. Armed with this greater understanding and getting back to this thread, I have a new question:

I read that in the single electron dual slit experiment, a collimator is inserted to direct the electron on a more parallel path, in other words, it deflects the electron. However, this does not seem to have an effect on the wave function as far as allowing for an interference pattern on the detection screen. Therefore, it causes no collapse of the wavefunction.

Well that got me thinking that there is a certain amount of deflection allowed without affecting the outcome. In a thought experiment I envision 2 types of black boxes. One I call a "passive collimator" and is nothing more than a standard collimator (which is mayby just a passive magnetic field), and the other black box is an "active collimator" which is more complicated. It measures the distance of an incoming electron from the center of the collimator using a detector and guides an electrostatic field to deflect the electron according to the same math that describes a "passive collimator". This "active collimator" therefore (at least on the surface) looks to an outside observer as being nothing more than a "passive collimator" The question is, will this collapse the wave function and if so why?

 

[DrChinese]

Well that got me thinking that there is a certain amount of deflection allowed without affecting the outcome. In a thought experiment I envision 2 types of black boxes. One I call a "passive collimator" and is nothing more than a standard collimator (which is mayby just a passive magnetic field), and the other black box is an "active collimator" which is more complicated. It measures the distance of an incoming electron from the center of the collimator using a detector and guides an electrostatic field to deflect the electron according to the same math that describes a "passive collimator". This "active collimator" therefore (at least on the surface) looks to an outside observer as being nothing more than a "passive collimator" The question is, will this collapse the wave function and if so why?

It wouldn't really matter how the particle gets to the double slit as long as it otherwise has suitable properties. The issue is that there is nothing to indicate which slit the particle is passing through. In such case, there will be path interference reflected in the result.

 

[anuttarasammyak]

According to https://en.wikipedia.org/wiki/Collimator, collimator is a tool to make beam of particles. I do not take the meaning of passive and active. They both seem to make same result.

Do you mean double-slit like double-collimator situation of one passive and the other active ?

 

[PeterDonis]

I read that in the single electron dual slit experiment, a collimator is inserted to direct the electron on a more parallel path, in other words, it deflects the electron.

Where did you read this? Please give a specific reference.

 

[Buckethead]

https://en.wikipedia.org/wiki/Collimator
"A collimator is a device which narrows a beam of particles or waves. To narrow can mean either to cause the directions of motion to become more aligned in a specific direction (i.e., make collimated light or parallel rays), or to cause the spatial cross section of the beam to become smaller (beam limiting device). "

 

[PeterDonis]

https://en.wikipedia.org/wiki/Collimator
"A collimator is a device which narrows a beam of particles or waves. To narrow can mean either to cause the directions of motion to become more aligned in a specific direction (i.e., make collimated light or parallel rays), or to cause the spatial cross section of the beam to become smaller (beam limiting device). "

Where does this article say a collimator is used on the electron beam in a double slit experiment?

 

[Buckethead]

https://iopscience.iop.org/article/10.1088/1367-2630/15/3/033018

From section 2 "Experimental setup"

"The experimental setup is shown diagrammatically in figure 1(a). An electron beam with energy of 600 eV, which corresponds to a de Broglie wavelength of 50 pm, was generated with a thermionic tungsten filament and several electrostatic lenses. The beam was collimated with a slit of 2 μm width and 10 μm height placed at 16.5 cm. The double-slit was located 30.5 cm from the collimation slit. The resulting patterns were magnified by an electrostatic quadrupole lens and imaged on a two-dimensional microchannel plate and phosphorus screen, then recorded with a charge-coupled device camera. For a more detailed description of the setup see supplementary information (available from stacks.iop.org/NJP/15/033018/mmedia)."

 

[PeterDonis]

The beam was collimated with a slit of 2 μm width and 10 μm height placed at 16.5 cm.

This doesn't seem like either of the collimators you were describing in post #16. This "collimator" is just a slit that blocks any electrons that aren't moving in the right direction to within a good enough approximation. The "collimator" is actually part of the "electron gun" in the diagram in Fig. 1(a) of the paper. It's not "deflecting" any electrons; it's just filtering out the ones that aren't moving in the right direction.

As for "collapse", all of the electrons that don't get through the collimating slit are "collapsed"; they hit somewhere else inside the "electron gun" apparatus and are absorbed. The electrons that get into the actual experiment are just the ones this didn't happen to.

 

[Buckethead]

This doesn't seem like either of the collimators you were describing in post #16.

When I copied and pasted that quote, I realized my over site. When I originally saw the word collimator, I just looked it up instead of actually reading what they were using. I'm surprised they called it a collimator. I would have called it an aperture. On the other hand the "electrostatic quadrupole lens" grabbed my attention since that device is before the detector and therefore potentially can affect the wave function although in this case not in the way that would cause the detector to not see an interference pattern.

It seems feasible that we could put in a collimeter of the type I describe, one that would simply deflect the electrons into a more parallel path for example, and then I wonder what would happen.

It wouldn't really matter how the particle gets to the double slit as long as it otherwise has suitable properties. The issue is that there is nothing to indicate which slit the particle is passing through. In such case, there will be path interference reflected in the result.

So it seems that in this case the collimeter of the type I describe (the "passive" collimeter) would not have an effect.

 

[PeterDonis]

the "electrostatic quadrupole lens" grabbed my attention since that device is before the detector

No, it isn't; it's part of what is labeled as the "detector". The pattern formed on the detector screen without the lens would be too faint to be recorded; the lens magnifies it so it can be recorded. The lens is magnifying light (the light emitted by the detector screen when electrons hit it); it's not doing anything to the electrons.

 

[PeterDonis]

It seems feasible that we could put in a collimeter of the type I describe, one that would simply deflect the electrons into a more parallel path for example

Can you find any reference to an experiment where such a thing has been used?

 

[vanhees71]

Isn't in a sense an electron microscope just that, i.e., a set of electron-optical elements (i.e., em. field configurations) constructed such as to get the "optics" you want?

One of the nicest real-world versions of a double-slit-equivalent experiments is the one with single electrons described here, using an electron-optical biprism:

A. Tonomura et al, Demonstration of single-electron buildup of an interference pattern
American Journal of Physics 57, 117-120 (1989)
https://doi.org/10.1119/1.16104

A literal one-, double-, etc. slit experiment with electrons was done by Jönsson in the 1960ies. There's an English translation of the corresponding paper also in AJP. Also this experiment of course uses some electron optics to get nice interference patterns:

C. Jönsson, Electron Diffraction at Multiple Slits
American Journal of Physics 42, 4 (1974)
https://doi.org/10.1119/1.1987592

Another nice article about electron diffraction/double-slit experiment is in Physics World:

https://physicsworld.com/a/the-double-slit-experiment/