Why do photons follow specific path after reflection from a mirror surface if they can be emitted in any direction by electrons of mirror surface?

Although a single photon can only be absorbed and emitted by a single electron, it leaves that electron in exactly its original state. There is no record, and no way of knowing, which electron absorbed and emitted the photon. According to quantum theory, to calculate the result when any electron could have absorbed and emitted the photon, we must form a superposition of all the processes which could have taken place. The calculation in quantum mechanics takes the form of wave mechanics -- this does not have to mean that there is actually a wave, only that the mathematical theory behaves as though there was a wave. Since a wave would reflect at a particular angle, that is also what happens for photons.

The reasons for this strange quantum behaviour are deep and subtle. Quantum mechanics is actually a theory of probabilities, not a theory of physical waves. The mathematics tells us that the probability of a particular angle of reflection is actually a certainty; we can even show that this mathematical behaviour is necessary to a consistent probabilistic interpretation of quantum theory. It is much harder, probably impossible, to conceptualise exactly what is physically taking place at the level of elementary particles leading to these kinds of results.

The actual working-out on the level of single photons is quite tricky.

I can only give you this hint: There is a coherent interaction of the photon with a lot of electrons in the silver layer. Silver or aluminium coatings work, because these materials have free electrons (which is also why the are excellent conductors)

If there was (mainly) absorption+emission by single atoms and their electrons, it would not be a mirror but just an ordinary surface.

You are confused because you try to imagine mirror reflection as absorption and reemission. Though on this site, this might be helpful for certain reasons, in this case you need to understand that reflection is elastic scattering.

Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modified (by interaction with other particles and/or potentials). Furthermore, while the particle's kinetic energy in the center-of-mass frame is constant, its energy in the lab frame is not. Generally, elastic scattering describes a process in which the total kinetic energy of the system is conserved.


Now what is very important to understand is that you are asking two questions:

  1. why is the relative angle of the photons kept in mirror reflection

  2. why is the angle of the photons the same (mirrored) as the incident angle

The answer to 1. is elastic scattering, which is specular reflection. This is the only way to keep the relative energy levels and relative angle of the photons, and build a mirror image.

Specular reflection, also known as regular reflection, is the mirror-like reflection of waves, such as light, from a surface. In this process, each incident ray is reflected at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal in the plane formed by incident and reflected rays. The result is that an image reflected by the surface is reproduced in mirror-like (specular) fashion.


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Now the answer to 2. is the law of reflection.

The law of reflection states that for each incident ray the angle of incidence equals the angle of reflection, and the incident, normal, and reflected directions are coplanar.

Now the opposite is diffuse reflection, like a non-shiny surface, where the photons are re-emitted or reflected in random directions.