The 3dplot package has been extended to handle the plotting of user-specified functions in spherical polar coordinates. In this example, a spherical harmonic is rendered, where the fill hue represents the complex phase angle. More info about spherical harmonics can be found at http://en.wikipedia.org/wiki/Spherical_harmonics.
The 3dplot.sty package can be found at http://www.heinjd.com/dev/latex/3dplot/3dplot.sty.
Documentation for the 3dplot.sty package can be found at http://www.heinjd.com/dev/latex/3dplot/3dplot_documentation.pdf.

Edit and compile if you like:
%harmonics.tex: produces spherical harmonic plots using the 3dplot package% Author: Jeff Hein\documentclass{minimal}\usepackage{tikz} %for TikZ graphics\usepackage{tikz-3dplot} %for 3dplot functionality\usepackage[active,tightpage]{preview} %generates a tightly fitting border around the work\PreviewEnvironment{tikzpicture}\setlength\PreviewBorder{2mm}\begin{document}\tdplotsetmaincoords{70}{135}\begin{tikzpicture}[line join=bevel,tdplot_main_coords, fill opacity=.7]\tdplotsphericalsurfaceplot[parametricfill]{72}{36}%{sqrt(15/2)*sin(\tdplottheta)*cos(\tdplottheta)}{black}{\tdplotphi}%{\draw[color=black,thick,->] (0,0,0) -- (2,0,0) node[anchor=north east]{$x$};}%{\draw[color=black,thick,->] (0,0,0) -- (0,2,0) node[anchor=north west]{$y$};}%{\draw[color=black,thick,->] (0,0,0) -- (0,0,2) node[anchor=south]{$z$};}%\end{tikzpicture}%Here's some more examples.%L = 0%\begin{tikzpicture}[scale=2,line join=bevel,tdplot_main_coords, fill opacity=.7]%\tdplotsphericalsurfaceplot[parametricfill]{72}{36}%%{1}{black}{0}%% {\draw[color=black,thick,->] (0,0,0) -- (2,0,0) node[anchor=north east]{$x$};}%% {\draw[color=black,thick,->] (0,0,0) -- (0,2,0) node[anchor=north west]{$y$};}%% {\draw[color=black,thick,->] (0,0,0) -- (0,0,2) node[anchor=south]{$z$};}%%\end{tikzpicture}%%L = 1, M_L = -1%\begin{tikzpicture}[scale=2,line join=bevel,tdplot_main_coords, fill opacity=.7]%\tdplotsphericalsurfaceplot[parametricfill]{72}{36}%%{sqrt(3/2)*sin(\tdplottheta)}{black}{-\tdplotphi}%% {\draw[color=black,thick,->] (0,0,0) -- (2,0,0) node[anchor=north east]{$x$};}%% {\draw[color=black,thick,->] (0,0,0) -- (0,2,0) node[anchor=north west]{$y$};}%% {\draw[color=black,thick,->] (0,0,0) -- (0,0,2) node[anchor=south]{$z$};}%%\end{tikzpicture}
Click to download: spherical-polar-pots-with-3dplot.tex • spherical-polar-pots-with-3dplot.pdf
Open in Overleaf: spherical-polar-pots-with-3dplot.tex