# Reflection, at a quantum level?

1. Dec 14, 2011

Hello everyone, im new to this forum, and to physics really, so i apologize in advance for silly mistakes.

Bascially, im a keen A-level student, and i spend much of my spare time reading and trying to learn physics, far above me, it would be fair to say, i've jumped in at the deep end.

That aside, i was thinking the other day, and came across the idea of reflection at a quantum level, i've asked both my physics and my maths teacher (who has a PhD in nuclear physics) and none could come up with a useful answer. I was just wondering, exactly HOW does it work? I assume it has little to do with electron energy levels? Because photons released are of discrete energy levels?

I'd be most grateful if someone could try to explain it, without being too complex

Thanks

2. Dec 14, 2011

### PatrickPowers

It isn't that obvious, and it is a bit much to describe in a post. I'd recommend Richard Feynman's book QED, in which he goes into this in detail with no math. I had no idea how it worked until I read that.

3. Dec 14, 2011

is it not possible to sum it up? Include maths if it helps, I'm comfortable up to A-level further maths.

4. Dec 14, 2011

### Ken G

First of all, any time you ask a question like "what is really happening in situation X", the correct answer is always "no one can really say." But what you probably mean is "how does theory Y provide insight into what is happening in situation X", then we can give a good answer (probably a few!). Since you used the phrase "at a quantum level", I surmise that you want an answer involving the theory of quantum mechanics, but even then it's not so obvious what "theory Y" is here-- because we have ordinary quantum mechanics, and relativistic quantum mechanics, and quantum field theory, etc., and they all give different answers to your question! But I'll give an answer from the most ordinary quantum mechanical point of view I can.

Quantum mechanics basically takes the wave mechanics of classical physics and applies it to individual particles. There are a few differences that crop up when you do this, but I think the main concept here carries over pretty well-- the concept of interference. In classical wave mechanics, we have Huygens' principle, which says that every part of a wave acts like sources for how the wave evolves forward in time. Also, we have the "superposition principle", which says that every solution to the wave equation that comes from one source just adds up with the solutions that come from all other sources. This means in classical waves, the waves get a high total amplitude whereever there is constructive interference, and low total amplitude whereever there is destructive interference.

In quantum mechanics, we also have a superposition principle, except now it applies to individual particles. It basically says that anything that we can describe as something that can happen to an individual particle can happen in superposition, so what 'actually happens" is a kind of constructive sum over all these "possible happenings." This is the spirit of the Feynman path integral approach, for example. So using this mathematics, what has a high probability of happening is what receives constructive interference over this sum, and what has low probability is what destructively interferes. This is a key point-- each individual term in the sum starts out equally likely, even ones that correspond to absurd behavior, but the absurd behaviors cancel each other out, sort of like monkeys voting in an election.

In the case of reflection, we find that the presence of the mirror allows a certain behavior to receive constructive interference, which would not if the mirror were not there. The new behavior is "angle of incidence equals angle of reflection", and the mathematical property of that type of solution is that it exhibits "stationary phase." Stationary phase means the phase of the type of process (so how quickly the amplitude of the process varies over the set of very similar processes) is not varying over the physically allowed process, but does vary rapidly as soon as you test a non-physical process. Thus, the "angle of incidence equals angle of reflection" gives an extremum in the phase as you vary over a range of possible processes-- in this case, it is the minimum time to get between specified points A and B. Without the mirror, the only minimum time is the straight line between them. With the mirror, a second possibility emerges, the local minimum in time that comes from an equal angle of incidence and angle of reflection path between A and B that glances off the mirror. (It takes more time than the straight shot, but less time than all its neighboring paths, so it exhibits stationary phase and so constructive interference when you add the neighboring amplitudes.)

