## Monday Math 53

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).