Revisiting the electron double-slit experiment

We wrote about the significance of the 2012 University of Nebraska-Lincoln double-slit experiment with electrons before—as part of our Reading Feynman blog, to be precise. However, we did not have much of an understanding of matter-waves then. Hence, we talked about the de Broglie wavelength (λL = h/p) and tried to relate it to the interference pattern without any idea of what the concept of the de Broglie wavelength actually means. We, therefore, feel it is appropriate to revisit this subject as one of our very first entries for this new blog, which wants to probe a bit deeper.

Let us recall the basics of the model. We think of an electron as a pointlike charge in perpetual light-like motion (Schrödinger’s Zitterbewegung). The anomaly in the magnetic moment tells us the charge is pointlike but not dimensionless. Indeed, Schwinger’s α/2π factor for the anomaly is consistent with the idea of the classical electron radius being the radius of the pointlike charge, while the radius of its oscillation is equal to the Compton scattering radius of the electron. The two radii are related through the fine-structure constant (α ≈ 0.0073):

re = α·rC = αħ/mc ≈ 0.0073·0.386 pm (10−12 m)  ≈ 2.818 fm (10−15 m)

 It is good to get some sense of the scales here—and of the scale of the slits that were used in the mentioned experiment (shown below).

electron-double-slit-set-up

The insert in the upper-left corner shows the two slits: they are each 50 nanometer wide (50×10–9 m) and 4 micrometer tall (4×10–6 m). The thing in the middle of the slits is just a little support. Please do take a few seconds to contemplate the technology behind this feat: 50 nm is 50 millionths of a millimeter. Try to imagine dividing one millimeter in ten, and then one of these tenths in ten again, and again, and once again, again, and again. You just can’t imagine that, because our mind is used to addition/subtraction and, to some extent, with multiplication/division: our mind is not used to imaging numbers like 10–6 m or 10–15 m. Our mind is not used to imagine (negative) exponentiation because it is not an everyday phenomenon.

The second inset (in the upper-right corner) shows the mask that can be moved to close one or both slits partially, or to close them completely. It gives the interference patterns below (all illustrations here are taken from the original article—we hope the authors do not mind us popularizing their achievements). The inset (upper-left corner) shows the position of the mask vis-á-vis the slits. The electrons are fired one-by-one and, of course, few get through when the slits are closed or partly closed.

Interference 1

The one-by-one firing of the electrons is, without any doubt, the most remarkable thing about the whole experiment. Why do we say that? Because electron interference had already been demonstrated in 1927 (the Davisson-Germer experiment), just a few years after Louis de Broglie had advanced his hypothesis on the matter-wave. However, till this 2012 experiment, it had never been performed in exactly the same way as Feynman describes it in his 1963 Lectures on Quantum Mechanics. The illustration below shows how the interference pattern is being built up as the electrons go through the slit(s), one-by-one.

electron-blobs

The challenge for us is to explain this interference pattern in terms of our electron model, which may be summarized in the illustration below, which we borrow from G. Vassallo and A. Di Tommaso (2019). It shows how the Compton radius of an electron must decrease as it gains linear momentum. Needless to say, the plane of oscillation of the pointlike charge is not necessarily perpendicular to the direction of motion. In fact, it is most likely not perpendicular to the line of motion, which explains why we write the de Broglie relation as a vector equation: λL = h/p. Such vector notation implies h and p can have different directions: h may not even have any fixed direction! It might wobble around in some regular or irregular motion itself!

Celani and Vassallo

The illustration shows that the Compton wavelength (the circumference of the circular motion becomes a linear wavelength as the classical velocity of the electron goes to c. It is now easy to derive the following formula for the de Broglie wavelength:de Broglie wavelengthThe graph below shows how the 1/γβ factor behaves: it is the green curve, which comes down from infinity (∞) to zero (0) as goes from 0 to c (or, what amounts to the same, if β goes from 0 to 1). Illogical? We do not think so: the classical momentum p in the λL = h/p is equal to zero when v = 0, so we have a division by zero. Also note the de Broglie wavelength approaches the Compton wavelength of the electron when v approaches c, and that 1/γβ factor quickly reaches reasonable values: for β = 0.2, for example, 1/γβ is equal to 5, more or less. For higher velocities, the de Broglie wavelength is just three or two times the Compton wavelength—or less. Of course, a = 0.2c velocity is substantial but not uncommon in such experiments.

de Broglie wavelength

These are remarkable relations, based on which it should be possible to derive what we refer to as the equivalent of the Huygens-Fresnel equations for electron interference.