This still doesn't answer what the mirror is doing that allows for this new stationary phase solution. The classical answer to that is that the mirror acts like a source of waves that cancel the incident wave within the mirror, and constructively interfere to make a reflected wave. Quantum mechanically, the mirror creates a boundary condition on the photon wavefunction that forces the wave function to go to zero at the surface of the mirror, and this constraint suffices to give the reflected wave when you apply the superposition of wave functions analyzed in terms of all the modes that obey that constraint and have the energy of the incident wave. You could even do it with quantum field theory, and one way to picture that would be to say that the incident photon is destroyed by the mirror, but its energy must be accounted for, and since the mirror is elastic, it must use the energy to promote a "virtual photon" to the status of a real photon. What's more, the virtual photon not only has to have the energy of the original photon, it must also have a wave function that resonates constructively with the original photon-- in other words, it is indistinguishable from the original photon, so is ruled by the same wave function, and that wave function must experience constructive interferece (for all the above reasons) to have a reasonable probability of actually happening.

So you may be surprised to find there is not just one answer to your question, and there might be new possible answers in a few more centuries, but the key idea is constructive interference set up by the way the mirror is required to prevent the photon from crossing its flat surface, yet energy is also required to be conserved. Generally, we're happy if we have one way to think about it that gives good results and seems simple enough for us to understand.

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5. Dec 14, 2011

Wow! Thanks! Could take me a while to digest all of that, reading it once or twice might help too!

Thanks a lot though, that is VERY helpful.

6. Dec 14, 2011

### MrRobotoToo

In the case of reflection, the most appropriate approach is a semiclassical treatment in which the radiation is described classically using Maxwell's equations and the atoms are described quantum mechanically using Schrodinger's equation. The radiation then becomes a perturbative term in Schrodinger's equation.

There are two basic categories of reflection: reflection from metal surfaces and partial reflection by nonmetals such as glass.

In the case of metals, the mechanism goes as follows: the atoms in the metal are arranged in a regular lattice. This lattice is held together because the outer electrons are loosely bound and thus free to roam between atoms. When an electromagnetic wave impinges on the surface of the metal it sets the surface electrons into motion. The eddy currents thus established then reradiate the energy back in the direction from which it came.

The mechanism for nonmetals such as glass is somewhat different. In this case, the outer electrons of the atoms are not free to roam. What happens instead is this: when an electromagnetic wave impinges on the surface it sets the electron orbitals of the surface atoms into oscillation. Because the orbitals are oscillating, they reradiate the energy in all directions. Basically, the energy gets scattered. One can then reconstruct the reflected wavefront using Huygen's Principle. The oscillation of the orbitals can be visualized with this applet http://www.falstad.com/qmatomrad/

7. Dec 14, 2011

My understanding of an eddy current, is that it is simply a current induced by a change in a magnetic field? is that correct? If so, how does this change come about, and by what process, does the current, convert to energy, in the form of photons, of EM waves, if it's more simple?

8. Dec 14, 2011

### MrRobotoToo

Yes, you're right. I probably shouldn't have used the phrase 'eddy current' since it is more appropriately applied to the phenomenon you mention. But the surface currents that I wrote about are quite similar to eddy currents. They come about because an electromagnetic wave will set the local EM field into oscillation. It is these oscillations that set the surface electrons into motion. Because the changes in the local EM field are oscillatory in nature, so are the currents that they induce. An oscillating current means that there are charges that are accelerating/decelerating. Whenever a charge accelerates/decelerates it radiates energy in the form of EM waves. Information about the photons can extracted from the classical EM waves as follows: the probability of finding a photon at a particular point is proportional to the sum of the squares of the electric and magnetic field vectors at that point; the energy of any photon will be proportional the frequency of one of the vibrational modes of the EM wave.

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9. Dec 14, 2011

Okay, that makes a lot of sense, although, i need to spend yet more time reading it again and again! What field is this? Would it fall under QED?

10. Dec 14, 2011

### MrRobotoToo

Yes, it ultimately all boils down to QED. Although what I described is more of an approximation to QED. I would describe it as 'nonrelativistic semiclassical electrodynamics'. 'Nonrelativistic' because it doesn't factor in Einstein's principle of relativity, and 'semiclassical' because the atoms are treated quantum mechanically while the radiation is treated classically. But when you do factor in relativity and make all the entities quantum mechanical, what you end up with is QED.

11. Dec 14, 2011