Feynman paths and double slit experiment

[hammock]

I am just inhaling "The Grand Design" and am stuck in the chapter on the "buckyballs" double slit experiment.

The authors say that in case of the experiment, a particle may take any possible way ("perhaps to Jupiter and back"), which then Feynman depicts as adding vectors to a result vector (as I understand).

However, I wonder how this can be real, as the buckyball (or photon) has a definite speed s (or c) on the result vector path. But in case the particle takes the path to "Jupiter and back" the length of the path it has taken cannot fit the speed of the particle on the result vector, resulting in the (presumably false) supposition, that it had a speed greater than s (or c).

 

[Drakkith]

I believe the book I read about QM stated that almost all of these different paths cancel each other out in probabilities so that something like that doesn't happen.

 

[hammock]

So, the particles that travel to Jupiter interfere all together with themselves but they don't travel at a speed greater than s (or c). Right?

[Edit] After reflecting that I assume that any path which would result in a speed greater than s (or c) might be eliminated, which then leads to the conclusion that only the direct paths remain [?] But in this case, there wouldn't be an interference, would it?

 

[Jacques_L]

So, the particles that travel to Jupiter interfere all together with themselves but they don't travel at a speed greater than s (or c). Right?

You are self-contradictory, here. Concurrent paths only apply to ONE quanton at the same time. Simply the laws of physical optics continue to apply to fermions, too.

 

[Drakkith]

No I mean that the possible paths interfere with each other similar to the interference pattern on the double slit. Where they destructively interfere the particle(s) have a much less chance to take that path, if they have one at all.

 

[hammock]

I am not sure if I fully understood your answer.

Does that mean, that there may be some particles traveling to "Jupiter and back" according to the interference pattern?

 

[Jacques_L]

I am not sure if I fully understood your answer.

Does that mean, that there may be some particles traveling to "Jupiter and back" according to the interference pattern?

You are just dreaming with dreams, not more.
The theorist may compute such an aberrant path, but the conclusion will not yield another result than null. So why to exhibit magic sentences when the real result remain null ?

When a photon is emitted by an emitter, its history begins, as seen from our laboratory.
When a photon is absorbed by an absorber, its history ends, as seen from our laboratory.
Meanwhile, it is tight by the laws of physical optics (the Maxwell equations), as long as lasts the synchroneous transfer from emitter to absorber. This leaves not so much room for theoritical and magic fantasies.

More strange is the noise before any transaction succeed, but alas beyond reach of most experimentations.
However the hope is not null : someones claim that some radioactive decays depend on external conditions. Stay tuned for eventual confirm.

 

[hammock]

Hi Jaques,

well, I don't have any problems to wonder how interference may appear by applying the law of optics. From the book I understand that there ist some kind of "model dependend realism" which should let anyone choose the best fitting model in order to explain realism.

BUT in the current chapter basically the different histories of a particle are taken as basic argument in order to explain that the universe has infinite histories (and perhaps infinite futures). And the authors say that understanding that is very important for understanding the chapters afterwards.

Otherwise the book (from Hawking and Mlodinow) would be science fiction?

 

[ZapperZ]

From my quick reading of this thread, this has more to do with trying to understand Feynman's path integral. Unless someone has some extraordinary capability with written communication, trying to illustrate this principle is almost impossible on a public forum such as this.

So maybe a readable source that introduce what a Feynman path integral is might be useful. Try this one:

http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=CPHYE2000012000002000190000001&idtype=cvips

Zz.

 

[Jacques_L]

This famous work of Feynman was reinventing the wheel, but less practical, with heaps of unuseful mathematical fatigue.
Why ? Just because of american arrogance : what is not published in english is thought not being, for american physicists.
So Feynman simply ignored the periodic character of any quanton that has a mass, its two intrinsic frequencies :
Broglie frequency for all of them : $\frac{m.c^2}{h}$, proved in 1924 (published in french).
Dirac-Schrödinger electromagnetic frequency for fermions, such as the electron : $\frac{2.m.c^2}{h}$, proved in 1930 (published in german).

