The Quantum Spring.

Sorry for the delays! As it turns out this was becoming challenging to make an animation for, but I finally got it!

Alright, so here it is, the quantum spring. First and foremost, this is one of the more useful solutions to Schrodinger's equation.  Why? As it turns out, a lot of things behave like little springs! Anything that is held in place by some kind of force vibrates very slightly, the physics that describes that is the quantum harmonic oscillator. This means the quantum spring can be used to make calculations about solid materials. Understanding materials improves our every day lives!

Just as all quantum mechanics, it comes from a normal physical theory made weird by Schrodinger's equation. Remember the potential energy part? Well here's where it comes from in the case of the spring. With the infinite square well, I just said that it would take infinite energy to penetrate the walls, but in this case, I'm tying my particle to a little spring. This could be describing an electron wiggling around near an atom, or anything else that is small and vibrating! If you've ever studied physics in high school, then you've surely heard of Hooke's Law. This is a physical law that describes the motion of something attached to a spring. It simply says:

F = kx

The force on the particle is proportional to the distance it is from where it feels no force.

If this is confusing, think about this. Say you hang something from a spring. It'll bounce up and down going above and below a center point. We call this point the equilibrium point. I know you've all seen it in real life, but here's what I'm referring to.

Now what happens to the quantum picture? Well, we take Hooke's law and using some fancy math, turn it into a potential energy equation that we can shove into Schrodinger's equation. When we solve the equation, we find out what happens to a particle that we attach to a spring. I made these videos with Python using scipy and matplotlib, two open source modules that are easy to use and very fast! Feel free to ask me if you want my code, all my code is open source. Here's my videos, I added color this time! What you will see is the particle bouncing back and forth due to the spring pushing and pulling it.

Probability Density - Remember, I started the particle out as a bell curve, meaning I'm guessing it's right about the center of the peak, but there's some uncertainty in its position.

Wave Function. The "real" part is blue, the "imaginary" part is red.

Pretty trippy, eh?

These don't really get old for me. I think quantum mechanics is cool, and as we go on, it'll get weirder and cooler (for me, and I hope, for you.). Can anyone name some possible applications for a particle on a tiny "spring"? I've intentionally left some out. I'll give you a hint, your gps wouldn't work nearly as well without it!

Also, next up: Expectation Values, Delta Potential, Quantum Tunneling, and Quantum Theology (yeah, you heard me correctly!)