We do not see the lens through which we look.
~Ruth Benedict

I was recently asked to explain refraction using quantum mechanics. To really understand this on the quantum level requires understanding a field called “quantum electrodynamics,” which was invented independently by Richard Feynman, Sin-Itiro Tomonaga, and Julian Schwinger (and for which they all shared a Nobel prize). Unfortunately, I don’t know very much about quantum electrodynamics, so I can’t explain this the way a particle physicist or condensed matter physicist might. I can however, give a “pseudoclassical” model that was invented around the turn of the twentieth century…right before the discovery of quantum mechanics began. Like the Bohr model of the atom, it’s not quite accurate, but it gets the right idea across.

Light and Electromagnetism

The following discussion closely mirrors my discussion of Maxwell’s equations. If you remember this discussion, feel free to skip to the next section.

You’ve probably heard that electricity and magnetism can be described by the interaction of electric and magnetic fields. An electric field is just an arrow at each point in space that tells me where a charged particle (say, an electron) would move if I placed it in the area where the field was. Similarly, a magnetic field describes where a magnet would go. (Magnets are bit more complicated because there’s no such thing as “magnetic charge,” but the principle is the same.)

At first glance, it might seem like these fields are just theoretical abstractions. We imagine that these arrows permeate space so that they can tell us where an object affected by an electric or magnetic force might go. However, the experimental physicists Andre-Marie Ampere and Michael Faraday discovered otherwise.

Ampere discovered that he could create magnetic fields with electric current. (Such an electricity-created magnetic field is called an electromagnet.) We now call this principle Ampere’s law. Faraday discovered something oddly symmetric: that he could induce a current in a wire by moving a magnet near it. We now call this principle Faraday’s law, and it explains electric generators, electric motors, and railguns.

Ampere’s and Faraday’s discoveries mean that these strange invisible fields are very much real. They oscillate, warp, and shape each other. Indeed, these fields form the building blocks of light itself. In 1864, Scottish physicist James Clerk Maxwell expanded upon Ampere’s and Faraday’s laws. He hypothesized that magnetic fields aren’t just produced by electric current–they can be produced by any changing electric field. Similarly, Maxwell hypothesized that any changing magnetic field will produce an electric field. By combining these hypotheses with two more physical laws (Gauss’ Law and the law stating that magnetic monopoles cannot exist), he produced the founding axioms of classical electrodynamics. We now call these laws Maxwell’s equations.

By manipulating these equations, Maxwell discovered that these fields can feed into each other and become self-sustaining. A changing electric field produces a magnetic field, which changes and produces an electric field, and so forth. Moreover, when these fields oscillate like this, they behave exactly like light–meaning that light is a wave made up from these fields! (I am, of course, glossing over the fact that light is both a particle and a wave. From the quantum perspective, the electromagnetic wave describes the probability of detecting a photon at a given place and time.)

Atoms, Electrons, and Springs

As we know from Bohr, the atom is made from a positively-charged nucleus surrounded by negatively-charged electrons. In insulating materials, electrons are tightly bound to their respective nuclei. What happens if you were to somehow grab onto and gently pull one of these electrons away from the nucleus? Assuming you didn’t pull too hard before you let go, the nucleus would pull the electron back in, and it would oscillate around the nucleus like a mass on a spring.

Of course, the electron was (to a good approximation) orbiting in a circle around the nucleus, and it doesn’t stop orbiting after we perturb it. Because it keeps overcorrecting for the perturbation, the electron yo-yos back and forth between two elliptical orbits.

When light passes through a material, it pulls on the electrons in that material. (Electrons are, of course, affected by electric fields. And, as we just learned, light is partly made out of an electric field.) The electrons pull back on the light waves, damp them, and slow them down. How much the light is slowed down depends on how tightly each electron is bound to its atomic nucleus; this is why light travels more slowly in some materials than in others. The ratio of the speed of light in vacuum compared to the speed of light in a given material is called the “index of refraction” of that material. It’s denoted :

   

where is the speed of light in vacuum and is the speed of light in a given material.

Bending at the Boundary

So what happens when a beam of light enters a piece of glass? Glass has a higher index of refraction than air, so the beam will slow down. However, how the beam slows down is important. Let’s imagine that the beam hits the glass at an angle, as shown below. Part of the beam hits the glass first, before the rest of the beam, and thus slows down first.

Maxwell’s equations tell us what happens to the light in a complicated way. However, there’s a simple analogy that gives us a good intuition. Imagine that a plant stem is growing faster on one side than on the other. The faster-growing side needs more space and it pushes on the rest of the stem, causing the stem to bend. Similarly, the faster-moving light causes the beam to bend towards the slow-moving light. (WARNING: Take this analogy with a grain of salt–it breaks down easily! The way to really understand what’s going on here unfortunately involves a lot of math with electric and magnetic fields.)

An Alternative Way of Thinking About It

There’s another way to think about refraction. Imagine that a beam of light is emitted from point A, passes through a piece of glass, and then is detected at point B, as shown below. We know that light travels less quickly in glass, so we can try to figure out which path between the two points will take the light the least time. Unbelievably, this calculation will tell us that light bends just as we would predict from refraction. This method is called the calculus of variations, and it has a wide variety of applications in physics. Calculus of variations is the base technique of Lagrangian mechanics, which (along with its lesser-known cousin, Hamiltonian mechanics) offers an alternative to Newton’s method of solving physics problems. The idea is that every object moves to expend the least energy. (For experts: to make the action extremal.) This is called the principle of least action.

This is incredibly weird, because it means that the light somehow knows what the quickest path will be. Since this seems to attribute both intelligence and prescience to photons, it’s extremely unintuitive. Of course, we don’t actually believe that photons have either of these traits. However, the fact that calculus of variations works brings an important idea to light.

Because we are part of the universe, we can never really observe it and understand it from the outside. Our models and descriptions reflect our human nature and are limited by it. Every description or explanation we can come up with is based on human ideas, senses, and preconceptions. Calculus of variations is a beautiful example of this–it’s completely unintuitive and makes no sense, but it describes refraction just as well as Maxwell’s picture of electromagnetic boundary conditions. Which model is correct? Probably neither. Over time, we build a better and better picture of the world, but it will always be incomplete.

Further Reading

Refraction is covered in every undergraduate physics textbook without much effort to explain it intuitively. As a result, there aren’t many other resources available.

• Hyperphysics has an article on refraction.
• The online physics textbook has an article, too.

EDIT: QM has helpfully informed me that the Feynman Lectures on Physics has a very beautiful explanation of refraction. Feynman was famous for making difficult ideas seem intuitive and this is no exception. The relevant chapter is volume 1: section 26-2.

Questions? Comments? Hatemail?

If anyone can give a description of refraction from a quantum electrodynamics perspective, I’d love it if you could make a guest post. Shoot me an email! And, as always, if you have any questions, comments, corrections, or insults, please don’t hesitate to let us know in the comments!