This fact (the Broglie intrinsic frequency) drastically reduces the alternative paths to mathematically explore, as they very very quickly become unphysical : the interferences become destructive.EDIT.
Err, there is a frequency in the text cited above, but is not the good one, it is much lesser, not relativistic, not intrinsic :

This fundamental and underived postulate tells us that the frequency f with which the electron stopwatch rotates as it explores each path is given by the expression :

$f=\frac{KE-PE}{h}$.

So with this inappropriate tool, Feynman explores much much broader paths than necessary, much much broader than the physical real paths.

 

[Drakkith]

Sounds like you just have a problem with Feynman and his work, not that it is incorrect or not.

 

[Jacques_L]

Hi Jaques,

well, I don't have any problems to wonder how interference may appear by applying the law of optics. From the book I understand that there ist some kind of "model dependend realism" which should let anyone choose the best fitting model in order to explain realism.

BUT in the current chapter basically the different histories of a particle are taken as basic argument in order to explain that the universe has infinite histories (and perhaps infinite futures). And the authors say that understanding that is very important for understanding the chapters afterwards.

Otherwise the book (from Hawking and Mlodinow) would be science fiction?

In some meaning, yes, it seems pure fiction (I have not seen it yet), as most of the books designed by an editor to the broad public. So there is a terrible temptation to play circus tricks.

For instance I have caught this dismaying book, written by a Nobel laureate (dead since) and a theorist : http://citoyens.deontolog.org/index.php/topic,887.0.html
Alarmingly stupid. And they pretended to have read the Integral of paths, of Feynman...

As long as I trust your description of what you have read, it seems to me fairy tales, without solid physical grounding. We know so little on "Universe", it is so presomptuous to tell so much on its "future" or "futures". But it is so good for the selling figures...

They found their tales on an idea of "particle", terribly flawed. It is usually taught and believed that "particles" are some very tiny, even punctual "corpuscles", but with goblinish and poltergeist behaviour. It yields terrible contradictions, of course...

So Feynmans and followers carry their ficticious particles up to Jupiter and back, without any physical grounding. This is absurdly streched out of an already absurd confusion : Feynman made no distinction between to distinct steps :
1. The quest for handshakes,
2. The transfer of a photon, an electron, a particle.

The path of the real transfer is necessarily narrow - for human distances - in comparison to its length. Of course it depends of the wavelength. The shorter the wavelength, the narrower the path. Though on astronomical distances, the mid-journey width of a photon becomes astronomical too.

But before the successful hanshake, both the demand from the potential absorbers and the offer from the potential emitters are broad. It is a broad palpation of the surroundings, both orthochrone (emitters mainly) and retrochrone (absorbers mainly).

The fairy tales of your book mix the two steps. This mixing may be fascinating for the lay men, but remains completely un-physical. As it has no real meaning, it is a good trick for hypnosis...

 

[nnnm4]

Feynman set out to formulate quantum theory in terms of the particles and not the fields. (Read his Nobel Lecture for a wonderful personal history). He was at a party and another physicist asked him what he was working on. He responded by asking if the physicists who had asked had ever seen a method of incorporating the lagrangian into a quantum theory and he suggested he look up the paper of Dirac where it is suggested. Feynman path integrals are useful for calculating paths of quantum systems through configuration space -- not necessarily for the motion of a single particle. With a single particle there are much more efficient methods, but for many-body systems or fields path integrals are often more useful. Also, substituting formally time in the integral for imaginary time gives powerful methods of calculating partition functions in statistical mechanics.

It's funny that you come up with American arrogance in relation to the path integral. Feynman recalls a European being hung up with the fact that "Americans always ask how something is useful."

