Schrödinger's Equation - Part 2

Okay, so last post we briefly introduced Schrödinger's equation. Mainly, we introduced the left hand side, which consists of the Hamiltonian, and the wave function. As a reminder, here is the equation once more.

On to the right hand side!

To begin, let's look at the first letter. This is i, which is commonly called the imaginary number. This is because it's defined as a number whose square is equal to -1. You can't go out into the world and measure an imaginary number, never the less, physics is full of them. If you aren't used to complex number, just ignore it, it really isn't THAT important to understand them. Next to the i, we see the familiar "hbar". Recall hbar is a very small number and only appears in quantum mechanics. It's not important to know much about it, but it's important to the calculations and a very small number (but it's just a number, nothing more). d/dt is the next part of the equation. This is something called a derivative. Essentially it tells you something about how fast something changes with time. the i, hbar, and d/dt is called the time evolution operator. Recall how I said you could write a quantum state as something inside brackets (e.g | something >). Well, in this case we define the quantum state as psi, that trident looking letter. the (t) just means that it's different at different times.

So let's sum up what the equation means. On the left we have the Hamiltonian operator (don't worry about the word operator, it isn't critical to the understanding.) This "acts" on the wave function, or quantum state, and this is equivalent to propagating the wave function forward in time. All the equation does is take an inital state and, depending on what the system is, it moves it forward in time!

This may sound really complicated, and believe me, it is. This is why this equation is so important. This equation tells us how to get the best guess at where a particle is, and what it's doing at any future time!

Let's talk for a second about the classical physics version. Let's define our system as me, the earth, and a ball.

 

Look familiar? I recycle.  When I drop the ball, it falls, and with some fancy calculus, I can figure out where it is and how fast it's going (or it's momentum) at any time. This is the classical analog to solving Schrödinger's equation. 

The difference is, what I get out of solving Schroedinger's equation is psi, the quantum state (or wave function).  So what is psi? To tell you the truth, no one knows. I know, this saddens me too, but there is some good news! If you square psi, you get the probability density for the state! Using this, we can calculate the probability densities for position, momentum, and well as the average location of the particle, and other cool things. 

So what else is this stuff good for? Why should YOU care about quantum mechanics?

Well, the next article is going to go through the most simple example in quantum mechanics. The system is a particle that is stuck on a line, trapped in one dimension. Imagine this as a wire, and the two ends are fixed. The particle can't go past the ends, so we say there is extremely high potential there (actually, we say it's infinite!). Think of this as a big huge wall at the end. This sounds like the results would be simple, and in the grand scheme of things, they are, but it's actually a lot cooler than it sounds. At least, I think it is!

But to why you care... Well, the simple example I'm going to show in the next post can be extended to a particle that is trapped in a box (instead of just a wire). This is often used to model simple metals! In the future, after we've developed quantum mechanics a bit more, I'm going to show how to calculate properties of materials with quantum mechanics. Using a slightly modified version, I'm going to discuss a simple model of semi-conductors, which are what all of our modern electronics are based on! 

As always, comments are encouraged, see you soon!