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Physics Notes

Schrödinger's Hydrogen Atom

The Wavefunction

The hydrogen atom's electron is described by ψ(r,θ,φ) — a solution to the Schrödinger equation. It encodes all measurable properties of the electron's state.

ψ = R_nl(r) · Y_lm(θ,φ)

Probability Density

|ψ|² gives the probability density — the likelihood of finding the electron at any given point in space. Dense clouds = high probability regions.

P(r,θ,φ) = |ψ(r,θ,φ)|²

Quantum Numbers

n sets the energy level, l determines the orbital shape (s, p, d, f...), and m controls orientation in space.

n ≥ 1, 0 ≤ l < n, −l ≤ m ≤ l

Radial Function

The radial part R_nl uses associated Laguerre polynomials and determines how ψ varies with distance from the nucleus. Nodes appear where R = 0.

R_nl = N · e^(−ρ/2) · ρ^l · L_{n-l-1}^{2l+1}(ρ)