The discussion revolves around the relationship between spherical harmonics and orbital wave functions, particularly in the context of hydrogen and other spherically symmetric potentials. Participants explore the similarities and differences in their representations, focusing on the angular parts of these functions and the implications of the radial components.
Participants express differing views on whether spherical harmonics and orbital wave functions can be considered the same or different, particularly regarding their visual representations. The discussion remains unresolved, with multiple competing perspectives on the matter.
Limitations include the lack of clarity on how the radial part quantitatively influences the overall wave function and the potential implications of different potentials on the representations discussed.
jtbell said:The angular part of an orbital wave function for hydrogen (or any other spherically symmetric potential) is a spherical harmonic:
\Psi_{nlm}(r,\theta,\phi) = R_{nl}(r)Y_{lm}(\theta,\phi)
The angular parts look the same, because they are identical (see jtbell's comment). Unlike spherical harmonics, orbital wave functions, however, do not consist only of an angular part. They also have a radial part. And this radial part is non-trival and comes from solving the Schroedinger equation for some potential (e.g., in hydrogen the nuclear attraction of the proton, in higher spherical atoms from nuclear attraction and the mean field of the other electrons (Fock potential)). But this has no influence on the angular part. E.g., 2p and 3p orbitals have the same angular part, not only in a single atom, but across all atoms (in the nonrelativistic case etc.).Chemist20 said:yes but for a 2p for example, what's the difference in representation between the orbital itself and the spherical harmonic? they look the same to me.
cgk said:The angular parts look the same, because they are identical (see jtbell's comment). Unlike spherical harmonics, orbital wave functions, however, do not consist only of an angular part. They also have a radial part. And this radial part is non-trival and comes from solving the Schroedinger equation for some potential (e.g., in hydrogen the nuclear attraction of the proton, in higher spherical atoms from nuclear attraction and the mean field of the other electrons (Fock potential)). But this has no influence on the angular part. E.g., 2p and 3p orbitals have the same angular part, not only in a single atom, but across all atoms (in the nonrelativistic case etc.).
Note that multiplying the radial wave function by constant factor changes the size, not the shape, of the "drawing" of the orbital, which is really just a drawing of the surface of maximum probability density. By looking at how more general changes in the radial wave function affects this surface, you can see why spherical harmonics look so much like these surfaces.Chemist20 said:yes, but when drawing the 2p orbital and the 2p spherical harmonic what's the difference? THEY ARE THE SAME!