Indeed, as far as we know, that has not been done yet. We are not quite sure if it can be done: an analysis of the interactions between the incoming electron and the electrons in the material of the slits must be hugely complicated, and we need to answer several difficult questions—first and foremost this: how does the pointlike charge – as opposed to the electromagnetic oscillation which keeps the charge in its orbit – go through the slit(s)? It must do so as a single blob—as opposed to the electromagnetic fields, which may or may not split up so as to produce the interference pattern.

What? May or may not split up? They should split up, right? Maybe. Maybe not. We are not so sure. We are not so sure. If we refer to interference in the context of two slits and diffraction when only one slit is open, then it is pretty obvious that the interference pattern that is produced when the two slits are open looks very much like the superposition of the two diffraction patterns that are produced by the electrons coming out of the individual slits. So, no, we do not buy the standard story here. Sorry.

So… What to say? We’ve got good ideas here—a good explanation but, in physics, the question is not (only) how but: how, exactly? The Zitterbewegung interpretation of an electron explains how diffraction and interference of an electron (with itself and/or with other electrons) might work but Zitterbewegung theorists still have some work to do to explain the how exactly. We think it can be done, however, and we therefore hope this post may inspire some smart students! The math is probably quite daunting, but then it is a rather nice PhD topic, isn’t it? And a decent quantitative explanation (as opposed to our qualitative explanation here) would sure make waves! 🙂

Post scriptum: We should, perhaps, also add a few remarks on some of the likely technicalities for the calculations. The shape of the wave combines the characteristics of transverse and longitudinal waves. It may, therefore, be very difficult to model this. The combination of linear and circular motion probably also involves some combination of plane, cylindrical and spherical wave geometry. In our paper on the geometric interpretation of de Broglie wavelength, we actually distinguished three different wavelength concepts which can be related through Menaechmuslatus rectum formula. To this, one should then add the intricacies of diffraction and interference.

Fortunately, there is a lot more quantitative analysis material now: this is a link to a good 2019 article which, in turn, has a good bibliography with links to many other good articles. I find the research by Frabboni, Gazzadi, Grillo and Pozzi particularly interesting. The point is: you are probably not going to produce a decent classical mathematical model of what’s actually going on overnight! 🙂 But it should be possible: the fact that this 1/γβ factor quickly reaches reasonable values, is very encouraging!

At the same time, one has to carefully relate scales and electron energies. The kinetic energy of the electrons in the Nebraska-Lincoln experiment was 600 eV only, so the electrons were quite slow (to accelerate electrons to a velocity of 0.2c, you need to apply something like 11,000 eV). Again, the analysis is not going to be easy, but if you want to be a physicist, you should surely try your hand at it! 🙂

Oh—one more thing: you will say this blog post is all about QED—as opposed to the stated objective of this blog. Well… You are right, of course, but then my thought processes are not exactly linear. 🙂

9 thoughts on “Revisiting the electron double-slit experiment”

  1. […] this and my other blog (I have two so as to separate simple from more complicated stuff). It was on the double-slit experiment with electrons—one of these experiments which is supposed to prove that only an analysis in terms of probability […]

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  2. Of course, a slow-moving electron (and its deform-able charge sphere) is deflected by the electron cloud of the atoms that make up the slit. An electron’s coulomb charge field is larger then 50nm slit (The adjacent slit may also alter the dynamic because the charge sphere drops at 1/r^2 from the point charge but is not zero at slit #2). The slit deflects the electrons (unlike the c speed photons obeying Huygens’s Principle) –especially as the random electrons veer very close to this boundary.
    The double-slit establishment of wave-particle duality is possibly the biggest fallacy in modern science. It may have established an incorrect philosophical proposition at the foundation of quantum physics. The patterns look the same but the electron’s explanation was never thoroughly tested.
    I have always been suspect of the wave-particle duality principle taught in modern physics. This truly unexplained phenomenon bestows a “religious dimension” to physics. Debunking the religious aspect of quantum theory would allow us to bury Schrodinger’s cat. Einstein has always been correct. The Copenhagen interpretation has always been wrong. God does not play dice. It may be time to “reset” quantum theory

