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Maybe an illegal question. Sorry. Having read that a photon has both wave and a light characteristics it is natural for a novice like myself to wonder "what is the amplitude? " Naturally I have looked on wiki and other sources and there are a number of answers some of which appear not so much to contradict one another of varying opinions of which is the most meaningful in physics.

I seem to get a higher number of physics sources saying the amplitude depends on how many photons you have and if you have one the question has no meaning. Of course I have not idea if this is accurate or how one would even test this.

Maybe there are some physics teachers in the audience that can give me a layman's intuition with a little math tossed in.

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Photons are the excitations of a quantum field. For such a field, you have to distinguished between $\hat{\mathbf{E}}$ which is the electric field itself (and described by a mathematical object called an operator), and observables/expectation values such as $\langle\hat{\mathbf{E}}^2\rangle$. If you want to define an amplitude from the field itself, you would have to deal with the math of quantum mechanics directly, and I'm not sure it would even work out (see: Operator norm of creation and annihilation operators). The averages are the things we can actually measure - and these only make sense statistically, when we measure them repeatedly. That's why the amplitude of a single photon does not make sense.

You might want to look into number states and coherent states however, to broaden your view on photons, amplitude etc.

I think the top answer to this Reddit question also answers this well:

https://www.reddit.com/r/askscience/comments/19vwre/does_a_single_photon_have_an_amplitude/

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Plan's answer is better than this one, but just in case it misses anything, this is a more basic view, and I can learn myself from any comments on this.

I seem to get a higher number of physics sources saying the amplitude depends on how many photons you have and if you have one the question has no meaning. Of course I have not idea if this is accurate or how one would even test this.

I have never myself ever seen this linkage between particles and amplitudes, (although it may depend on the sophistication of the text).

The advantage of the wave/ particle is that it explains observations such as interference, if we treat electromagnetic radiation as waves, and the photoelectric effect, if we treat em radiation as photons.

Neither interpretation is more meaningful than the other, they are both methods that are useful in explaining particular observations that cannot be easily understood using a mental picture that is not appropriate to a particular experiment.

Behind both photons and waves is a mathematical framework (which is the real explanation) that allows us to calculate and predict physical results and effects in terms of photons or waves.

Normally, it's the energy and momentum associated with a physical event that we are concerned about.

If you use photons, you don't have to worry about amplitudes, energy is calculated using Photon Energy, and this reference gives you the math.

But amplitude is more a wave related concept than any application to particles.

If you use waves to describe electromagnetic radiation, the energy is given by this reference, Energy of an electromagnetic wave, which will tell you that the energy carried by a wave is proportional to its amplitude. Again, the math behind the wave picture is included.

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You are getting confused between the quantum and classical pictures.

In quantum mechanics, configurations of the electric field are described by complex numbers, probability amplitudes, at every point in spacetime. A collection of photons is described by very special superpositions of those types of numbers (formally it's a vector in a tensor product space built out of the single particle theory). Amplitudes in the quantum theory are probability amplitudes for certain field configurations. One might say something sloppy like this type of amplitude might "not make sense," for a single photon if it is in a well-defined state, since in that case it's just an overall phase of the state (a non-observable in QM).

In the classical theory, the field is in only a single configuration, and the amplitude one speaks of is linearly related to the amount of energy stored in the field (which is also linearly related to the number of photons os various wavelengths)

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