– Maths Wizards – What is/are Spherical Harmonics?
In: Mathematics
This is going to be ELI have taken basic linear algebra, not quite ELI5:
The way that I like to frame it is that the spherical harmonics are a particular set of basis functions that span the set of all angular functions (functions on a sphere). A proper mathematician can probably qualify my use of the word *all* – I don’t recall if it’s basically all smooth functions, or if it’s a more subtle point.
What’s useful about the spherical harmonics is that they are eigenstates of both the angular momentum operators L^2 and L_z. L^2 describes the total angular momentum and L_z describes the angular momentum in a specific direction.
In quantum mechanics, we understand eigenstates of an operator to represent wavefunctions with a definite value of the observable corresponding to that operator. This means the spherical harmonics describe wavefunctions with definite values of the quantum angular momentum numbers, usually denoted *l* and *m* (corresponding to L^2 and L_z, respectively).
Now go back to the first point: *any* angular function can be written as a sum of spherical harmonics (basis states). Again, I’ll let a proper expert qualify that “any”. But we conclude from this that, in general, we can describe the solutions to a spherically symmetric potential like the hydrogen atom, as a quantum superposition of angular momentum states. It’s often useful to decompose it into these states for related calculations, and we use the spherical harmonics to do so.
If you pluck a guitar string, and watch it while it vibrates, the string moves too fast for you to see and it looks all blurry.
The shape of that blur is a nice little curve. (It’s actually a sine curve). There are two points that aren’t vibrating, those are the two ends of the string.
You might know, if you play guitar, that you can make the string vibrate at a higher pitch, by gently touching the middle of the string when you pluck it.
Then, the blur has a different shape. It’s still a sine curve, but now there are three points that aren’t vibrating: the two ends of the string, and the middle.
By gently touching different points on the string when it’s plucked, you can get it to ring at higher and higher pitches, and each pitch gives a different shape to the blur.
These special shapes are the harmonic functions for the string. They are the natural ways the string vibrates, and they are important in a whole lot of other areas of physics, for example if you wanted to model heat flow along the string, or quantum mechanics in a quantum particle confined to the string.
The same principle applies to higher dimensional shapes, like the surface of a drum. Or the surface of a sphere.
If you could somehow hear the vibrations of a soap bubble, you could get different pitches by having it vibrate in different patterns. The patterns of vibration for these pure tones are called “spherical harmonics”.
Just like for the guitar string, spherical harmonics are also useful when analyzing heat flow on the surface of a sphere, or the behaviour of spherical quantum objects such as hydrogen atoms.
Edit: just for fun, here’s a cool video showing the harmonics of various Tetris pieces: https://youtu.be/Qh3lRZFnUJ4?si=LhnFqGOpiqzZ0VKg
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