What You Are Seeing

The cloud is electron probability density, not a literal path. Each dot is a random sample from where an electron is likely to be measured.

Hydrogen

Hydrogen has one electron, so its nonrelativistic Schrodinger equation separates into radial and angular parts:

psi(n,l,m)(r,theta,phi) = R(n,l)(r) Y(l,m)(theta,phi)

The quantum numbers control the shell size, orbital shape, and orientation. The simulator computes hydrogen-like radial functions with associated Laguerre polynomials and real angular functions based on spherical harmonics.

Probability

The program samples points using the squared wavefunction:

P proportional to |R(n,l)(r) Y(l,m)(theta,phi)|^2 r^2

Denser regions mean higher measurement probability. The two colors in hydrogen mode show opposite wavefunction phase.

Phase A and Phase B

In hydrogen cloud mode, Phase A and Phase B are not two electrons. They show the sign of the real-valued wavefunction: Phase A is positive and Phase B is negative.

Phase A: psi >= 0
Phase B: psi < 0
Probability: |psi|^2

Both phases can have high probability. The sign matters for interference, bonding, and antibonding behavior. In helium mode, these colors distinguish the two interleaved 1s electron clouds as a visualization aid.

Helium

Helium has two electrons, so the electron-electron repulsion term makes the exact problem nonseparable:

H = -1/2 nabla_1^2 - 1/2 nabla_2^2 - 2/r_1 - 2/r_2 + 1/r_12

The app uses a classic variational approximation for the ground state: two hydrogen-like 1s electrons with an effective nuclear charge:

Z_eff = 27/16 = 1.6875

This captures screening and the more compact helium cloud, but it does not model full electron correlation, spin, or precision quantum chemistry.

Bohr Mode

Bohr mode is a historical comparison. It is intuitive, but the quantum cloud is the better scientific picture.

Full write-up: SCIENCE.md