Wow. Sometimes work really is worth doing.

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.

6 thoughts on “Wow. Sometimes work really is worth doing.

  1. That is where I think that Maths for the sake of Maths falls down – our AMV course centres around most of the stuff in that Wikipedia article, but most of us will never get to see where it’s applied. Looking back over past years’ papers, they’ve taken more and more context out of the module each time it’s been taught. This doesn’t bother me enough to want to take Physics, but I can understand the satisfaction of seeing some seemingly complex theoretical mathematical relation in ‘the real world’ fall out so perfectly – Fourier series took on a new light when I looked up about how they relate to different sound tones in Music!

    On a not unrelated note, Green’s functions. Urgh…

  2. That is where I think that Maths for the sake of Maths falls down – our AMV course centres around most of the stuff in that Wikipedia article, but most of us will never get to see where it’s applied. Looking back over past years’ papers, they’ve taken more and more context out of the module each time it’s been taught. This doesn’t bother me enough to want to take Physics, but I can understand the satisfaction of seeing some seemingly complex theoretical mathematical relation in ‘the real world’ fall out so perfectly – Fourier series took on a new light when I looked up about how they relate to different sound tones in Music!

    On a not unrelated note, Green’s functions. Urgh…

  3. This is precisely why I’m a scientist. The laws of nature are beautiful and I think life is more than we will could ever understand. Nice post. :)

  4. This is precisely why I’m a scientist. The laws of nature are beautiful and I think life is more than we will could ever understand. Nice post. :)

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