quantum chemistry - Difference between radial, planar, angular and spherical nodes

What is the difference between them?

I think radial nodes and spherical nodes are the same, and angular and planar nodes are the same.

Reference

Finally, how many spherical nodes are there in a $2p$ orbital?

Answer

The wave function $\Psi(r,\theta,\phi)$ of a one-electron (hydrogen-like) atom is seperable as the product of a radial function $R(r)$ and an angular function $Y(\theta,\phi)$

$\Psi(r,\theta,\phi) = R(r)Y(\theta,\phi)$

If $R(r_1) = 0$, there exists a radial node. The radial node is a sphere with radius $r_1$. Therefore the terms "radial node" and "spherical node" are the same.

$Y(\theta,\phi)$ is further seperable as

$Y(\theta,\phi) = P(\theta)F(\phi)$

If either $P(\theta)$ or $F(\phi)$ is zero for a given respective angle value, there is an angular node. However, a angular node is not necessarily a planar node. An angular node could be a planar node or a conical node.

$F(\phi)$ being zero corresponds to a planar node.

$P(\theta)$ being zero corresponds to either a conical node or a planar node (some think of the planar case as a specical case of conical, with apex angle being 180 degrees)

Overall, there will be $n-1$ nodes.

$l$ nodes will be angular

$n-l-1$ nodes will be radial (spherical)