 

[Drakkith]

Jacques, your posts do nothing but show your ignorance and misunderstandings on what Feynman was doing and why. And about QM itself. And remember that you have 40+ years of history that Feynman didn't have when he was coming up with theories and such.

 

[Born2bwire]

I am just inhaling "The Grand Design" and am stuck in the chapter on the "buckyballs" double slit experiment.

The authors say that in case of the experiment, a particle may take any possible way ("perhaps to Jupiter and back"), which then Feynman depicts as adding vectors to a result vector (as I understand).

However, I wonder how this can be real, as the buckyball (or photon) has a definite speed s (or c) on the result vector path. But in case the particle takes the path to "Jupiter and back" the length of the path it has taken cannot fit the speed of the particle on the result vector, resulting in the (presumably false) supposition, that it had a speed greater than s (or c).

Ok... going back to the OP...

Feynman's path integrals are not stating that the particle is physically taking the path out to Jupiter and back. It is probably best to think of these paths as an abstraction from the mathematics. However, you can glean physical insights into the paths by how they behave over a "bandwidth" of paths. It would help if you took a quick look at steepest descent and stationary phase. Basically the path integral is an exponential integral. Each "path" has the same magnitude but what differs is the phase. If the phase is highly oscillatory between adjacent paths, then what happens is that the oscillations cancel each other out when you sum them together over the integral.

Take for example the integral of a sineusoidal function:

$$\int_0^\phi \cos \theta d\theta$$

What happens when we take \phi to be any multiple of 2\pi? The result is zero due to the cancellation regardless of how many periods we take. This is what happens in these unlikely "paths." But how does this relate in any physical sense? Mainly by the fact that the classical trajectory is the limit of the steepest descent path. That is, the trajectory or trajectories where the phase varies the slowest (or not at all). That is because where the oscillations are very slow are the ones that contribute the most to the overall result. And as we take the limit of the path integrals to the classical limit, the oscillations become faster and faster until the only paths that contribute are the steepest descent paths.

Still, keep in mind what the path integral is saying. It is only stating the wave amplitude for a particle going from position A and time t_a to position B at time t_b. It isn't stating how it from A to B, just the probability that it will do so. But it does fall out that if we take the limit of the path integral to the classical limit that the integrating paths take on a more physical picture. But this is to be expected of course, you can say the same thing about taking the limit of the Schroedinger equation to the classical limit. On the one hand, path integrals will replicate the Lagrangian picture while Schroedinger will replicate the Hamiltonian picture.

EDIT: Take a look at the link given by ZapperZ. You may also find Feynamn and Hibb's text to be insightful as well (and it recently got a long overdue reprinting, in cheap softcover too!). Regarding the talk about how oscillations and stuff cancel, take a look at steepest descent, stationary phase, Laplace's method stuff.

 

[Jacques_L]

Jacques, your posts do nothing but show your ignorance and misunderstandings on what Feynman was doing and why. And about QM itself. And remember that you have 40+ years of history that Feynman didn't have when he was coming up with theories and such.

Maybe you should have to consider that "science" and "obedience" are not exactly synonyms, and that "ignorant" and "not believer" (or "skeptical") are not exactly synonyms, too.

 

[Jacques_L]

It's funny that you come up with American arrogance in relation to the path integral. Feynman recalls a European being hung up with the fact that "Americans always ask how something is useful."

Nevertheless, you still have to deal with, and to explain why during all his life, Richard Feynman remained ignorant of the periodic characters of any quanton that have a mass.
Though they were published in 1924 and 1930 respectively.
The periodic character of the photon is not discussed, but alas its frequency is not intrinsic but depends on the frame.

 

[Drakkith]

Maybe you should have to consider that "science" and "obedience" are not exactly synonyms, and that "ignorant" and "not believer" (or "skeptical") are not exactly synonyms, too.

It is CLEAR that everything you have posted in this thread has an enormous bias against Feynman and apparently Americans. I don't even need to know QM to see that. I ask you to please construct your posts with a little more thought before posting again.

