Lab 01 — Quantum Mechanics

Hydrogen Electron Clouds

Solve the Schrödinger equation in the electric pull of the nucleus — the Coulomb potential — and you get a wavefunction ψ for each set of quantum numbers (n, l, m). The point cloud below is 26,000 samples drawn from the probability density |ψ|²: the distribution of where a measurement would actually find the electron. Flip through the chips from 1s to 4f and watch the cloud change shape.

-\dfrac{\hbar^{2}}{2m}\nabla^{2}\psi \;-\; \dfrac{e^{2}}{4\pi\varepsilon_{0} r}\,\psi = E\psi \qquad P(r,\theta,\varphi) = \bigl|\psi_{nlm}\bigr|^{2} \qquad E_{n} = -\dfrac{13.6\ \mathrm{eV}}{n^{2}}

Drag to rotate, scroll to zoom

protonneutron

Energy eigenvalueE₁ = −13.61 eV

MeasurementsN = 0 / 26,000

phase +phase −

Green and gold mark the sign (phase) of the wavefunction. Each point is one "measurement"; as N grows, the shape of the cloud — the probability distribution — comes into focus. The lump at the center is the nucleus (gold = protons, gray = neutrons), drawn about 100,000× larger than life. Switching isotopes leaves the cloud unchanged: the real shift from the heavier nucleus is under 0.03%, far too small to see. That is why chemistry barely notices the neutron count.