Previously, we introduced the spherical harmonics. We note that as written, the spherical harmonic is real for *m*=0, but complex for *m*≠0. However, one can find (normalized) linear combinations of and for *m*≠0 that are real, namely

for *m*>0 and for *m*<0. With , we have that the first few real spherical harmonics are:

Let us examine where these are zero. First, is a non-zero constant, and so is zero nowhere. For *l*=1, we have

, which is zero when , which is the plane ; , which is zero when , which is the plane ; and , which is zero when , which is the plane . Thus, when *l*=1, we will have a single surface (a plane) on which the harmonic is zero; such a surface is called a node.

We see that for , we have , which gives the cones and ().

when , or on the planes *x*=0 and *z*=0.

when , or on the planes *y*=0 and *z*=0.

becomes , which becomes the planes *x*=*y* and *x*=-*y*.

Lastly, becomes , which becomes the planes *x*=0 and *y*=0.

In fact, the real harmonics always have *l* nodes, all of which are vertical planes (constant *φ*), the plane *z*=0 (*θ*=π/2), or cones with the *z*-axis as their axis (constant *θ*≠π/2).

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Tags: Math, Monday Math, Nodes, Spherical Coordinates, Spherical Harmonics

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