# Can you explain Snells' law

## How are classical optical phenomena explained in QED (Snell's law)?

Hwlau is right about the book, but the answer is actually not that long, so I think I can try to make a few basic points.

### Path integral

One approach to quantum theory called the path integral is that you need to sum probability amplitudes (I assume you have at least some idea of what probability amplitude is; QED cannot really be explained without this minimal level of knowledge). through all possible paths the particle can take.

Now for photons the probability amplitude of a given path is exp (iK.L) where K. is a constant and L. is a length of the path (note that this is a very simplified picture but I don't want to get too technical like that that this is fine for the moment). The basic point is that you can think of this amplitude as a unit vector in the complex plane. So when making a path integral, add lots of short arrows (this terminology is of course thanks to Feynman). In general, for any given trajectory, I can find many shorter and longer paths so that we get non-constructive interference (you'll be adding lots of arrows pointing in random directions). However, there may be some special paths that are either longest or shortest (in other words extreme) and these will give you constructive intervention. This is called Fermat's principle.

### Fermat's principle

So much for preparation and now to answer your question. We will proceed in two steps. First we will give the classic Fermatian answer and then we will have to deal with other problems that will arise.

First, let's illustrate this with a problem where light moves between points A and B. in free space. You can find many paths between them, but if it is not the shortest it will not add to the path integral for the reasons given above. The only one who wants is the shortest, so restoring the fact that the light moves in straight lines. The same answer can be restored for reflection. For the refraction you have to take into account the constant K. The above mentioned depends on the refractive index (at least classically; we will explain later how it arises from microscopic principles). But here, too, you can only arrive at Snell's law by following Fermat's principle.

### QED

Now to current microscopic questions.

First, a refractive index arises because light moves more slowly in materials.

And what about reflection? Well, we're actually getting to the roots of QED, so it's time to introduce interactions. Amazingly, there is actually only one interaction: the electron absorbs photons. This interaction is again given a probability amplitude that you have to take into account when calculating the path integral. Let's see what we can say about a photon emanating from there A then hits a mirror and then goes to B.

We already know that the photon moves in straight lines between both A and the mirror and between mirror and B. What can happen in between? Well, the full picture is of course complicated: photon can be absorbed by an electron, then it is emitted again (note that even if we are over here **the** Speaking photon, the emitted photon is actually different from the original one, but it doesn't matter) then it can travel inside the material for some time, absorbed by another electron, re-emitted and finally fly back B.

To simplify the picture, let's only consider the case that the material is a 100% real mirror (if it were glass, for example, you would actually get multiple reflections from all the layers in the material, most of which would be destructive and you ' I would be left with reflections from the front and back of the glass; of course I would have to make this long answer twice longer :-)). There is only one main contribution for mirrors: the photon is scattered (absorbed and re-emitted) directly on the surface layer of the mirror's electrons and then flies back.

**Quiz question:** And what about the process where the photon flies to the mirror and then changes mind and flies back? B. without interacting with electrons; This is certainly a possible trajectory that we need to consider. Is this an important contribution to the path integral or not?

### Sklivvz

### Marek

### Marek

### unsym

### TheQuantumMan

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