So today I sat down with the intention of figuring out how to solve ∇²u = 0, otherwise known as the Laplace equation, in spherical polar co-ordinates. Because it’s part of my course.
It may sound as a task somewhat obscure, but it’s really not. It governs any kind of potential, like gravitational, or fluid, or electrical, whatever.
Solving the equation in spherical polar co-ordinates gives insight into any problems in which potentials are important in a spherical environment, like the hydrogen atom. As it turns out, the various solutions to this equation are what create the energy levels in atoms, what makes a metal like copper behave differently from a gas like argon. It’s kinda fascinating that you are just going in solving this equation, and this kind of really fundamental stuff just leaps out of the mathematics.
Like the basis of energy levels is that a component of this differential equation has a series solution, a long chain of terms. If this chain of terms is allowed to go off to infinity, it’ll be unbounded – the sum of the series will itself be infinite. So you have to impose an artificial cut-off to the sequence for the solution to exist. The series of terms has to be finite. The really odd part is then this cut-off number, known as L, actually is something physical.
If you ever studied chemistry, you’ll know about s, p, d, and f orbitals, and how different numbers of electrons can fit in each. Well, if an electron is in the p orbital, then the L number I mentioned is 1. d, the L number is 2. You can probably guess what f is!
The reason that chemistry is the way it is all falls out of the solutions to this kind of equations. That really boggles my mind that the way the world is seems to be an inevitable result of the equations that govern it. Amazing.