Contents
1. Introduction
In this tutorial, we’ll be introducing quantum mechanics in a simple, experimentally motivated way. From the apparently unremarkable fact that you can see through sunglasses, we will learn about the quantum nature of light, and end by exchanging quantum secrets and teleportation!
2. Fun with photons
According to Maxwell’s laws, light is a self-propagating electromagnetic wave: a changing electric field creates a changing magnetic field, which in turn gives rise to a changing electric field. It’s like a perpetual motion machine! Unlike their mechanical counterparts, this “machine” is allowed by the laws of nature. The wave-like nature of light explains common phenomena like rainbows, transparency, or the fringes you see when you shine a pocket laser at a thread of hair. But there is more to light than meets the eye.
2.1. Light is a particle
In the early 20th century, experiments began to suggest there was something amiss with this wave picture. Most dramatically, assuming light is a wave implies that a lump of hot coal emits an infinite amount of energy, and would destroy anything and everything around it. This is crazy! Clearly, this can’t be true, and Max Planck figured out that in order to get the right answer, light needs to come in discrete bundles of energy called quanta. For light of frequency $f$, these quanta or photons have energy
\[E = hf, \quad h \sim 6.34 \times 10^{-34} \text{ J s},\]where $h$ is called Planck’s constant.
A few years later, Einstein invoked photons to explain how light creates electricity. This photoelectric effect is the basis of solar panels, for instance. Briefly, the idea is as follows. At an atomic level, a chunk of metal looks like a rigid array of positive nuclei surrounded by a free-flowing sea of negative electrons. Electrons are not bound to any nucleus in particular, just to the chunk as a whole. To pull an electron out of the chunk, we need to give it some minimum energy $\epsilon_\text{bind}$ called the binding energy. We can create a sort of electrostatic hoover which sucks up any free electrons and tells us when they have been liberated.
We can deliver energy by shining a flashlight on the metal. Physicists expected this to work the same way ocean waves deposit energy on the shore. Imagine the electron as a beach ball sitting in a small dip in the sand. An arriving wave will continuously deliver energy to the ball until it gets enough ($\epsilon_\text{bind}$) to be dislodged from the dip. The waves could be very high, or arrive one after the other in quick succession, but it makes no difference to the beach ball. It just needs to receive enough energy to get out of the dip.
For light, the equivalent of the height of waves is the amplitude $A$ and the rate waves come in is the frequency $\omega$. In fact, the energy delivered is proportional to the intensity $I = A^2$. If light is like the waves trying to dislodge the beach ball, we expect that we could liberate electrons either by making $A$ big or $\omega$ big. But experimentalists noticed that, weirdly, for a given metal, below a certain frequency electrons would never be liberated, no matter what the intensity of the flashlight! This is very surprising, almost like a low-frequency tsunami failing to dislodge the beach ball. What gives?
Einstein’s explanation was simple. He took Planck’s idea that light carried energy in discrete lumps $E = hf$ depending only on frequency, and gives it to the electrons as a lump sum. The electrons will begin to flow as soon as the frequency $f$ satisfies
\[E = hf \geq \epsilon_\text{bind}.\]In this picture, the intensity is proportional to how many photons are coming out of our flashlight per second, so increasing intensity will just liberate more electrons. It’s like the waves at the beach are themselves made of beach balls, whose energy depends on the frequency of waves, and increasing the amplitude just stacks more balls on the wave. Balls can only collide one at a time, so even a tsunami wave of low-energy beach balls cannot dislodge the ball from the dip.
Exercise 1. Something?
2.2. Flashlights and sunglasses
Planck and Einstein teach us that light comes in discrete lumps. But there’s nothing to prevent these lumps from behaving like little, classical beach balls. We will see, however, that quantum mechanics is fundamentally different. To see how this happens, let’s remember that according to classical electromagnetism, light has an electric component which wobbles up and down. The plane in which this wobbling happens is called the polarization.
Although wobbling is a classical notion, it is still a valid property associated with a photon. Let’s set up a coordinate system with the $z$-axis coinciding with the path of the photon, and $x$ and $y$ perpendicular to it. The polarization makes an angle $\theta$ with respect to the $x$-axis, as pictured below. A polarizer is a filter which allows only certain polarizations to pass through. A simple example is the polarizing sunglasses that you can find at the chemist.
Hidden inside the phrase “only allows certain polarizations to pass through” is a revolution in physics. To see why, consider the photons coming out of a flashlight. The polarizations are random, for the simple reason that the flashlight just heats up a filament, and creates light using random atomic collisions. What should we expect when the randomly polarized light approaches the polarizer? If the polarizer only lets through photons with precisely the right polarization, then almost no light will get through, since the chance of getting this exact polarization is miniscule. We wouldn’t be able to see through polarizing sunglasses at all! Since we can see through sunglasses, something else must be happening. We can use intensity to help us understand.
Exercise 2. Remember that intensity tells us how many photons there are. You can install a lightmeter app on your smartphone that measures intensity, so you can tell how many photons are getting through. Now, let’s do an experiment!
(a) Measure the intensity of the flashlight, $I_0$. Pass it through the sunglasses, and measure the intensity, $I_1$. What is the ratio $I_1/I_0$?
(b) Now pass the light through two aligned pairs of sunglasses, and measure the intensity, $I_2$. What is the ratio $I_2/I_1$?
Your experiment should show that the intensity drops by half after the first pair, and should be approximately the same after the second. In other words, half the photons are blocked, and then most get through!
2.3. The law of Malus
Why do half get through, and then none? The simplest explanation for