Here are some more trajectory plots of Hydrogen in various combinations of Eigenstates. Some facts about them:

- As in other posts, the points (shown in blue here) represent a scatter plot of electron probability density.
- Additionally, I’ve drawn yellow tracers on 1 out of every 20 points to let the overall motion become more visible.
- The Eigenstates involved are shown at the top of each one. (If you aren’t familiar with QM, Eigenstates are special configurations such that certain quantities are uniform throughout the state. These quantities correspond to those that would be conserved in a classical system with the same potential. In the case of Hydrogen, we have a -1/r central potential, so the conserved quantities are Energy, total Angular Momentum, and the Angular Momentum about a given axis (we’ve chosen the z axis as our special “given axis”). Check out chapters 2, 3, 5, and any intro to QM textbook for more details). For example, |3,2,-1>, one of the eigenstates from the first video, is in the 3rd energy band, with total angular momentum of 2, and angular momentum about the ‘z’ axis of -1.
- You should notice that certain combinations of eigenstates produce regular motion. Combinations of different energies create expansion and contraction. Differing total angular momenta produce up and down motion. And combinations with different angular momenta about the z axis produce vortices spinning in opposite directions.
- You’ll also see a lot of little “tornadoes” spinning around very rapidly. These occur when a point is near a region with very low probability density due to destructive interference between the eigenstates. They tend to have pretty interesting personalities.
- Each video takes place over a time span of 500 atomic time units, which is about one 80 trillionth of a second.
- Resolution is 900×900 pixels. I can make some higher res versions if people are interested.

Wanting to see these Hydrogen states animated is what got me started with this project in the first place. To me, they really motivate trying to understand the mathematics and provide a glimpse of the beauty that underlies everything.

(My apologies if these videos take awhile to load or don’t play at the proper frame rate (60 fps). You might have better luck if you save the files locally and open them with your video player. You can do this by either right clicking each video and doing a “Save Video As”, or try “Downloadthemall” if you’re using Firefox.).

A long, “3D” version to start with.

Youtube Video

I also made a torrent for this file, which you can download here.

Other videos:

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