I’ve mentioned before that I’m answering the occasional question over at the Physics Stack Exchange site, a crowd-sourced physics Q&A. When I’m particularly pleased with a question and answer, I’ll be promoting them over here like, well, now. Yesterday, somebody posted this question:
Consider a single photon (Î»=532 nm) traveling through a plate of perfect glass with a refractive index n=1.5. We know that it does not change its direction or other characteristics in any particular way and propagating 1 cm through such glass is equivalent to 1.5 cm of vacuum. Apparently, the photon interacts with glass, but what is the physical nature of this interaction?
I didn’t have a ready answer for this one, but I’m pretty happy with what I came up with on the spot, so I’ll expand on it a little bit here. I think it’s an interesting question not only because the issues are a little bit subtle, but because it also shows the importance of understanding classical models as well as quantum ones. The key to understanding what’s going on here in the quantum scenario is to recognize that the end result is the same as in the classical case, and adapt the classical method accordingly.
So, how do you explain this classically, that is, in a model where light is strictly a wave, and does not have particle character? The answer is, basically, Huygens’s Principle.
To understand the propagation of a wave through a medium, you can think of each component of the medium– atoms, in the case of a glass block– as being set into motion by the incoming wave, and then acting as a point source of its own waves. In the picture above, you can see that each of the the little yellow spots in the gap in the barrier is at the center of its own set of concentric rings, representing the emitted waves.
When you work this out, either by drawing pictures like the above, or by doing out the math, you find that these waves interfere constructively with one another (that is, all the peaks line up) in the forward direction, but that the waves headed out sideways to the original motion will interfere destructively (the peaks of one wave fall in the valleys of another), and cancel out. This means that the light continues to move in the same direction it was originally headed.
When you work out the details, you also find that the wave produced by the individual point sources lags behind the incoming wave by a small amount. When you add that in, you find that the wave propagating through the medium looks like it’s moving slightly slower than the wave had been moving outside the material. Which is what we see as the effect of the index of refraction.
This model of light propagation through a medium is fantastically successful, so our quantum picture should reproduce the same features as long as you’re at a frequency where quantum effects don’t play a role. So, how do we carry this over to the quantum case, thinking about light in terms of photons?
This is a tricky question to answer, because in many ways it doesn’t make sense to talk about a definite path followed by a single photon. Quantum mechanics is inherently probabilistic, so all we can really talk about are the probabilities of various outcomes over many repeated experiments with identically prepared initial states. All we can measure is something like the average travel time for a large number of single photons passing through a block of glass one after the other. We can come up with a sort of mental picture of the microscopic processes involved in the transmission of a single photon through a solid material, though, that uses what we know from the classical picture.
To make the classical picture quantum, we say that a single photon entering the material will potentially be absorbed and re-emitted by each of the atoms making up the first layer of the material. Since we cannot directly measure which atom did the absorbing, though, we treat the situation mathematically as a superposition of all the possible outcomes, namely, each of the atoms absorbing then re-emitting the photon. Then, when we come to the next layer of the material, we first need to add up all the wavefunctions corresponding to all the possible absorptions and re-emissions.
Thus, we more or less reproduce the Huygens’s Principle case, and we find that just as in the classical case, the pieces of the photon wavefunction corresponding to each of the different emissions will interfere with one another. This interference will be constructive in the forward direction, and destructive in all the other directions. So, the photon will effectively continue on in the direction it was originally headed. Then we repeat the process for the next layer of atoms in the medium, and so forth.
It’s important to note that when this picture is valid the probability of being absorbed then re-emitted by any individual atom is pretty tiny– when the light frequency is close to a resonance in the material, you would need to do something very different. (But then, if the light was close to a resonant frequency of the material, it wouldn’t be a transparent material…) while the probability of absorption and re-emission is tiny for any individual atom, though, there are vast numbers of atoms in a typical solid, so the odds are that the photon will be absorbed and re-emitted at some point during the passage through the glass are very good. Thus, on average, the photon will be delayed relative to one that passes through an equal length of vacuum, and that gives us the slowing effect that we see for light moving through glass.
Of course, it’s not possible to observe the exact path taken by any photon– that is, which specific atoms it scattered from– and if we attempted to make such a measurement, it would change the path of the photon to such a degree as to be completely useless. Thus, when we talk about the transmission of a single photon through a refractive material, we assign the photon a velocity that is the average velocity determined from many realizations of the single photon thought experiment, and go from there.
The important and interesting thing here is that the effect that we see as a slowing of a particle– a photon taking a longer time to pass through glass than air– is actually a collective effect due to the wave nature of the photon. The path of the light is ultimately determined by an interference between parts of the photon wavefunction corresponding to absorption and re-emission by all of the atoms in the material at once. And since we know the photon has wave characteristics as well as particle characteristics, we can use what we know from classical optics to understand the quantum processes involved.
This is, as I said, an explanation invented on the spot yesterday, when I started thinking about the question, but I think it’s fairly solid. As always, if you see a major hole in it, point it out in the comments.
And if you have physics questions, I encourage you to take them to the Stack Exchange site. I’ve got dozens of other things I’m supposed to be doing, so I won’t necessarily have time to address specific questions, but that’s the beauty of the crowd-sourced option– there’s bound to be somebody out there who isn’t too busy to answer…