 

[phinds]

Hammock, my perhaps overly simplistic view is that yes, theoretically, there IS a path to Jupiter and back, but there is ALSO theoretically an equal path in the opposite direction and the two cancel in the total sum of possible paths.

Where this view DOES leave me with a problem is that if it's true then why don't ALL paths cancel and there is always NO resulting motion, which is just silly.

 

[ZapperZ]

I'm not sure how this thread has degenerated into Feynman/American bashing, but if it doesn't stop immediately, this thread will be locked and several of you will go down with it.

Zz.

 

[hammock]

@phinds: Yes, that is exactly my problem from my (own, too) simplicistic view. However, further in the book, the autors say something about "normalization" as adding just the Fineman paths together would result in an infinite number.

If i understood correctly (especially Born2bewire) then the normalization (or however it is called) has something to do with the kind of results you receive when calculating the different paths.

I will study the article referred by ZapperZ but I am not sure if I find there the answer for my initial question.

However, without "bashing" anyone, I still wonder, if then most "paths" cancel each other, and there remain mathematically explained paths according to the interference pattern, how can Hawking and Mlodinov argue this as base for an infinite number of histories.

However, it might be that my question ist to less practical and too much philosophic for this forum, so I apologize for that. I am a great fan of Hawking, and I like his books.

 

[Born2bwire]

@phinds: Yes, that is exactly my problem from my (own, too) simplicistic view. However, further in the book, the autors say something about "normalization" as adding just the Fineman paths together would result in an infinite number.

If i understood correctly (especially Born2bewire) then the normalization (or however it is called) has something to do with the kind of results you receive when calculating the different paths.

I will study the article referred by ZapperZ but I am not sure if I find there the answer for my initial question.

However, without "bashing" anyone, I still wonder, if then most "paths" cancel each other, and there remain mathematically explained paths according to the interference pattern, how can Hawking and Mlodinov argue this as base for an infinite number of histories.

However, it might be that my question ist to less practical and too much philosophic for this forum, so I apologize for that. I am a great fan of Hawking, and I like his books.

Normalization is a bit different. Many problems in QED have infinities in them. For example, if you calculate the energy density of the electromagnetic vacuum it is infinite. This poses a lot of problems when we wish to work with these equations to garner results. One way around this is to renormalize the divergent behavior. The idea being that after some point the contributions towards our desired result become insignificant despite the problem diverging. Take the Casimir effect which states that objects, like metal plates, will experience forces due to their disturbing the vacuum energy. When we place these objects in the vacuum, the fluctuating electromagnetic fields of the vacuum change to conform to the boundary conditions imposed by the objects. Now the energy is still divergent, however, we note that electromagnetic waves of very high energy interact less and less with objects (think of gamma rays being able to penetrate far into even the densest of materials). So the idea is that at extremely high frequencies the fields do not interact with the objects and thus we can impose a high frequency cutoff to our energy density.

Another way to deal with this is to take a reference energy that is also divergent and subtract that off from our calculated energy to renomalize it. We can do that by calculating the energy of the system when the objects are infinitely separated. Both the original Casimir energy and the normalization energies are infinite by when we subtract the two we get a finite energy. Renormalization still gives us physical results here because it is the change in energy that gives rise to the Casimir force. Thus, any constant offset that we may apply to the Casimir energy (like we did when we renormalized it) does not affect the forces that arise.

This is different from why we state that the paths out to Jupiter and back cancel. Even if we do path integrals we may have divergence which requires renormalization (in fact we can calculate the Casimir effect using path integrals and we find that we need to renormalize). So these are two different things. The cancellation when we take the path integral is different because this cancellation is occurring in the calculation of the wave amplitude. Renormalization occurs when the observables (like energy) of the system contain infinities that do not contribute to the physical behavior that we seek. Renormalization is a big deal in QED as it posed a stumbling point for a long time until people like Dyson, Schwinger, Feynman and Tomanaga worked it out.

