Drinking Tea is a Lot Like Quantum Mechanics

One of my favorite subjects in physics is Quantum Mechanics. I'm going to tell you about a very famous quantum mechanics problem called the Finite Quantum Well, also called Particle in a Box.

If you have ever drank out of a cup with a flat bottom and vertical walls then you have observed this problem in real life.

If you pull out a Celestial Seasonings tea bag (Any kind of "Zinger" is an excellent choice as well as most of their herbal teas. I'm not much of a green/black/white tea kinda guy) and put it in hot water, here's something to do while it's brewing: Tap the cup and watch the water. You'll see different "modes". One mode that you'll see just has one bump in the middle, going up and down. That's the first order. Second order is one with two oscillating bumps to each side. It get's really complicated really fast because fluid mechanics is ridiculous, but I'll just talk about the first 2 modes.

When you tap the cup, you give the liquid a kinetic energy. If you give it too much energy then the water will splash out of the cup, burning you with scalding hot water. You'll scream in pain, such on your hand for a while, and maybe even go over to the sink to rinse it with cold water. The more energy you give the water, the higher order modes you will get and the higher the water gets.

The cup provides potential energy. The further out you go, the higher the potential energy gets. Since the energy goes up as you go out, you get a stable equilibrium, meaning the water stays in the cup pretty well. If you put a cup upside down it won't have a stable equilibrium anymore since the energy goes down as you go out. The only way for water to get out of the cup is if it has more kinetic energy than the walls have potential energy, thus causing the burning hot splash.

In quantum, instead of having water in a cup, you have a particle in a box. The box has potential energy walls just like the cup and keeps the particle inside the box pretty well. You have modes just like the water in the cup. The lowest energy mode says that you will most likely find the particle in the center of the well. The second order mode says that you will either find the particle on one side or the other, just like the two oscillating bumps.

The most significant difference between these two problems is that quantum says that there's a probability you will find the particle outside the box for a single particle in a box with thin walls. If you had a quantum well shaped exactly like a coffee cup sitting on a table, then eventually the particle in the well will leak out, which is called tunneling. In terms of the water in a cup problem, this is essentially saying that if your cup sits there long enough then the water will leak out the sides (note that I didn't say it would spill over the top, which is the only way for your tea to leave your cup).

It sounds crazy, but it's true. And don't worry, it won't be long till your tea is done brewing.

In quantum, particles are described by a "wave function." If you square the wave function you get the probability distribution of the particle. You have all seen a bell curve, so when I say probability distribution, think about a bell curve. A very famous equation called Schroedinger's equation tells you all the wave functions for a given quantum well. If you write out an equation for the shape of a coffee cup and plug it into Schroedinger's equation, then you can solve for the wave function of that particular well (assuming you know how to solve differential equations). If you did this then you would see a bell curve for the first order mode. The tails of the curve would extend outside of the well, telling you that there's a probability of finding the particle outside of the well. What I'll mention in passing is that if the walls of your coffee cup are much thicker then a few nanometers, then the probability is mind bogglingly low. I assure you that nanoscientists are currently working on building a coffee cup with 3nm walls, so pretty soon you can buy your own quantum well coffee cup.

So now you know what a quantum well and a wave function are. They're pretty cool. You may drink your tea now.