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In optics course, in one hand, we learned how the light could be created by spontaneous emission, illustrated conveniently by the following figure.

enter image description here

In this process, we absolutely treat light as particles, namely so-called the photon.

But on the other hand, we also talk about the propagation, interference, diffraction of light in the air or in many other media.

In this process, we, of course, treat light as electric field, namely so-called wave.

So my question is: How can one fill the gap from photon to the electric field? I mean what's the reason (or existing approximation) in both description for light between creation process and propagation process? Or is there any way we can transform the equation governing the dynamics of the photon to the equation dominating the dynamics of the electric field (from photon to electromagnetic wave)?

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To understand the mathematics connecting photons, which are a quantum mechanical entity/particle, with the classical electromagnetic wave built up by photons, one needs quantum mechanics and quantum field theory, i.e. quantum electrodynamics, QED.

Quantum mechanically the photon has a wavefunction which is a solution of the quantized Maxwell's equation. This is a complex function which has the frequency of the wave built up by many such photons , in a complex numbers formulation. The free photon is observed with a probability given by the complex conjugate squared of the wavefunction, and the only measurable quantities are its energy=h*nu and its spin (+/-1h) to its direction of motion.

When more than one photons are in the same space-time the wavefunctions add and this addition, allows the build up of the classical electric and magnetic fields of the light beam. The mathematics of this is given in this blog entry but it needs QED to be understood.

Maybe this link as an example could give an intuition:

enter image description here

Left and right handed circular polarization, and their associate angular momenta.

The red circulating arrows are the direction of the electric field

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  • $\begingroup$ Thanks for your answer. So can we derive Maxwell's equations (describe electromagnetic wave) from the Klein-Golden equation (describe the photon) in quantum electrodynamics? $\endgroup$
    – Jack
    Commented Feb 3, 2017 at 9:35
  • $\begingroup$ the link gives that the solutions of Maxwell's equations emerge from the confluence of the solutions of the QED quantized maxwell's equations. The photons obey a version of quantized maxwell equations. for example arxiv.org/ftp/quant-ph/papers/0604/0604169.pdf $\endgroup$
    – anna v
    Commented Feb 3, 2017 at 13:02

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