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Maxwell's theory of electromagnetism describes light as a wave of the electromagnetic field.

Quantum mechanics associates a probabilistic wave (my own interpretation of the function wave) from which we reproduce results similar like the model of electromagnetic waves, e.g. diffraction. The same goes with electrons and other particles

What happens in this interpretation with the electromagnetic field that travels in space? Is it only valid to speak about it when the number of photons is big (a statistic property), or it is wrong thing in this model and you have to forget it when you work with quantum mechanics?

Could somebody explain how to reconcile all the things that you learned classically about electromagnetism with this idea?

DanielSank
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Mahl-Deneb
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2 Answers2

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Conceptually one can make a connection from the "probabilistic wave" of photons to classical Maxwell EM wave in the following way:

To explain the idea, let's consider a double slit experiment. As you said, quantum mechanically all the particles travel through certain kinds of probabilistic wave. Photon is no exception. For both electron and photon, one would see a dot on the screen on each shot, but the interference pattern will show up after shooting a lot of electron/photon.

The special thing for photon here is that:

  1. it's a boson, so many photons can stay at the same state

  2. there is no interaction between photons in vacuum.

This means that it's possible to shoot a lot of photons at the same time, and the interference pattern will show up directly if you count the number of photons received at every point on the screen. In the old day, people always shoot a lot of photons, and they call the number of photons "intensity", and the interference pattern from the probabilistic wave can be directly interpreted as the classical EM wave.

One can easily generalize the idea to arbitrary configurations, though I personally find it easier to visualize by path-integral formalism. In case you are interested in it later, refer to "QED The strange theory of light and matter" by Richard Feynman.

(P.S.: The reason that the wave, both the amplitude and the phase, suddenly becomes measurable (like classically) is because there are a lot of photons in the same states, which form coherent state so that both amplitude and phase are roughly known with small error.)

Yen-Ta Huang
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Maxwell's theory of electromagnetism describes light as a wave of the electromagnetic field.

Yes it does.

Quantum mechanics associates a probabilistic wave (my own interpretation of the function wave) from which we reproduce results similar like the model of electromagnetic waves, e.g. diffraction. The same goes with electrons and other particles.

As far as I know quantum mechanics doesn't describe light as a probabilistic wave. It refers to the "quantum", the photon instead.

What happens in this interpretation with the electromagnetic field that travels in space? Is it only valid to speak about it when the number of photons is big (a statistic property), or it is wrong thing in this model and you have to forget it when you work with quantum mechanics?

IMHO you consider the photon to be a singleton "soliton" electromagnetic wave.

Could somebody explain how to reconcile all the things that you learned classically about electromagnetism with this idea?

Can you tell me where you're getting your information from? I've obtained my information from a variety of sources. For example see Steven Weinberg's 1997 essay what is quantum field theory, and what did we think it is? He refers to the famous "Dreimännerarbeit" paper on quantum mechanics II, which was written in 1925. Weinberg says "in one of the very first papers on quantum mechanics, Born, Heisenberg and Jordan presented the quantum theory of the electromagnetic field".

Also see Pascual Jordan’s resolution of the conundrum of the wave-particle duality of light by Anthony Duncan and Michel Janssen dating from 2007. On page 8 they quote Jordan saying this: "what the Dreimännerarbeit says about energy fluctuations in a field of quantized waves is, in my opinion, almost the most important contribution I ever made to quantum mechanics". On page 47 they quote Jordan saying this: "Einstein drew the conclusion that the wave theory would necessarily have to be replaced or at least supplemented by the corpuscular picture. With our findings, however, the problem has taken a completely different turn. We see that it is not necessary after all to abandon or restrict the wave theory in favour of other models; instead it just comes down to reformulating the wave theory in quantum mechanics. The fluctuation effects, which prove the presence of corpuscular light quanta in the radiation field, then arise automatically as consequences of the wave theory. The old and famous problem [of] how one can understand waves and particles in radiation in a unified manner can thus in principle be considered as solved". Jordan had taken care of wave-particle duality, but people didn’t appreciate it.

The Dreimännerarbeit pdf above is from Bartel van der Waerden’s 1967 book Sources of quantum mechanics See page 375. Jordan said the same will apply if we consider the vibrations of an elastic body idealized to a continuum, or the vibrations of an electromagnetic cavity. He said this on page 376: "So strong an association between the eigenvibrations of a cavity and the light quanta postulated formerly can nonetheless be drawn that every statistic of cavity eigenvibrations corresponds to a definite statistic of light quanta, and conversely". He was saying light quanta are electromagnetic waves rather than corpuscles. Hence you consider the photon to be a singleton self-contained "soliton" wave that propagates through space without dispersing.

John Duffield
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