Physics
Atomic Orbitals
The exact hydrogen orbitals ψnlm drawn as glowing clouds of probability, with their spherical shells and nodal planes, the phase winding of m and the signed lobes of chemistry. Measure the electron and it turns up at one point; measure hundreds of atoms and the dots draw the orbital. Superpose two states and the cloud starts to slosh: the origin of emitted light.
probability density |ψ|²phase arg ψ (hue)ψ > 0ψ < 0measured position
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Energy—
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State
Preset
Mix
Show
Plane
thindense
Values
energy En—
⟨r⟩ = (3n² − l(l+1))/2—
⟨r⟩ on the grid—
most probable r—
nodes: radial / angular—
∑|ψ|²ΔV on the grid—
⟨z⟩ (dipole)—
box half-width—
time t—
measurements—
mean measured r—
What to observe
- Cut a quarter out of 3s and two dark spherical shells split the cloud: the radial nodes, n − l − 1 = 2 of them. Across each one ψ changes sign, so the colour flips.
- With m = +1 the 2p cloud is a doughnut round the z axis and its hue turns once around it: that winding eiφ is the angular momentum. Tick Real orbitals and the same states become the px and py dumbbells, with a nodal plane between their two lobes.
- Each measurement finds the electron at one point. Measure 200 atoms and the dots draw the orbital, and their mean distance from the nucleus closes in on ⟨r⟩.
- An orbital is stationary: |ψ|² never moves, only its phase turns. Mix 1s with 2pz and the cloud sloshes up and down once every 0.41 fs, a charge oscillating at the frequency of the 10.2 eV light it emits (Lyman α, 122 nm).
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