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This page shows the result of a Saturday afternoon spent in the pursuit
of a better visualization of the simplified hydrogen atomic states. While
one can find plots of the spherical harmonics and radial wave function in
most competent textbooks on the subject, and occasionally even a surface
contour of the wave function or probabilty density, I have never seen a
good rendering of the hydrogen wave functions -- if one could
look at such things, how would they appear?
The problem is essentially this: to render an image where the opacity is a complicated function of position. Pixar's PRman falls flat on its face. Rayshade and POVray can only handle constant densities. So, I wrote my own renderer. The approach I took was ray-tracing. For each eye-ray, the attenuation is computed as a numerically-approximated line integral along the ray path. The local density (whether the magnitude of the wave function or the probability density, depending on output desired) is treated as the local linear attenuation coefficient. Intra-cloud shading is disabled, which speeds rendering and gives the electron cloud the appearance of glowing. A black background was used to provide contrast. The problem of image quantization must also be addressed. For simplicity, each image is normalized separately, meaning that "white" for one image does not usually correspond to the same density as "white" on another image. This has the disadvantage of removing the comparison of the relative densities of different states.
N = 1 State
N = 2 States
N = 3 States
MPEGsHere is a first attempt at an animation based on the ray-traced hydrogen atom. It is the (n=4, l=2, m=0) state, as one goes from 10 Bohr radii to 40 Bohr radii from the atomic center. The animation file is only 19KB in size, so feel free to download a copy. It took approx. 20 hours to render the frames on a 25 MHz 68040. All images (c) Andy Dunn, 1998. All rights reserved.
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