clearscience:

If you want to calculate the position of a particle, in other words its mechanics, the equation you use is the Schrödinger Wave Equation. Say you have an electron, and it is oscillating back and forth a certain distance from a nucleus. This is an example of the harmonic oscillator, mentioned previously. Now, if it were a baseball on a spring oscillating (same idea), you could use some relatively simple equations on it—maybe ones you would see in an advanced high school physics class. Instead, the  Schrödinger Equation is required. Let’s explain some aspects of it:

What is H?: H is an operator, called the Hamiltonian, which describes the physics of the particle. Hamiltonians are a very fancy way to do physics, not peculiar to quantum mechanics. Basically, H will contain derivatives with respect to location.

What is E?: E describes the energy eigenvalues of the system. Long story short: there is not just one solution to this equation, but an infinite number of solutions. But they are regularly placed, and can be numbered “solution #1,” “#2,” etc. So we write E in such a way that we can enter in which solution we want (e.g. n = 0 for solution #1, etc).

Why is it a “wave equation”?: The quantity you have to solve for is Ψ (psi), and that is called the wave function. It gives not the location of the particle, but the probability the particle is at a particular place. The reason it’s a “wave equation” requires a little calculus or differential equations to understand, but here goes: H contains derivatives and E doesn’t. So when you have derivatives of something (Ψ) equalling that something (Ψ), the solution is often in the form sin, as in a sine wave. So, an equation that looks like that makes waves.

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