4-D Spheres

[circa 1991]

Question:  I was wondering if electron orbitals are really 4 dimensional (or 
higher) spheres because d and p orbitals look similar to pictures I have seen 
of 3-D projections of 4-D spheres.  Any comments?
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Well, there might be something to your observation, but I have 
never heard of this before.  The equations describing p-orbitals, though, are 
really pretty simple - basically the wave function amplitude is proportional 
to  x *f(r) (or y, or z) where r = sqrt(x^2 + y^2 + z^2) and f(r) is something 
like an exponential e^(-a*r), although it has oscillations or higher n values.  
The pictures normally shown are surfaces of constant amplitude; so solutions 
of an equation of the form  x * f(r) = constant.  Since f(r) has spherical 
symmetry, (it depends only on distance r from the nucleus) the angular 
dependence of the surface is due to the factor x - the surface pokes out along 
the x-axis, and comes in to the origin when it meets the x=0 plane.!  Also, d-
orbitals are kind of similar, but instead of x have a factor x^2 - y^2, or 
z^2, or xy, etc, so their extensions tend to be narrower (due to the higher 
power) and more complicated.

A. Smith
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