Undergraduate · 20 min

Seeing what quantum numbers do to an orbital

Dial the quantum numbers n, l and m of a hydrogen-like orbital and watch its electron cloud change shape, lobes and nodes.

Goal

Link each quantum number to a visible feature: n sets size and radial nodes, l sets the number of angular nodes and the orbital family, m sets the orientation.

Apparatus and reagents

No chemicals — only the Schrödinger solutions for the hydrogen atom, sampled as an electron-density cloud.

Procedure

Set n = 1, l = 0, m = 0 to see the spherical 1s orbital, then raise n to 2 and 3 keeping l = 0 and count the radial nodes.
Switch to l = 1 (p orbitals) and slide m through −1, 0, +1 to compare the orientations along x, y and z.
Try l = 2 at n = 3 to see the cloverleaf d shapes; toggle the phase display to switch the lobe colours on and off.
Raise the point count to 60000 for a dense cloud and note where the density is highest relative to the nucleus.

What to observe

A 1s orbital is a featureless sphere; 2s and 3s add one and two radial nodes where the density vanishes.
Every l = 1 orbital has one nodal plane through the nucleus; the lobe colours invert across that plane.
For a given l, different m values give the same shape rotated; the total number of nodes is always n − 1.

Explanation

An atomic orbital is a one-electron wavefunction $\psi_{nlm}(r,\theta,\varphi)$ that solves the Schrödinger equation for the hydrogen atom; the dots show $|\psi|^2$ , the probability density of finding the electron. The wavefunction factorises into a radial part with $n-l-1$ nodes and an angular part (a spherical harmonic $Y_l^m$ ) with $l$ nodal planes or cones. Inside the Born–Oppenheimer picture, these one-electron functions become the building blocks that fill electron configurations in many-electron atoms.

Chemists behind it

Linus Pauling