Schrödinger's Equation - Part 1

This is the start of where all the real quantum mechanics happens. Schrödinger's equation equation is what we use to calculate useful information about a quantum object or system of objects. Here it is...

 

Don't be afraid of it, it is difficult to solve, but not too difficult to understand. Through the next couple articles, I'm going to explain what it means, and why you should care.

Before go into details, I should say a thing or two about what this gets us. See the symbol in the equation that looks a little bit like a trident? That's the Greek letter psi. That symbol represents what's called the wave function.  The wave function is what allows us to calculate the probability density of a system. This allows us to calculate stuff, like momentum, position, energies, and other exciting quantities that will make much more sense as I develop this topic. Let's examine this equation.

Let us begin. Starting from the left hand side of the equation, you see H(t).  This is called the Hamiltonian. Remember back in our first post on quantum mechanics we defined a quantum mechanical system? Well, this is how you define it mathematically. Basically, the Hamiltonian represents how the energy in a system is changed and transferred. At a fundamental level, there are two types of energy. The one we are most familiar with is called kinetic energy. This is the energy of motion. Anything that is in motion has kinetic energy. Mathematically, kinetic energy is defined by the momentum squared (multiplied times itself) divided by twice the mass of the object in motion. Kinetic energy is generally written with a capital T (and momentum with a p).

The other form of energy that is important is called potential energy. You experience this every day as well, but it isn't as tangable. The best example I can think of of potential energy is a spring, or possibly a magnet. If you take a spring and compress it (or bring to north poles of a magnet near each other), you are increasing the potential energy of a system. You press harder and harder on the spring. The spring now has a lot of potential energy (I'm just going to say potential when I mean potential energy). 

 

Of course, what happens when you let go of a spring? 

 

So, as it turns out if I have a fixed amount of energy in my system, it can be transferred between potential and kinetic energy. Make sense?

Now, you may ask, what the hell does this have to do with quantum mechanics? Aren't we kinda off track. As it turns out, we really aren't off track at all. The kinetic energy term in the Hamiltonian pretty much always takes the same form, and the potential term is how we define a system. Take, for example, our spring. This is known as a simple harmonic oscillator (just a fancy way of saying spring.) Interesting, many many real world potentials can be approximated by a spring as long as there isn't too much motion. The spring is crucial to many quantum mechanics problems! Now, when I say potential, there's a lot I can mean. Consider the system that includes just me. If I wrote the hamiltonian for me, it would have a T (kinetic term) in it (meaning the Hamiltonian would just depend on how fast I'm going.). If I add a potential term (the spring), now the more I compress the spring, the less kinetic energy I have, and the more potential the system has. Generally, the potential terms with be due to something like another atom causing an electric force on an electron. 

That's essentially what the Hamiltonian (H(t)) is. The mathematics behind it is a little tricky, so I won't go into it. If you're curious, feel free to ask questions and discuss!

My next article will take a look at what the right hand side of the equation means. Before I end, I'm going to go a little into why you should care. As it turns out, knowing psi, the wave function tells you a lot about a system. Let's say our system is an atom. If we solve for psi, we automatically get with it the eigenvalues, which is a fancy word for the energies of the atom. If you know the energies of an atom, it tells you exactly what colors of light the atom will emit if heated up. Ever wonder why certain things burn green? Quantum mechanics has the answer. I'll go into much more detail later, and there is much MUCH more that quantum mechanics has to offer.

If you have any questions or comments, please feel free to post! I like feedback.