Spherical Harmonics are obtained by solving schrodinger equations with spherical symmetry, for example: hydrogen atom.

which has the following solution:

The following figures are illustration of probability density of some low energy orbitals.

I use gnuplot to draw these functions, here is an example:

reset
set term pngcairo font "Arial,60" size 1600,1600
set output "dx2-y2.png"
set style line 1 lc rgb '#157545' lt 1 lw 0.6 # --- green lines
#The following color settings are from: http://www.gnuplotting.org/klein-bottle/
set pm3d depthorder hidden3d 1
set hidden3d
set style fill transparent solid 0.65
set palette rgb 9,9,3
unset colorbox
unset autoscale
set parametric
set angle degree
set urange [0:360]
set vrange [0:180]
set isosample 100,100
set ticslevel 0
set lmargin 3
set xr [-0.40:0.40]
set yr [-0.40:0.40]
set zr [-0.40:0.40]
unset tics
set xlabel 'x' offset 2
set ylabel 'y' offset -3
set zlabel 'z' offset 6
fx(u,v)=cos(u)*sin(v)
fy(u,v)=sin(u)*sin(v)
fz(v)=cos(v)
a(u,v)= 0.4 * ( fx(u,v)**2-fy(u,v)**2 )**2
set view 70,20
set view equal xyz
splot a(u,v)*fx(u,v),a(u,v)*fy(u,v),a(u,v)*fz(v) t 'dx^2-y^2 orbital' w pm3d
set output

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