$egingroup$ The math behind the proofs linked in the above comments is somewhat straightforward; I'm not sure you're saying what you typical with 'prove this (mathematically)'. It's much more likely that you're hoping for a derivation from first principles and an essential characteristics that light and also matter; wording which suggests the exact same would be helpful. $endgroup$

This is beautifully explained by Feynman using his route integrals.

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I cannot hope to do it better, but just a rapid non-mathematical overview. What is mind-blowing about the theory is the you assume the individual photon (on quantum electrodynamics level) is in reality "reflected" in *each* possible direction through *each* atom that the winter surface. If girlfriend calculate how all this "reflections" interfere with each other, girlfriend will view that the wouldn"t result in chaos, due to the fact that most that them have tendency to silence each other, *except* for one calculation angle. The silencing is due to the fact that depending on time of each possible path, the phases can be opposite at a place. According to the concept it way that the photon wouldn"t probably appear there. What is an excellent about it, is that "summing" (integrating) the phases of all these zillions routes doesn"t call for a supercomputer, yet can be done in couple of minutes by drawing little pictures ~ above a blackboard - see the video.

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answered jan 3 "19 at 17:06

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ACuriousMind Indeed. However a ray at -5° would certainly not it is in reversible through time ~ above a plane surface. It is the symmetry the underlies all the proposed mechanisms (of i beg your pardon this answer was the best). $endgroup$

–user137289

jan 6 "19 in ~ 9:58

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The price by harshit54 is very concise and clear and gives friend answers in multiple layers of understanding. However, to quote Leonardo DiCaprio: *we must go deeper*. Not because we must, but because we can! There"s a TL;DR below.

A beam that light deserve to be assumed of as a currently of energy packets (photons, which are the quanta of irradiate - too many of amazing words to look up in the dictionary already). Currently let"s zoom in and also look at what happens once the photon hits any type of material. That runs right into a wall surface of atom - several nuclei surrounded by electrons (also power packets - there"s an ext to it but let"s not compose out all of quantum mechanics here). Once a photon hits an electron, its energy gets took in and the electron goes right into a higher energy state. This does not last long; the electron left an "empty" power state listed below it, i beg your pardon is one energetically more favorable position for it. Thus, there is a possibility that it spontaneously jumps ago to a lower energy state. This chance rises over time, therefore it"s pretty specific that it will jump back quite quickly. Once it does, it needs to eliminate its extra energy. This energy is released as a photon!

If this is the only electron in the neighbourhood the releases a photon, it will certainly go in any random direction. HOWEVER! There"s a catch. *Hint: this is where the wave nature of irradiate comes right into play*. Let"s assume that the beam that light hits the showing surface directly from above, for this reason the angle of incidence is 0 degrees. Currently you have plenty of electrons that are being bombarded by even more photons, all emitting photons in numerous directions. The photons that room emitted at an angle however, will be the end of phase with eachother (since there is a distance in between the electrons, if 2 photons space emitted at any type of angle at the exact same time, there will be a slight delay between them). Photons that are out of phase will tend to cancel eachother out. Photons that are in step (all the photons that are emitted upwards) will constructively interfere v eachother.

Now something interesting happens - other that also explains why lasers work. **When one electron emits a photon, and also there are countless photons roughly it who all have actually the exact same phase and direction, the emitted photon will certainly copy the phase and direction of the photons about it!** So an extremely quickly, all photons that space emitted in arbitrarily directions die out and also only photons that space emitted perfect in phase with eachother remain.

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Now tilt her light beam in ~ an angle. No much longer the photons that space emitted upwards are in phase through eachother, but only the photons that are emitted in ~ the exact same angle together the incident photons are in phase. So they remain!

So why walk this not happen at any type of surface? Well, the above only uses to surfaces with lots of electrons, discovered in materials where electrons are complimentary - for instance metals! Surfaces where all electrons are bound will not absorb the photons instantly - they"ll pass through the first few layers of atom unhindered until by possibility they room absorbed. As soon as a brand-new photon is emitted, it will run right into other atoms (it"s not at the surface ar anymore!) and keep the reaction going until at the surface, photons room reflected in random directions. Integrate this through the reality that without free electrons that is VERY daunting to smooth a surface, that will give you no chance for a kind (specular) reflection.

**TL;DR**

Photon power is absorbed by electron

Energy is emitted by electron in the form of a new photon

Photons that room out that phase with eachother die out

Only photons that are emitted at the same angle as the edge of incidence are in phase