Young's slits with incandescent light source

[PeterDonis]

I imagine that, as electrons are fermions and photons are bosons, the effects of any real aperture could be different.

Electron diffraction experiments, and AFAIK electron double slit experiments, have been done, and AFAIK they show the same general interference phenomena. There might be differences in the fine details, yes.

 

[gentzen]

Unless the source is emitting Fock states (which it isn't in any double slit experiment that has been done to date), there are not photons passing through the experiment. There is light (the quantum electromagnetic field) passing through the experiment, but you can't usefully describe it as photons. It will get detected as individual impacts on the detector screen, but that does not mean it is photons before then. It's not.

Would everyone's answers change if I said 'electron' instead of 'photon'?

Yeah, good question. I didn't give an answer, but I guess my answer would change. But why? I think it is easy to produce single electron states in practice. Or more precisely, states which can be described by single electron states with excellent accuracy. Do I mean the same thing by "single electron state" as PeterDonis means by "Fock state"? Not exactly, because a "Fock state" could also be a state with two electrons, or three electrons. What is not allowed (for a Fock state) is a superposition of a two electron state and a three electon state, i.e. superpositions between states with different numbers of photons or electrons.

So in 'double slit' experiments using electrons, what happens can be described with excellent accuracy by single electron states interfering with themselves.


However, there is also a different perspective: Weak light can often be described to excellent accuracy by a superposition of a zero photon state with a one photon state. And similar, there seems nothing wrong with seeing the single electron states mentioned above as superpositions between a zero electron state and a one electron state.
I guess the point is that I use non-relativistic QM for my mental picture of electrons, but a QFT description for my mental picture of photons. The crucial difference is that the QFT picture is a variable number of particles picture, while the non-relativistic QM picture is a fixed number of particles picture.

 

[Lord Jestocost]

Throwing another pebble in the pond...

Would everyone's answers change if I said 'electron' instead of 'photon'?
As I understand it, we get just the same interference effects in 'double slit' experiments using electrons. Is that just a coincidence? Maybe the effects aren't the same - has anyone done other optics-like experiments with electrons? There must be lots of diffraction patterns out there, for example.

There are also double-slit experiments with neutrons (“Single- and double-slit diffraction of neutrons” by A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe, Rev. Mod. Phys. 60, 1067, 1988).

The outcomes of all these experiments confirm the predictions of standard quantum theory.

 

[sophiecentaur]

From this link: (QSNP)
"Fock state is a quantum state that contains a precise number of non-interacting, identical particles,"

Does this imply that a beam of electrons, which will all interact with each other, will not have a Foch state? It confuses me (not a bit).

Also, diffraction of electrons in electron optic equipment is hard to understand because of the need for phase information - phase of what 'wave'`? We've all seen electron diffraction patterns in school demos but what actually is going on there?

 

[sophiecentaur]

I also have a problem with treating all photons arriving from a (series of?) emissions as if they have perfect coherence. What sort of oscillating equipment could have zero bandwidth? There has to be some degree of decoherence so, apart from what classical wave theory tells us, what would be the effect of finite bandwidth on the resulting pattern? (filling in of nulls etc. which is also affected by slit widths)

 

[PeterDonis]

there seems nothing wrong with seeing the single electron states mentioned above as superpositions between a zero electron state and a one electron state.

It is possible to construct what are called coherent states (which is where we get the idea of a very weak light source emitting a superposition of states with zero and nonzero photons) for fermions, but I don't know if they have all of the same properties. But more importantly, I don't think those are what electron sources like a cathode ray tube emit. I don't think there are meaningful superpositions of different electron numbers in those states; at any point in space, at any given time, I think the electron state will be an eigenstate of electron number. (Btw, I was using "Fock state" in general to mean any such state, not just the ones with eigenvalue 1.)

Talking about "states" in the context of relativity creates problems because, heuristically, trying to adopt such a viewpoint in QFT requires choosing a particular frame, and the "states" you get then become artifacts of that particular choice of frame. For experiments where relativistic effects are negligible, this works all right, you can just pick the rest frame of the lab and pretend you're just doing NRQM in that frame (this even works for many experiments involving photons, such as the double slit--you just formally write down the same sorts of states you'd write down for any non-relativistic uncharged massless spin 1 particle, and ignore the fact that there is no rigorous mathematical derivation of any such thing from the underlying QFT). That seems to be more or less what we're doing in this thread.

 

[gentzen]

Does this imply that a beam of electrons, which will all interact with each other, will not have a Foch state? It confuses me (not a bit).

For the typical electron beam in an electron microscope, the direct interaction between the electrons is negligible. And even in electron beam writers, which can use much higher currents, the direct (Coulomb) interaction is rarely ever a problem.

There is indirect interaction in the sample, because the damage, charging, or other changes to the sample induced by earlier electrons do influences electrons which hit the sample later. But I wouldn't call this "all interact with each other". And it doesn't really impact the quantum description, at least not conceptually.

Also, diffraction of electrons in electron optic equipment is hard to understand because of the need for phase information - phase of what 'wave'`?