As for arguing about infinite histories and what not. I'm not familiar with that although this sounds more like an explanation done for laypeople. Unfortunately, when the subject of QED and quantum field theories is distilled down for consumption by the general public a lot of the details, nuances, and so forth get lost. Feynman's QED book (the one for the general public not the path integral book he did with Hibbs) is a sad example of this. He talks about the effects of the phase of the path integral and how it cancels out and steepest descent paths and so forth. But he can't use the mathematics that demonstrate this. Instead, he just states how it is and uses a stopwatch (I believe the stop watch is also used in ZapperZ's paper) to explain how phase cancellation works between paths. It doesn't really explain it clearly and I think that people who know a modest amount of calculus and mathematics feel lost with such a simplistic explanation. So that could be the reason but it really is hard to say without knowing the actual context and words that were stated.

 

[Jacques_L]

... Feynman's QED book ... the path integral book he did with Hibbs...

If you have this book - I will not see it before september - maybe you can solve this mystery :
In the paper quoted by ZapperZ (http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=CPHYE2000012000002000190000001&idtype=cvips ) the distinction between group velocity and phase velocity only appears in note 16. Did Feynman and Hibbs proceed real and serious corrections ? And why do they not appear in this paper, where only the group velocity and a ficticious frequency are used ?

 

[jfy4]

I'd like to add also that one can define Hamiltonian mechanics from the path integral approach, which makes it quite powerful for two reasons. It possesses particular computational power for various situations, and a whole separate formalism of dynamics can be derived from it. Take
$$\langle\phi_b(x)|e^{-iHt}|\phi_a(x)\rangle=\int\mathcal{D}\phi \exp\left[i\int_{0}^{T}d^4x\mathcal{L}\right]$$
this can be differentiated with respect to $T$ and one can get the Hamiltonian dynamics from what results.

 

[Jacques_L]

From my quick reading of this thread, this has more to do with trying to understand Feynman's path integral. Unless someone has some extraordinary capability with written communication, trying to illustrate this principle is almost impossible on a public forum such as this.

So maybe a readable source that introduce what a Feynman path integral is might be useful. Try this one:

http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=CPHYE2000012000002000190000001&idtype=cvips

Zz.

Let's return to the real world.
In the real world, the electrons are not fairy beasts, but are used by industry and scientific research for more than a century. Vacuum diode exists for 1904, triode for 1907. Cathode ray tubes exist for 1907. X-Ray tubes are in use in radiography for the first World war.
Electron beams are also used for microscopy, by transmission or by scanning, in microprobes, so on.
They are not as good as X-Ray are in crystallography, for their poor parallelism and poor monochromaticity. However we obtained Laue diffractograms of carbide inclusions on a Siemens electronic microscope, just by modifying the focalization and diaphragming on the inclusion.
And electron beams are also used in nuclear industry, for very deep and narrow weldings.

The next step will be to take the example of a beam of electron mimicking the ray $K \alpha_1$ of molybdenum, at 0.709300 Å (17.47934 keV). What are the speed and the potential difference for that ?

First : how many wavelengths for a 30 cm "flight" ?
30 cm / 0.709300 Å/cycle = 4.23 milliards of wavelengths.
Momemtum : 9.3417 . 10^-24 kg.m/s
Speed, rated as non-relativist : 10,255 km/s
v/c : 0.0342 (3.42 %). OK, it is not relativist, or very little.
Gamma : 1.00059
Corrected speed : 10,249 km/s.
Kinetic Energy : 4.790 . 10^-17 J.
Potential difference : 299 V.

Time of flight for 30 cm : 29.27 ns.
In the frame of the electron : 29.29 ns.
How many broglian periods during this flight ?
3,619 milliards of periods during this flight.

