Sampling |ψ|²…
- n
- 2
- ℓ
- 1
- m
- 0
- nodes
- 1
shell, size
shape (s, p, d)
orientation
0 radial + 1 angular
An orbital is not a freehand blob. It is a solution of the hydrogen Schrödinger equation. n and ℓ pick the subshell (size and the s/p/d letter). The magnetic quantum number m then picks an orientation. For the Cartesian clouds chemists actually draw (pₓ, d_xy), that orientation is a real mix of +m and −m.
n — how far out
The principal quantum number n = 1, 2, 3… is the shell. Higher n means a larger cloud. In hydrogen it also means a higher energy (E ∝ −1/n²). 1s hugs the nucleus. n = 2 sits farther out. 3d is larger still. In hydrogen, 2s and 2p have the same energy. That degeneracy, however, breaks in every other atom. The Bohr model’s rings are a cartoon of n. That is why they still help you count electrons on the inspector.
ℓ — the shape, and the letter
ℓ runs from 0 to n − 1. Chemists rename the values: ℓ = 0 is s, 1 is p, 2 is d, 3 is f. ℓ is also the number of angular nodes. s has none (a ball). p has one (a plane). d has two. That is the whole visual difference between the letters on the table.
m — which way it points
For a given ℓ, m runs from −ℓ to +ℓ. So you get one s orbital, three p, five d, and seven f. 2p_z is the m = 0 member. 2p_x and 2p_y are real combinations of m = +1 and m = −1, which is why the HUD labels them ±1. In a free atom those three p orbitals are degenerate. Ligands, crystal fields, and magnetic fields are what then split them. 3d_xy is the real combination of m = ±2 that sits between the x and y axes — one of five d orientations.
The node rule
Total nodes = n − 1. Split them as radial (n − ℓ − 1) plus angular (ℓ). 2s (n=2, ℓ=0) has one radial node: a spherical shell of zero at r = 2 a₀. 2p (n=2, ℓ=1) has no radial node and one angular node: the plane you can drag edge-on in the viewer. If a drawing of 2p has a hollow sphere inside the lobes, someone mixed up 2s and 2p.
Allowed triples in this viewer
| Orbital | n | ℓ | m | Nodes |
|---|---|---|---|---|
| 1s | 1 | 0 | 0 | none |
| 2s | 2 | 0 | 0 | 1 radial node |
| 2pₓ | 2 | 1 | ±1 | yz nodal plane |
| 2pᵧ | 2 | 1 | ±1 | xz nodal plane |
| 2p_z | 2 | 1 | 0 | xy nodal plane |
| 3d_xy | 3 | 2 | ±2 | xz and yz planes |
Hydrogen also has a spin quantum number m_s = ±½. It does not change the cloud’s shape. It does, however, double the occupancy of each orbital. That is why the block widths are 2, 6, 10, and 14.