Quantum mechanics · WebGPU · Exact
The electron in a hydrogen atom, drawn from Schrödinger's exact solution: every orbital up to n = 4 as a cloud of probability, coloured by its quantum phase. Mix two states and the cloud starts to move, sloshing at the frequency of the light it would give off.
Hydrogen is one proton and one electron, and it is the only atom whose Schrödinger equation can be solved with pen and paper. Schrödinger did it in 1926. The solutions are labelled by three whole numbers. The shell n sets the energy, −13.6 eV/n². The shape l runs from 0 to n − 1 and is written s, p, d, f. The turn m runs from −l to l and says how the electron circulates around the vertical axis.
Nothing here is simulated or approximated. For every pixel, the GPU walks a ray through the atom and evaluates the exact wavefunction ψ at 144 points along it. The glow is |ψ|², the probability of finding the electron there. The sphere is sized to hold 99.5 percent of that probability, so its radius is about 5 bohr radii for 1s and nearly 50 for 4s. The readout in the corner gives the real size.
ψ is a complex number, and its angle is called the phase. The colour shows it: green is a phase of 0, magenta is half a turn, with blue and gold in between. The real orbitals that chemistry uses, such as px or dxy, are real numbers, so they come out in just two colours. The lobes of a p orbital are opposite in sign, one green and one magenta. Switch to complex and pick m = +1. The colour now winds once around the axis, which is the electron's angular momentum. Each real orbital is an equal mix of +m and −m.
Choose Slice to cut the atom through its vertical axis. The brightness is on a log scale, so the nodes show as dark lines: places where ψ is exactly zero and the electron is never found. Every state has n − 1 of them. There are l angular nodes, which are planes and cones through the nucleus, and n − l − 1 radial nodes, which are spheres. 2s has one sphere, at exactly 2 bohr radii. 3p has one sphere and one plane. 3dz² has two cones and no sphere.
A single state is called stationary for a reason. Its phase turns at a rate set by its energy, but its cloud never moves. Mix two states with different energies and their phases drift apart. The cloud then moves, repeating with a period of 2πħ/ΔE.
Sloshing mixes 1s with 2p. The cloud swings up and down through the nucleus every 0.41 femtoseconds (Slice shows it best). That swinging charge is an oscillating dipole, like a tiny radio antenna, and it radiates at that frequency. This is the Lyman α line at 121.6 nm, in the ultraviolet. The model computes 121.568 nm, against 121.567 nm measured. The real atom loses the mixture this way in about 1.6 nanoseconds.
Breathing mixes 1s with 2s. The cloud pulses in and out but stays centred, so it has no dipole and cannot shine a single photon. That is why 2s is metastable: it lives about a tenth of a second, nearly a hundred million times longer than 2p. Circling mixes 2p with 3d in the complex basis, and a lump of charge orbits the axis. Standing still mixes 2s with 2p, which have the same energy in this model. The cloud is lopsided, the shape chemists call an sp hybrid, and it never moves.
Each state is stored as a handful of numbers: e−r/n times a polynomial in r, times a polynomial in x, y and z of degree l. Together these are the Laguerre and spherical-harmonic parts of the textbook solution, rewritten so the GPU never divides by r. The browser builds them when you pick a state, and the shader multiplies them out at every step of every ray. A check script integrates the same numbers. Every state up to n = 4 is normalised to within 10⁻⁶, every pair is orthogonal, and the mean radius matches the exact (3n² − l(l + 1))/2.