The Double-Slit Experiment (quantitative) - Part 2

(click on the picture for a larger view)

Now that we know what a sine wave is, we can understand the double-slit experiment. I need to start with a few definitions that I probably should have put in the last post: the wavelength is the length between two successive peaks in the wave (often represented by λ) and the amplitude is the height of the wave (we'll call it A). There is a symmetry property of the wave; if you shifted the wave to the right or left by its wavelength, it would look exactly the same. In fact, you could shift the wave by any integer times the wavelength, and you wouldn't be able to tell the difference. This will be important later on.

To understand the double slit experiment, we need to ask what happens when two waves overlap. The answer depends on their "phase," or where each wave is in its oscillation relative to the other. For example, suppose two waves are perfectly "in phase," so that when one wave is peaking, so is the other. When you add these two waves together, you'll get a wave that is twice as big in amplitude.

What about when the waves are "out of phase" so that one is all the way up when the other is all the way down? In that case, the waves destructively interfere so that the addition contains no wave at all.

This interference is the key to the double-slit experiment and allows us to predict the shape of the light pattern on the screen. When the light impinges on the slit, the waves that come come out the other side are initially in phase. If you look at the the screen directly across from the slit, you would see a dark spot. However, that bright spot will be banded by bright spots, which will in turn be banded by two more dark spots in a fringe pattern. To understand why, let's zoom in on the slit right where the light passes through (on this lovely diagram I stole from here).

Here, the distance between the slit and the wall is L and the slit separation is d. Where on the wall do you expect to see bright or dark spots? If we want there to be a bright spot on the wall, then we know the two waves must interfere constructively (the first case discussed above) or be in phase. A dark spot will appear when the waves are out of phase and interfere destructively. Let θ (again, angles are always θs) be the angle between the horizontal and the position of a given fringe on the wall). If we look high on the wall, the light that came out of the top slit doesn't have to go as far as the light that came out of the bottom slit (in other words, r₁ is bigger than r₂). This means that the bottom ray of light will have more time to trace out its wavelength and will drop out of phase with the top ray of light. The extra distance traveled by the bottom ray is equal to d*sin θ (remember that the sine function also related the sides of a triangle and notice that the light paths r₁ and r₂ form a triangle with the slit). Now, remember the symmetry of the wave - a wave that is shifted by its wavelength looks the same. So if the extra distance traveled by the bottom wave equaled exactly its wavelength, it would look identical to the top wave, and the waves would interefere constructively - a bright spot would appear on the wall. If, on the other hand, the extra distance traveled was exactly half a wavelength, so that the bottom wave had just enough time to get out of phase, the two waves would interfere destructively and a dark spot would appear on the wall.

This is exactly what happens - bright spots appear if d*sin θ = λ or some multiple of λ, while dark spots appear if d*sin θ = λ/2. Wave properties predict exactly the patterns that appear in the double-slit experiment, confirming that light is like a wave.

What is the nature of light?

The first non-introductory post was on the Doppler effect, which is relevant to dark matter because of its applications to galactic rotation curves and the evidence those curves provide for the existence of dark matter. I figure I can keep mining this particular bit of evidence for a series of posts explaining exactly how we measure galactic rotation curves and why that implies the existence of dark matter. To do that, I need to start with one of the mysteries of physics, the "wave-particle duality." I expect I'll need a few tries at explaining this (since I have some trouble understanding it myself), but we'll see how it goes.

A Google search for the wave-particle duality will give you many interesting discussions and it has been commented on throughout the years by people who are eminently smarter than I. Therefore, I will try not to be too clever, but instead just describe qualitatively what is going on. Essentially, fundamental particles like photons, electrons, etc. behave as both waves and particles. The debate between these two views of the world goes back as far as Newton (who suggested light was "corpuscular") and Huygens (who thought it was a wave), and the problem lies in the fact that there is evidence for both.

Light as a wave
In the 1800s, Young and Fresnel performed a series of double-slit experiments. In these experiments, a light source is directed towards a plate with either one or two very thin slits in it, and the experimentalist observes the resulting pattern that forms on a screen behind the slits. In the single-slit case, the picture that forms on the screen shows a diffraction pattern (i.e. alternating bright and dark spots) instead of a single bright spot as you might expect if the light was a point-like particle. In the double-slit case, the resulting pattern shows signs of interference (I got the nice illustration from the Wikipedia entry, which is very good as usual).

Both of these results can be explained if one thinks of the light as a wave - if a flat wave (like waves at the beach) comes across a barrier with a small hole, the part of the wave that passes through the hole begins propagating outward in all directions, with circular wave fronts (you can test this in a bathtub if you are interested). Therefore these results from experiments performed in the early 1800s seemed to prove definitively that light was a wave.

Light as a particle
Unfortunately, Einstein came along and in one year, 1905, published three of the most famous papers in physics history. And while his work on special and general relativity are monumental accomplishments, in the end, he received his Nobel prize for a theoretical description of the photoelectric effect that showed that light is a particle.

The photoelectric effect is the phenomenon that when light is shone on certain metals, the metal emits electrons. Naively, using the wave theory of light, one might think that the brighter the light (i.e. the bigger the wave), the more energetic the freed electrons would be. In fact, the energy of the electron is independent of the intensity of the light (although brighter light does correspond to more electrons). Instead, it is the frequency of the impinging light that determines the energy of the freed electron, and if the frequency is low enough, no electrons are freed at all, no matter how much light is blasted against the metal.

As we've already discussed, light has wave properties. It's wavelength, the distance between two wave peaks, determines the color of the light, which is how we see the many colors of the rainbow. It turns out that violet and blue light have shorter wavelengths while red light has a longer wavelength. Frequency is essentially how often a wave passes a given spot, and since the speed of light is constant, the frequency is inversely proportional to the wavelength, so violet and blue light have higher frequencies than red light. Therefore, by the photoelectric effect, one can imagine a metal for which the smallest amount of blue light would free electrons, but a Sun's worth of red light would have no effect.

Einstein figured out this dilemma by building on an idea of Max Planck's from a few years earlier. Basically, he suggested that light was actually a discrete particle (a quantum, or photon) with an energy proportional to its frequency (for the purposes of dark matter, this is really the only statement that you need to take away from this long discussion on light - it's energy is proportional to its frequency, which is in turn inversely proportional to its wavelength). If the photon has enough energy, an electron is released from the metal; if the photon does not have enough energy, then no electrons are released, no matter how bright the light. This proved that light was a particle.

Everything is both wave and particle
In the end, physicists have had to accept that light seems to have both properties. One might think that the slit experiments can be explained if the light was a wave made of multiple photons (like a water wave consists of many water molecules). However, experiments in the 20th century showed that even when the intensity of light hitting the slits was reduced to individual photons, the interference patterns still emerged. This is really, really weird. From our macroscopic experience, we would think that a single photon would pass through one slit or the other. Instead, it seems as if the individual photon actually passes through both. Basically, our physical intuition loses its way; individual light quanta are both waves and particles.