dz^2 orbital looks 4D to me

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dz^2 orbital looks 4D to me

Postby Secret » Fri Dec 17, 2010 3:59 pm

after learning orbitals
and thinking as if the world is 4D
The dz2 orbitals looks like a dxz rotated by 45o along the xy axis then 90o along the zy axis
Image
Image

But actually is this sort of perspective wrong?

P.S. I'm still preliminary in 4D interpretation
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Re: dz^2 orbital looks 4D to me

Postby PWrong » Mon Dec 20, 2010 1:48 am

Are you talking about atomic orbitals? Like solving Schrodinger's equation for Hydrogen?

The two pictures don't look similar to me, I'm not sure if that's what you're asking though. They do look like they could be cross sections of some 4D shape, but that's probably because they change through time and they're not solid objects but complex wavefunctions.
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Re: dz^2 orbital looks 4D to me

Postby Secret » Mon Dec 20, 2010 1:08 pm

The bottom picture gives me a feeling that if I rotate so that its two blue lobes are pointed into and out of 4D, and then project the resulting image back to 3D, you'll get the top picture
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Re: dz^2 orbital looks 4D to me

Postby PWrong » Sun Jan 09, 2011 6:24 am

Maybe. Hard to say for sure without the equations.
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Re: dz^2 orbital looks 4D to me

Postby quickfur » Mon Jan 24, 2011 3:33 am

Actually, the atomic orbitals aren't 4D, they are 6D, because they are complex wave functions (so each dimension is represented by a complex number, which is 2D). Of course, this is not Euclidean 6D, because while the complex numbers do behave somewhat like geometric 2D vectors, the correspondence is not 100%. So you do get effects that could be rationalized geometrically, but other effects can't.

The ring shapes of some of the d and f orbitals (and theoretically, the g orbitals) do indeed arise from toroidal loops in 6D (3D complex space) intersecting with 3D real space. But the underlying symmetry is not what you'd expect an Euclidean 6D geometry would have, since if it was actual Euclidean 6D (real space), and not just 3D complex space, it should be possible to get icosahedral orbitals by a suitable alignment of the 6 axes w.r.t. 3D real space, but as far as we know, these don't exist.
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