Mathematica(@mathemetica):The electron has no orbit. It has only a probability cloud. These surfaces mark the set of points where |ψ|² equals 0.04 times its maximum value for the hydrogen wave function ψ_nlm. The resulting spheres, dumbbells, and tori are fixed once n, ℓ, and m are chosen. ℓ = 0 produces the spherical s clouds. Nonzero ℓ and m generate the axial lobes of p_z and the rotationally symmetric rings of higher |m| states. All of them are exact solutions of the Schrödinger equation. Classical trajectories have been replaced by these precise geometric surfaces of probability.

2026.10.11 19:49

The electron has no orbit. It has only a probability cloud. These surfaces mark the set of points where |ψ|² equals 0.04 times its maximum value for the hydrogen wave function ψ_nlm. The resulting spheres, dumbbells, and tori are fixed once n, ℓ, and m are chosen. ℓ = 0 produces the spherical s clouds. Nonzero ℓ and m generate the axial lobes of p_z and the rotationally symmetric rings of higher |m| states. All of them are exact solutions of the Schrödinger equation. Classical trajectories have been replaced by these precise geometric surfaces of probability.