B Interference patterns of electrons vs. photons

[entropy1]

I understand that fermions are subject to the Schrödinger equation, but photons are not. I understand that interference patterns of electrons are governed by the Schrödinger equation, but with photons it is different. If I understand correctly, then what is the nature of this difference?

 

[secur]

The question is discussed here also:
https://www.physicsforums.com/threads/double-slit-experiment-with-photons-vs-electrons.725657/

Photons, in the aggregate (like a beam of light), obey Maxwell's equations (vacuum solution) which is a simple classical wave equation. BTW the vacuum solution predicted that EM radiation travels at the speed of light, a great discovery at the time.

The thread referenced goes into some details which may or may not be of interest. For one thing Maxwell's equations are not for single photons; you need a QM equation for that (as in QED). But of course one photon can't show an interference pattern; if you repeatedly send them at a screen it will produce the interference pattern predicted by the vacuum solution. They also mention that you can use Schroedinger's for photons also, with an appropriate Hamiltonian, so when you say "photons are not subject to Schroedinger's eqn" that's not entirely correct. But I don't understand that (considering Schroedinger's is not relativistic) so for further info take a look at that thread if interested, or wait for someone more knowledgeable.

Anyway, I think the answer to your question is, there is no difference qualitatively between the interference patterns produced by Schroedinger's (which is usually applied to fermions) and Maxwell vacuum solution. In spite of the fact that Schroedinger's involves complex numbers and is a QM equation, not classical, the solutions in both cases boil down to sin / cos wave functions and produce the same type of interference pattern. (Assuming no complications like, for instance, Schroedinger solutions for charged particle, like electron, being modified by an electrostatic field present at the detector.) Although the actual coefficients will in general be different, if you scale the two patterns appropriately they can be superimposed so as to match one another.

Of course I could be wrong!

[EDIT]

Having thought about it ...

As I mentioned Maxwell's vacuum solution doesn't apply to the non-classical single-photon case (although lots of these fired one after the other will generally produce the Maxwell interference pattern). Actually I should extend that comment to cover any non-classical EM manifestation - in particular, lasers, which of course emit coherent EM radiation. I think in any such case we must use QED. As I understand OP's question he's interested in the simple, classical case, so Maxwell vacuum solution is appropriate. Beyond that, there are more details that I'm not competent to answer.

 

[entropy1]

I understand that for individual photons we need QED to explain the interference pattern, due to the emergence of the quantum nature. What I don't understand is what the essence of the difference is between

1. The wave nature of electrons, given that electrons are particles, and
2. The particle nature of photons, given that photons behave like waves.

That is to say: photons seem to emerge "out of nothing" at the interference pattern, while electrons (correct me if I'm wrong) are exactly located when standing still for instance, but become a wave at relativistic speeds, thus producing an interference pattern. What is the essence of this difference (if it can be easily explained ).

 

[Nugatory]

while electrons (correct me if I'm wrong) are exactly located when standing still for instance, but become a wave at relativistic speeds, t

That's not correct; it sounds as if you're holding too tightly to the easily misunderstood notion of wave-particle duality here.

An electron that is "standing still" (you might want to take a moment to think about what it means to say an electron is "standing still" - can that mean anything other than has zero momentum and kinetic energy?) does not have a definite position and is not exactly located unless and until you measure its position; the way you measure its position is to observe that it has interacted with something else at a particular point in space. But this is the same situation as with photons, so the difference that is bothering you goes away.

It is true that electrons can be treated using the methods of non-relativistic quantum mechanics if the energies involved are sufficiently small, whereas photons cannot. That can tempt us into forming different models of electrons and photons - that temptation has to be resisted.

 

[jtbell]

electrons [...] become a wave at relativistic speeds, thus producing an interference pattern.

Electrons do not need to be moving at relativistic speeds in order to produce interference patterns. A common undergraduate lab apparatus for studying electron diffraction is a small CRT which produces a distinct diffraction pattern at a KE of about 2 keV, which gives v = 0.1c approximately. At such speeds the non-relativistic formulas for momentum and KE work very well, and we use them in our analysis. (Then of course we ask students to use the relativistic formulas also, to see how small the difference is!)