I guess I see your conceptual problem: The single electron state gets described by a time-harmonic wave, so in a certain sense an infinitely spatially extended and timeless description. It is a solution of a 3D Schrödinger (or Dirac) equation. One way to get used to this description is to study the scattering of a plane wave at a planar potential barrier/step. Some confusing aspects of this description already occur in this simple scenario, and can also be resolved. Sometimes these aspects still confuse me, while trying to understand more complicated multislice or Bloch-wave computations. Sometimes I have to go back to this simple scenario to clear my conceptual confusions.

Also confusing is the interaction of such an "incoming electron wave" with the phonons, plasmons, inner shells, ... in the sample. However, I once decided to not reply to [...] so I guess I should stick to that. So now I saved that part of my reply locally. It contained references to
https://github.com/elena-pascal/Thesis
https://github.com/EMsoft-org/EMsoft
Budhika G. Mendis
but was not easy to understand. It did describe my "rationalizations" of what they are doing. You could argue that it was kind of "original research".

 

[PeterDonis]

From this link: (QSNP)
"Fock state is a quantum state that contains a precise number of non-interacting, identical particles,"

I don't think it's necessary for the particles to be non-interacting for there to be eigenstates of particle number. But it is true that there are plenty of traps for the unwary lurking here, such as:

https://en.wikipedia.org/wiki/Haag's_theorem

Does this imply that a beam of electrons, which will all interact with each other, will not have a Foch state?

I think that, as I said in #36, at any point in the space occupied by the beam, at any given time, the state will be an eigenstate of electron number. Whether the term "Fock state" is strictly correct, given that yes, electrons interact because they're charged, I'm not sure. But I think that in any case the state is very different from a coherent state of light, or for that matter an incoherent state like that emitted by an incandescent source. The differences just don't happen to matter for the double slit experiment.

phase of what 'wave'`?

The electron's wave function.

 

[flippiefanus]

You seem to be implying that classical field theory cannot be used for any phenomena.

No, I said it cannot explain all phenomena. There are scenarios where classical electromagnetism is perfectly adequate. There is also scalar and vector diffraction theory.

 

[flippiefanus]

Throwing another pebble in the pond...

Would everyone's answers change if I said 'electron' instead of 'photon'?
As I understand it, we get just the same interference effects in 'double slit' experiments using electrons. Is that just a coincidence? Maybe the effects aren't the same - has anyone done other optics-like experiments with electrons? There must be lots of diffraction patterns out there, for example.

For a while people amused themselves by producing electron beams (in electron microscopes) with vortices in their fields, similar to how it is done in optical beams. The results are very similar for electrons and photons.

 

[PeterDonis]

For a while people amused themselves by producing electron beams (in electron microscopes) with vortices in their fields, similar to how it is done in optical beams.

Can you give a reference?

 

[flippiefanus]

Can you give a reference?

There are lots, see for example:
Lloyd, S. M., Babiker, M., Thirunavukkarasu, G., & Yuan, J. (2017). Electron vortices: Beams with orbital angular momentum. Reviews of Modern Physics, 89(3), 035004.

 

[Cthugha]

The stabilisation of the two source makes this scenario much more complicated and thus cannot be compared with the single source scenario. This strictly speaking becomes a multiple-photon effect.

No, why should it? In a standard laser, you also cannot distinguish between photons coupled out of the resonator at some moment and photons coupled out exactly one round trip later. One would also not consider this a multi-photon effect. It just matters whether all the excitations are within a single mode (and therefore indistinguishable). The light field and the emitter are still entangled during the coherence time of the light field, which means that the excitation of the entire system is in a superposition of being in the field or in the emitter. It does not change anything if you spread a single excitation over a single light field and two emitters instead.

Yes, fields indeed, but unless one introduces the notion of quantised fields (photons are not classical point particles), one would just be doing classical field theory that cannot explain all phenomena. It is not sufficient to just use probability amplitudes without considering quanta. There are experiments (violation of Bell's inequality) where the quantised nature of the EM field plays an important role.

Replacing the intensity with a probability amplitude that is normalised under $L^2$ (modulus square integrates to 1), one implies a quantum of the field, i.e., a photon. So one can just as well call it by that name. Reading through the description of the calculation procedure, I get the feeling that it is still based on photons, just without calling it that.

This is a thread about the double slit, which means first order-coherence, which mathematically just involves a single creation and annihilation operator at different positions (or times for first-order temporal coherence). It is not sensitive to operator ordering and therefore not "quantum".
Of course there are effects that require quanta, but they are completely irrelevant to this thread and the double slit.
Also, assuming that a probability amplitude that normalizes to 1 implies a quantum of the field is fundamentally misguided. This only applies when you put the quantum in already, so you calculate the probability amplitude for detection of a single photon using a single-photon sensitive detector assuming that a single photon was emitted. Probability amplitudes are completely general and could be used for, e.g., quadrature detection in homodyne detection or detection of currents in detectors that are not singe-photon sensitive. One can introduce photons, but there is no need to.