Equations with meaning
Ψₙℓₘ(r, θ, φ) = Rₙℓ(r)Yℓₘ(θ, φ)Radial factor × angular factor.
Probability density = |Ψ|²Probability in a volume is its integral over that volume.
Radial probability = r²|Rₙℓ(r)|²Probability per radial interval after all directions are combined.
Radial nodes = n − ℓ − 1Angular nodes = ℓTotal nodes = n − 1Eₙ = −13.6 Z² eV / n²Hydrogenic energy; this lab fixes nuclear charge Z = 1.
Allowed values: n = 1, 2, 3…; ℓ = 0 through n − 1; mℓ = −ℓ through +ℓ. One Bohr radius a₀ is approximately 0.0529 nm.
Model boundaries
This is a one-electron, nonrelativistic hydrogen-like model. It is not a planetary atom, an electron path, or an exact model of a many-electron atom. The displayed real p and d shapes are Cartesian combinations; some do not have one definite mℓ.
The chamber, field coils, detector ring, and radial screen are an illustrative apparatus—not a calibrated instrument or a direct photograph. Cloud points and measurement markers are Monte Carlo samples from analytic hydrogenic probability densities. The continuous tails extend beyond the finite view.
The model shows selected 1s, 2s, 2p, 3s, 3p, and 3d orbitals. Hydrogen fine structure, Lamb shift, spin-orbit effects, external-field splitting, electron-electron repulsion, shielding, penetration, time-dependent transitions, and detector dynamics are excluded. One spatial orbital may hold at most two opposite-spin electrons, but occupancy is not simulated.