Plotting hydrogen wave functions This may sound a bit dumb but how do I plot the hydrogen wave functions? For example,  what is exactly being represented in this image? Is it just the norm-squared of the wave function and is the z-axis sticking out of the page?
I'm not sure how to use any other application but I'm using the Mac grapher tool. Say I wanted to plot $\psi_{200}$. The combined radial and angular equation for this state is $$\psi_{200}=\frac{1}{4\sqrt{2\pi}a_0 ^\tfrac{3}{2}} \left(2-\frac{r}{a_0}\right)e^{-\tfrac{r}{2a_0}}$$ where $a_0$ is the Bohr radius. As I said, i'm pretty clueless. I'm not sure how to plot this in spherical coordinates so I just converted $r=\sqrt{x^2+y^2+z^2}$. Basically, I plotted $$z=\frac{1}{4\sqrt{2\pi}a_0 ^\tfrac{3}{2}} \left(2-\frac{\sqrt{x^2+y^2+z^2}}{a_0}\right)e^{-\tfrac{\sqrt{x^2+y^2+z^2}}{2a_0}} $$When I do plot this, I get a flat plane with a sort of half-sphere in the centre, which is not what the image I linked above shows. I also tried graphing $\psi^2$ but I still did not get it. I feel like i'm missing something big.
 A: The plots you see in the Wiki images, are, as their title suggests, probability density plots ($\psi^2$, for Real $\psi$). Light shaded areas represent high probability areas, darker areas lower probability density.
Furthermore, they've been sliced, e.g. with an $x,y$-plane. For radially symmetric functions like $\psi_{2,0,0}$ (aka the $2s$ orbital) the specific slicing plane doesn't even matter: the $\psi^2$ value is the same in all directions as it only depends on $r$.
There's little to be gained from these plots for radially symmetric wave functions with regards to the simpler radial distribution plots like these.
If you really do want to plot radially non-symmetric probability density functions (e.g. for $2p$, 3$d$ or $4f$ orbitals) you'll need to find a tool to generate contour plots. I imagine advanced functions of Mathlab allow to do that.

Here's a *contour plot* of $\psi^2_{200}$ for $z=0$ (the $x,y$-plane):

The white areas are those of high probability density. The plot clearly shows the 'shell-like' structure of the hydrogen $2s$ orbital.
The plot was obtained by means of Wolfram alpha, by plotting the square of:
$$\psi=\frac{1}{4\sqrt{2\pi}a_0 ^\tfrac{3}{2}} \left(2-\frac{\sqrt{x^2+y^2}}{a_0}\right)e^{-\tfrac{\sqrt{x^2+y^2}}{2a_0}}$$
For $a_0=1$.
Here's a excellent resource for visualised wave functions: the Orbitron
A: For Mac there still is Atom in a Box. It can display hydrogen wave functions in different ways. I found the view mode with "phase as color" very helpful for my understanding. The wave functions are displayed in 3D and animated in time. One can also display superpositions of different eigenfunctions (for example the hybridizations from chemistry).
