Are Some Atomic Orbitals From a 4D Wave Function?

[Hornbein]

"To visualize the standing waves (or orbitals) of electrons bound to a positively charged nucleus in three dimensions, we will need a four-dimensional plot of the wave function vs. x, y, and z."
http://www.grandinetti.org/electron-orbital-shapes"The wavefunctions in the N=2 family are vectors in an abstract four-dimensional space. This can also be called a function-space and/or a Hilbert space." http://www.av8n.com/physics/wavefunctions.htm

So, what are those dimensions and wave functions? This must be fairly basic.

 

[blue_leaf77]

I am not sure what the author means by "four-dimensional plot of the wave function vs. x, y, and z" in the first link. For the second link, the author has specifically considered the n=2 subspace of a hydrogenic atoms where the eigenstates are 4-fold degenerate (one in 2s and three in 2p), therefore the subspace of n=2 has a dimensionality of 4.

 

[DrClaude]

For the first link, it goes like this. If you want to represent a one-dimensional function f(x), you need to produce a two-dimensional plot. Likewise, a two-dimensional function f(x,y) will require a 3D plot. So to plot $| \psi(x,y,z) |^2$, you would need to be in a 4D world, the 4th dimension representing the density at point (x,y,z). That's why you need other visualization techniques, such as shading. It has nothing to do with the wave function per se, but is true of any 3D function f(x,y,z).

 

[Hornbein]

For the first link, it goes like this. If you want to represent a one-dimensional function f(x), you need to produce a two-dimensional plot. Likewise, a two-dimensional function f(x,y) will require a 3D plot. So to plot $| \psi(x,y,z) |^2$, you would need to be in a 4D world, the 4th dimension representing the density at point (x,y,z). That's why you need other visualization techniques, such as shading. It has nothing to do with the wave function per se, but is true of any 3D function f(x,y,z).

Aha. Simple enough. Thanks!