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Before quantum theory was developed, Maxwell's equations were deemed to accurately model electromagnetism and light.

The wave nature of light had been observed, as in the double slit experiment, however, it was later found that instead of a continuum of values observed on the other side of the slit, the light would interact as if it were a single point, only showing the wave pattern when many points were recorded.

Thusly, since Maxwell's equations seemed to accurately describe the wave pattern, would it's description of light be equivalent to the probability distribution of a photon?

Or do the equations just approximate the quantum mechanical description?

Shacks
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So, in short, a proper answer to this question is a full graduate semester on quantum optics (https://en.wikipedia.org/wiki/Quantum_optics), and maybe more.

I recommend investigating the Nobel Prize winning Glauber State (https://en.wikipedia.org/wiki/Coherent_state), which explains how a classical plane (https://en.wikipedia.org/wiki/Plane_wave) wave (the eigen-solution to maxwells equations in a vacuum) is related to a superposition of photons (as Fock states, https://en.wikipedia.org/wiki/Fock_state).

Quantum optics is a non relativistic formulation (via the Hamiltonian) of electromagnetism. The relativistic formulation (via a Lagrangian) is QED (https://en.wikipedia.org/wiki/Quantum_electrodynamics), where the absolute minimum summary is that the quantum photon propagator is related to the classical Green's Function (https://en.wikipedia.org/wiki/Green%27s_function), and the next step is understand the S-Matrix (https://en.wikipedia.org/wiki/S-matrix).

So it's just too much to answer the question in https://physics.stackexchange.com

JEB
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The quantum theory of light was developed to describe phenomena that cannot be described by Maxwell equations. If both theories were complete equivalents, there would be no reason to develop QED or quantum optics.

However, it is a matter of fact that both should be equivalent in some regime. When a new theory is in development, a good sanity check is to test if it recovers all previous results already described by previous theories. If QED was not able to describe the same phenomena that Maxwell equations do, it would be a red flag that it was wrong. It could not be considered a more precise theory, but an alternative one. Happily, that's not the case, and we know QED is the correct generalization of Maxwell equations in the quantum relativistic regime.

Ruffolo
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Maxwell's equations describe the electromagnetic field in terms of the vectors $$ \begin{matrix} \mathbf{E}(\mathbf{x},t)=(E_x(\mathbf{x},t),E_y(\mathbf{x},t),E_z(\mathbf{x},t)),\\\mathbf{B}(\mathbf{x},t)=(B_x(\mathbf{x},t),B_y(\mathbf{x},t),B_z(\mathbf{x},t)) \end{matrix}$$ whose components are real numbers that also represent the result you would get if you measured the relevant field component at a particular point in spacetime.

In quantum field theory the equations of motion of fields are written in terms of Hermitian operator valued fields called observables. QFT predictions are usually written in terms of correlation functions among the values of the field observables. There are photon number observables whose eigenvalues represent the possible results of measuring the number of photons of a given momentum and polarisation with values corresponding to 0,1,2... photons. This is explained in many quantum field theory books such as "Quantum field theory for the gifted amateur" by Lancaster and Blundell.

Copying information out of a quantum system suppresses interference: this is called decoherence:

https://arxiv.org/abs/1911.06282

There are many papers on the decoherent limit of quantum field theories including QED:

https://arxiv.org/abs/quant-ph/9510021

https://arxiv.org/abs/quant-ph/0210013

For more on the classical limit of quantum field theory see Chapter 8 of "The conceptual framework of quantum field theory" by Anthony Duncan.

alanf
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