Phase velocity : 87.69 millions of km/s.

Another question not yet answered here is how many times during this flight, the electron was at + c speed forward ?
3,619 milliards x 2 = 7,238 milliards times.
And 7,238 milliards times backwards at speed -c.

How much more time forward than backward ?
51.71 % time forward.
48.29 % time backward.
So is the Zitterbewegung, according to the Dirac equation (1928), and the beating of negative and positive energy components.

So real electrons in the real world seem to have very very different properties than those postulated by Hawking and Mlodinow on one side, by Edwin F. Taylor, Stamatis Vokos, and John M. O’Mearac (and maybe Feynman himself) on other side.
Real electrons in a real flight in a CRT on Earth do not seem to have time to explore planet Jupiter...
Just compare to the physics of Santa Claus :
http://www.positiveatheism.org/writ/santa.htm
and draw your conclusion yourself.
In the fairy tales Santa Claus visits 967.7 houses per second, and the electron in a CRT tube on Earth explores the planet Jupiter.
In the real world, we are trying to have better experimental values of the real width of an electron, depending of its momentum and the length of its path. Here we seem to be in the 10-15 Å range, and if true, it is by far too narrow for a successful double slit experiment (too narrow or too fast, it is tight together).


Such equipments for Debye-Scherrer diffractograms with electrons are sold by Leybold Didactic for our classrooms :

Catalogue at : http://www.systemes-didactiques.fr/include/physiquechimiebiosvt/tube.pdf
This tube seems no more available at http://www.ld-didactic.de ([VP6.1.5.1] · Diffraction of electrons at a polycrystalline lattice (Debye-Scherrer diffraction)).

You see that they use higher voltages to obtain more Bragg reflections on a small screen.

The angular resolution is poor. We obtain far better results with $K \alpha$ of a metallic anticathode.

Now the question arises : what causes such a poor angular resolution ?

1. The crystallites in the metal or graphite sheet may be too small.
2. Each incident electron may be too small. However, it must be at least three to four interatomic distances long, and five to six interatomic distances wide, otherwise not any diffractogram could be seen at all.
3. Too much speed dispersion in the electrons beam.
4. Too much angular dispersion in the electrons beam.

To know more on the length of coherence of an electron in a beam, one must make the electron interfere with itself. So in Aharonov-Bohm type of experiments, the speed and the accelerating voltage are much lower. As far as I know, the so proved length of coherence is in the magnitude of ten wavelengths, at least.


To be continued.

 

[hammock]

Jaques, thank you much for your post.

As your arguments are a bit complex to me to understand, as I do not have full knowledge the terms, may you please be so kind and give me a try.

The "Zitterbewegung", is this a fluctiation in space or in time, or both? Having investigated a bit in WWW, this explanation seems to be an "alternative" therory.

Does this fluctuation then the cause the interference pattern in case of the double slit experiment?

 

[ZapperZ]

Let's return to the real world.
In the real world, the electrons are not fairy beasts, but are used by industry and scientific research for more than a century. Vacuum diode exists for 1904, triode for 1907. Cathode ray tubes exist for 1907. X-Ray tubes are in use in radiography for the first World war.
Electron beams are also used for microscopy, by transmission or by scanning, in microprobes, so on.
They are not as good as X-Ray are in crystallography, for their poor parallelism and poor monochromaticity. However we obtained Laue diffractograms of carbide inclusions on a Siemens electronic microscope, just by modifying the focalization and diaphragming on the inclusion.
And electron beams are also used in nuclear industry, for very deep and narrow weldings.

I'm not sure what you wrote in that post has anything to do with the topic. Still, since you're talking about the "real world", how about my checking if you also area aware of such a thing.

What you described above are not clear examples where QM of any kind (be it path integral, straight forward QM, etc..) is applicable. When dealing with free electrons, and when electron-electron correlations are negligible, many of these phenomena resort back to the CLASSICAL description! Don't believe me? Look in beam dynamics of particle accelerators. Particle tracking codes used such as PARMELA considers each electrons as classical particles!

