When given an electron with a certain amount of energy and angluar momentum, the orbital of that electron is known. The exact path of that electron is unknown since finding the exact position of the electron is impossible according to the Heisenberg Uncertainty Principle. It is possible is to calculate the probability of finding an electron in a certain volume and the goal of these visualizations is to create a proper 3D picture of the probability density of an electron in the hydrogen atom.
There have been many attempts to visualize the hydrogen atom. Earliest attempts involved cutting small chips of wood, spinning them and taking pictures as shown at http://phycomp.technion.ac.il/~phr76ja/viz1.gif/. Later attempts generally involved computer graphics to plot out the wave equation for Hydrogen, but are generally 2D graphics, as shown at http:/falstad.com/qmatom/, and proper 3D images had not been produced.
In these visualizations, the method known as smoke rendering was used, as inspired by Johnson, S., Potter, W., Malkin, K.,(1994) Visualizing the Quantum World: Volume Rendering Applied to Quantum Mechanics. By creating a cloud that is thicker in areas of higher density and thinner in areas of lower density, an accurate image of the electron probability can be created. Accordingly, the general volume where an electron can be found is known as an electron cloud, since its exact location is unknown. The cloud effect was created by plotting between 20 000 and 100 000 points. Different colours were used in volumes of different densities as to make it easier to distinguish between them.
The first step in the visualization is knowing how large of a volume needs to be scanned. By graphing the radial solution of the Schrodinger equation, as shown at http://library.wolfram.com/webMathematica/Physics/Hydrogen.jsp, a general idea of the limits of the high density volume can be found. Once this was known, a cube was scanned with its centre at the origin and its length given by the limits from the radial equation.
This volume being scanned was then broken up into many small cubes. Each small cube had the same volume and was spaced out evenly, side by side. The size of each cube determined the resolution of the final picture, so a balance between number of points and quality of the picture was needed to determine the size of each of the small cubes.
The wave equation is in spherical coordinates and the cubes' locations were spaced out in Cartesian coordinates, so a conversion between the two was necessary. At the centre of each cube, the probability was found from the Schrodinger wave equation. The probability is always going to be a number smaller then one, because the integral of the probability over all space is equal to one due to the fact that the electron must be found somewhere in the universe. This probability was then multiplied by some factor and rounded to give an integer. This intiger corresponded to the amount of points randomly distributed within the box and the colour of the points. The multiplication factor also affected the resolution and size of the file and was chosen to allow a proper distribution of points through space, as well as a variation of the amounts of points found within every small cube.
Once the number of points to randomly plot in every cube was known, this was performed using Mathematica and then plotted using Aviz. Aviz was originally designed to visualize atoms as spheres, but also can be used to visualize dots, as utilized here. More details can be found at http://phycomp.technion.ac.il/~aviz/
Here are some links examples of other visualization attempts for hydrogen wavefunctions: