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Photons do not seem to be very well defined in quantum mechanics like an electron or even a quark. My assumption is that the relationship between the quantum and classical views on radiation must be fleshed out to some extent. Ive read in several places that quantum mechanics converge into classical mechanics when expanding into the larger scales where particle wavelengths no longer matter. Details are thin in nontechnical descriptions of quantum theory so its possible that they are only talking about classical mechanical physics and the answer to this question is that there is no known relationship.

How does the concept of photons build up into something that looks like a classical electromagnetic wave?

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In essence, via the concept of a coherent state: the quantum state of the EM field which most closely resembles a classical field. It has a well-defined 'central' (expectation) value of the electric and magnetic fields, with a minimal uncertainty around them coming from the Heisenberg uncertainty principle, which is often called shot noise. At low intensities, this uncertainty is comparable with the expectation value, and needs to be taken into account carefully, but at high intensities its fixed value becomes negligible compared to the expectation value, and it can be discarded, giving you the classical picture.

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