And CRT? Yes, even that! Look at the codes used to model photomultipliers, especially the electron trajectory through the MCPs or the dynodes!

I'm not sure why these examples are being used here, and how exactly they are applicable. All I can see is a bunch of things taken out of context that were never based on QM's description in the first place!

And for your information, I routinely use SEMs, work at a particle accelerator, and am involved in a photodetector project to design and build new detectors. Would those qualify as "real world" experience?

Zz.

 

[Jacques_L]

Jaques, thank you much for your post.

As your arguments are a bit complex to me to understand, as I do not have full knowledge the terms, may you please be so kind and give me a try.

The "Zitterbewegung", is this a fluctuation in space or in time, or both? Having investigated a bit in WWW, this explanation seems to be an "alternative" therory.

Does this fluctuation then the cause the interference pattern in case of the double slit experiment?

The Zitterbewegung frequency is an electromagnetic frequency. Its presentation is reproduced in english from Schrödinger by P.A.M. Dirac at § 69 of his "Principles of Quantum Mechanics". It is based on the Dirac's wave equation of the electron (1928).
This Dirac-Schrödinger period and wavelength are those intervening in interactions between an electron and an electromagnetic wave, as in the Compton scattering of an X-ray photon by an extracted electron. In 1927, Erwin Schrödinger failed to correctly explain the Compton scattering by the Broglie's wave by a Bragg diffraction : Broglie wavelength (the only one known in 1927) gives twice the right equidistance required for first order Bragg reflection. So scattering could be explained only by a second order Bragg reflection, while the first (broglian) order is never observed. And all is now OK when you consider the Dirac-Schrödinger ("Zitterbewegung") wavelength.
You find inline the Schrödinger's attempt at http://www.apocalyptism.ru/Compton-Schrodinger.htm (otherwise in paper at your University).

But what rules the stiffness and the width of real paths of real electrons is the Broglie's wavelength and Broglie's frequency, the real and intrinsic one : $\frac{m.c^2}{h}$. When moving, the electron interferes with itself. Christiaan Huyghens had already explained and drawed this in the 17th century. What Huyghens had not in his time was the intrinsic frequency of an electron : the electron was invented only in 1891, observed in 1897, and its intrinsic frequency was only calculated in 1924 (by Louis Victor de Broglie), and proved in 1927 by Davisson and Germer (previous experiments had already shown the same diffraction effects, but were not correctly interpreted in time).

So an electron and any fermions have TWO intrinsic frequencies : the broglian one, which rules the stiffness and width of trajectories, and the interferences with itself, and the double, the Dirac-Schrödinger one, for electromagnetic interactions. Nobody clearly understands today why two related frequency, excepted that the second one flows from Dirac's wave equation. The common opinion is that it is related to spin 1/2. The fact remains that in one broglian wavelength of a moving electron, there are two electromagnetic wavelengths.

I have in house a copy of the original paper from Feynman in 1948 : it is at pages 321-341 of "Selected Papers on Quantum Electrodynamics", selected by Julian Schwinger (Dover ed.) : "space-Time approach to Non-Relativistic Quantum Mechanics". Buried in the lagranguan formalism is the hypothesis of a ficticious frequency, for a wave Feynman thought to be ficticious. Feynman reasoned as a corpuscularist, and believed that the phase was only an artifact. The dismaying consequences of his ficticious electronic frequency are that he and his readers considered far more soft, wide, bendy and extravagant paths than any real ones. And heaps and heaps of integrals doomed to give zero - with divergence... Much sweat for naught or nearly naught. Corpuscularism remains a blind alley.

Sorry "hammock", I was not clear enough with you last question : a double-slit experiment is making the electron interfere with itself, by two paths. So the competent wavelength there is the broglian one.