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  3. Your 3 1/2 hr YouTube video expands on “quantum confusion”. I think Einstein’s Nobel prize winning explanation of the photo-electric effect was the last rational idea in particle physics. From there, a 100 year odyssey of irrationality ensued. I don’t believe Feynman’s argument of electrons interacting with each other by passing a photon between them to communicate force. The most fundamental understanding of physics comes from the rational explanation of the electron. The electron is a mini-point-mass with a quantifiable electric field (this lepton is fundamental in “nature”). That electric field is like a sail–picking up and releasing photon energy and interacting with other electron force fields. It acts like a wave because 1) confined to an atom its potential/kinetic energy is defined by orbits (also fundamental to nature) in the “standing wave understanding” (the electron is like a particle on the rope’s path in the classical physical analog. The particle is on the rope–it is not “the” rope as wave theory prescribes) and 2) it interacts with light (photons) which are pure waves suitable to wave equation descriptions. The electron is fast (both in the atom and in a wire propelled by emf) but it never reaches light speed. It has a mass. No wave function can perfectly describe it. Schrodinger’s wave equation is not solvable. Perturbation theory methodology must be used to arrive at a suitably accurate answer. Perturbation theory, statistical analysis and computational methods are not “math”–which is perfect, by definition.
    I do not challenge any physical understanding gained from current high-energy physics experiments. Modern physics is fascinating. But the GUT sought through quantum theory is a dream. This is why quantum theory and relativity are incompatible. You can’t get the answers from one through the methodology of the other. But the differences are clear: relativity is pure math (which breaks down physically in singularities), quantum theory is an attempted explanation of dirty particle physics. We can’t get to God through the dirt.

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    1. I would not quite agree. Einstein did not re-invent everything and is surely not the last quantum physicist to make major contributions. Sommerfeld, Schroedinger, etc. did model real stuff. I agree on your remarks on perturbation theory – which led to quantum field theory (QFT). ‘New’ quantum physics is based on approaches inspired by QFT – which I don’t think are going anywhere. However, ‘old’ quantum physics looks fine to me. PS: I do not think the objective of physics is to get to God. 🙂

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  4. Can you derive the electron speed (as percentage of c) and distance from the nucleus for a hydrogen atom in its “base unexcited state”? Can you use the electron rest mass of .511 MeV and the electron escape energy of 13.6 eV (where, at escape, the electron speed is presumably zero) for a classic kinetic energy calculation in base orbit?
    I have done my calculations but I am only an enthusiast. You are a serious player in theoretical physics. I want to know how “relativistic” that electron really is. The farther it is from lightspeed the less the Lorentz transformation is relevant. The farther it is from lightspeed the lesser the connection to wave modeling in the original atomic conception. Is there another way to model hydrogen (with its spectral lines defined) than the “standing wave” theory (which beautiful but possibly flawed)? What is the “nature” of the electron and those orbits? Are those orbits “musically” related instead of “wave quantized”?

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    1. Hi David – Thank you for your interest. I would not say I am a serious player in physics but, yes, I can do those easy calculations. See chapters IV to VII in my manuscript (https://vixra.org/pdf/1901.0105vG.pdf). I am currently trying to revive the publishing contract I had with IOP and WSP to update this (lots of stuff that I would do away with – especially the excruciatingly long introductory chapters). You’ll find simple force and energy calculations based on the equations of motion there. As for music and math, it is funny you mention that: one of my very early posts (2015) is on that – and that one is still as simple and as correct it can possibly be. Enjoy: https://readingfeynman.org/2015/08/10/music-and-math/ Cheers – JL

