– 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.
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