# Spherical harmonics and wavefunctions

What's the difference in the representation of spherical harmonics and the orbitals themselves? they look exactly the same to me... unlike the radial part of the wavefunction though.

## Answers and Replies

jtbell
Mentor
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 part of an orbital wave function for hydrogen (or any other spherically symmetric potential) is a spherical harmonic:

$$\Psi_{nlm}(r,\theta,\phi) = R_{nl}(r)Y_{lm}(\theta,\phi)$$

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
Science Advisor
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.
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.).

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.).

yes, but when drawing the 2p orbital and the 2p spherical harmonic what's the difference? THEY ARE THE SAME!

If they are the same, then by the property of sharing the same identity they are not different.

yes, but when drawing the 2p orbital and the 2p spherical harmonic what's the difference? THEY ARE THE SAME!
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.