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  5. My “off the cuff” calculation for hydrogen base state electron speed was 2.7%c. Your more accurate statement of 0.0073c is of the same order of magnitude– clearly establishing the “nuclear-bound electron” as “Newtonian”. At that speed the electron has an ideal gas temperature we might baseline to zero Kelvin (I like it when we can baseline a theory to physical absolutes like zero Kelvin or c)
    Of all the characteristics I have self-listed for the mysterious electron Zitter Theory was not one of them. It is intriguing. The idea of an oscillating electric charge field (with its associated magnetic moment) moving at tangential light-speed gives the electron an “absoluteness”. The problem with this idea is that it looks suspiciously like a photon–but it is a “bound photon”–bound to a speck of matter. And the speck of matter cannot do light-speed. A neutrino is also a speck of matter–it is fast, but not light-speed fast. The electron “mass speck” appears similar to the “bound quarks” in a nucleon. Ups and Downs are 1-2 MeV. The rest of the 938 MeV nucleon is “bound energy” in the “gluon soup”. Gluons are not understood to be photons but they must be “light fast” to have zero mass. The bound gluon energy makes up most of the nucleon mass just like the “speck-bound (possibly oscillating) electromagnetic charge energy” is most of the electron mass.
    The electron is stable and can exist in free space–kicked around by photons like a soccer ball. I am not sure what happens when the experimentalists collide electrons with positrons to make gamma rays. The charge fields neutralize–what happens to the matter specks?
    Back to quantum. Rydberg did his calculations based on spectral lines long before the “fathers of quantum” laid down their theory. Rydberg’s methodology to “integerize” those spectral lines was mathematical wizardry. It looks like “quantum” followed his lead. Standing wave theory was the closest analogy to his “integerization”. The wave equations worked and physics “went along for the ride”.
    But what is going on in that hydrogen atom? The bound “possibly oscillating” electron charge cloud intercepts a photon (we must assume it approached from behind the “Newtonian speed” electron) and bumps its speed like a rocket. A small speed boost jumps it to orbital 2 where it immediately slows down in a higher coulombic potential (with its associated proton). It can do this a few more times and then it is off into free space. The thing that is interesting to me is that a hydrogen atom has 2 masses swinging around each other. The proton oscillates (vibrates) about the “system mass center” at the electron revolution rate. The “charge center” of the system rotates around midway between the nucleus and the electron orbit at electron angular velocity. I bring this up because helium would act differently. The helium nucleus has “barbel shaped charge pattern” which probably restricts the electron path to its “waistline”. Both electrons probably travel the same orbit 180 degrees apart. The system mass and the charge center are very near the nuclear center. Hydrogen is a reactive machine and helium is dead.
    Back to the electron. For Zitter to work the electron mass speck must be “inside” the oscillating “charge envelope”. The charge envelope might be doing tangential light-speed but the associated mass speck must not. If the “electron charge” is a Compton distance from the electron orbital path the oscillating speck might be half the distance–it would be doing half-light-speed. That would require a physical displacement of the speck and the charge. In free space the electron would have to “self-vibrate” in a symmetrical fashion with stable mass and charge centers.
    Electrons like to be bound. Obviously, “energy wells” favor this condition. In a wire, electrons move one way and “holes” move the other but when the emf stops they find their place. In nature, too many isolated electrons and their associated holes create arcing and lightning. Electrons do not like to “self-vibrate”. Shooting free electrons through a slit is “asking” for an experimental result not reflecting “atomic nature”. I will conclude that the electron slit experiment (compared to the light diffraction) is the greatest coincidence in physics.
    It’s time for you to pick up where Einstein left off. You can be like Dr Thomas Seyfried (who resurrected Dr Otto Warburg’s mitochondrial fermentation cancer hypothesis in his 2012 “Cancer as a Metabolic Disease” book) bringing disease analysis back to its roots after 80 years of butchering, radiating and toxic “chemotherapy-ing” people in the “wild goose chase” of the Somatic Mutation Theory. I probably saved my father from Stage 4 adenocarcinoma with $200 of phytonutrients and a ketogenic diet. I think you can save physics for an equal amount of money.

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    1. Hi David – Nice comment(s). Thank you. I earned a few hundred dollar with putting an ebook on amazon.com (but I took it off because it was not offering anything else beyond what is freely available on this site and the publications on Phil Gibb’s site) so, yes, that pays for the website and so I consider it all break-even (not counting my time, of course). I am sort of ‘nodding’ at most of what you wrote but it would take me too long to add/discuss so I won’t do that. Just two things: (1) I initially also thought the zitter charge inside of an electron should have a ‘speck’ of mass, but not any longer: my electron model is now a pure ‘mass without mass’ model (cf. Wheeler’s approach); (2) I do not see any reason to assume a photon should have some tiny rest mass (another ‘speck’ of mass you mention), and a neutrino is for me the ‘photon of the strong force’ (see https://vixra.org/abs/2005.0190 if you are not on ResearchGate). I should look at more recent news on the neutrino having some tiny non-zero rest mass but it does not keep me awake. I sort of switched of from physics but – whatever time I have left – goes to studying low-energy nuclear reactions (LENR aka ‘cold fusion’). […] Any, it looks like you are having fun with physics and that – as always – the most important with such intellectual pursuits. Cheers ! JL

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