More on the "Particle in a Box"

The infinite square well problem previously discussed as typically called the "particle in a box" problem. This doesn't really make much sense in terms of the previous problem as it's more like a "particle on a line". Well, this problem is called the particle in a box problem because that's exactly what it describes and is used for. In this post, I will talk about one of the applications of this simple model. But first a video. Remember in the last post how I had the particle sitting there, it's it's probability distribution kinda melted? Well, here's the same thing, but now the particle is confined to a flat sheet instead of a wire.

Now, I find that really cool. You can really see the wavelike nature of the particle bouncing off the walls and running back into itself! It kinda looks like ripple on a pond, but water waves are made out of trillions of parts, but this is just one! What happens if I give it some initial momentum? Take a look.

The wave function also looks like this. Here's a couple more videos, one moving fast, and one moving slowly.

Here's the slow mover.

So what are these good for? Why do we care how a particle trapped in a box behaves besides the fact that it looks cool?

Well here's the biggest application I know of. Have you ever studied the ideal gas law? If you haven't, don't worry. It's a rather simple relationship between pressure, temperature, and volume.

PV = NkT

If you've seen it in the past, that's probably written a bit different than you're used to. This is how physicists express it, and since I'm a physicist... Anyway, P stands for pressure, V for volume, N for number of particles, k is called Boltzmann's constant, and T is absolute temperature. This equation says if you put a gas, like air, into a container you can calculate the pressure if you know how much air is in there, the temperature, and the volume. Also if you know the other three variables, you can figure out the last one. This equation is used a lot in many engineering and science fields.

Since I bring this up, was this equation originally derived from quantum mechanics? The answer is no. It wasn't. So why do I bring this up? Read on. This equation can be derived by experimental measurements on gases that are nearly "ideal".  An ideal gas is essentially a gas that doesn't affect itself, even when it is smashed down to a small area. Gases like Helium, or other noble gases, are very well approximated by the ideal gas law in certain cases. Using a gas such as this, a experimental physicist or chemist can derive this law. This law can also be derived using classical physics methods called "statistical mechanics", which uses probability and statistics to determine the properties of large collections of stuff. The reason I bring this up though is that you need quantum mechanics to get a full thermodynamic description of an ideal gas. That is, you need quantum mechanics to accurately derive its entropy. Entropy is a measure of the disorder of a system. Once you have the entropy, you can calculate other thermodynamic properties. I think the most remarkable thing you can do is re-derive the ideal gas law! Once you have the entropy, it's relatively easy to get the ideal gas law out of it. And where does all of this come from? That's correct, the particle in a box problem. Next time you think you don't deal with quantum mechanics always, think again, it's everywhere!

That's it for the infinite square well for a while, I may revisit it in the future, as it's such an important problem. Next up, I'm going to talk about something a bit more current and down to earth, stay tuned!