Alternative linear comb. to vizualize hydrogenic orbitals p,d etc

[psand]

I'm teaching basic chemistry and got a question that I found very interesting. All textbooks we use depict hydrogenic p-orbitals (or spherical harmonics) by first making a linear combination of p+ and p- wave functions in order to get real orthogonal px, py, pz. But the thing that should be of (primary)interest is the probability density of the orbitals. i.e. the orbiltal wave function times its complex conjugate. Taking this of the p+ or p- wave functions result in doughnut shaped probability densities (allowing for a particle analogy with non-zero angular momentum, i.e. the Bohr orbit).

The question was:

Is the traditional vizualization of the p-orbitals just made in order to be able to depict a real wave function?

My extension of the question is:
Can another linear combination be found that results in 3 orthogonal wave functions whose probability densities are e.g. elongated doughnuts with identical shapes (ellipsoids)? Could anyone give me an example of such a linear combination? I simply do not have the maths required.

This would be a nice tramsition from the particle view to the wave view and show that the old atomic symbol of three crossed ellipses is not so crazy after all.

Respectfully
Peter S

 

[DrClaude]

Is the traditional vizualization of the p-orbitals just made in order to be able to depict a real wave function?

No. It is also used because p_x and p_y have a nice geometry, which is useful to understand chemical bonding (to which one can add hybridization).

My extension of the question is:
Can another linear combination be found that results in 3 orthogonal wave functions whose probability densities are e.g. elongated doughnuts with identical shapes (ellipsoids)? Could anyone give me an example of such a linear combination? I simply do not have the maths required.

I don't think this is possible.

This would be a nice tramsition from the particle view to the wave view and show that the old atomic symbol of three crossed ellipses is not so crazy after all.

It's not crazy, simply wrong. However, the Bohr-Sommerfeld model of the atom (which is what is pictured in those representations) was an important step in the development of quantum theory,