Hydrogen Orbitals: Spherical Harmonics and the Shapes of s, p, d States

A spherically symmetric potential separates cleanly

The hydrogen atom is one electron bound to a proton by the Coulomb potential V(r) = −e²/(4πε₀r), which depends only on distance r, never on direction. That spherical symmetry means the Hamiltonian commutes with the total angular momentum operators L² and L_z, so all three share a common set of eigenstates. Looking for solutions of the product form ψ(r,θ,φ) = R(r)·Y(θ,φ) then splits the full Schrödinger equation into one purely angular equation and one purely radial equation, tied together only by a separation constant l(l+1).

The angular part: spherical harmonics

The angular equation is solved by the spherical harmonics Y_l^m(θ,φ) — the same family of functions that describes multipole fields in electromagnetism and the vibration modes of a sphere. Each one factors into a θ-dependent associated Legendre polynomial and a φ-dependent phase e^imφ. The quantum number l = 0, 1, 2, 3… is labelled s, p, d, f by 1890s spectroscopic convention (sharp, principal, diffuse, fundamental), and for each l, m ranges over 2l+1 integer values from −l to +l — the origin of "one s orbital, three p orbitals, five d orbitals."

The radial part and the −13.6 eV/n² energy ladder

Solving the radial equation for the Coulomb potential requires substituting R(r) = e^−r/na₀·(r/a₀)^l·L(r), where a₀ is the Bohr radius. Demanding that the wavefunction stay finite as r → ∞ forces L(r) to be an associated Laguerre polynomial — and that same finiteness condition fixes the energy:

Eₙ = −(m e⁴) / (2(4πε₀)²ℏ²n²) = −13.6 eV / n²

Remarkably, this depends only on the principal quantum number n, never on l or m — an "accidental" degeneracy unique to the pure 1/r potential. It vanishes for any other central potential, and it is lifted in real multi-electron atoms by electron-electron screening, which is exactly why heavier atoms need the full periodic-table structure rather than hydrogen's simple ladder.

Quantum numbers n, l, m and orbital shapes

The three quantum numbers fully label an orbital: n (1, 2, 3…) sets the energy shell, l (0 to n−1) sets the shape, and m (−l to +l) sets the orientation. s orbitals (l=0) are spherically symmetric with non-zero density right at the nucleus; p orbitals (l=1) are two lobes of opposite sign split by a nodal plane; d orbitals (l=2) form four-lobed cloverleaf shapes with two angular nodal surfaces. Every orbital has n−1 total nodes split between radial shells and angular planes — the same standing-wave counting that governs the harmonics of a vibrating string, just in three dimensions and bound by the Coulomb attraction instead of a fixed length.

Frequently asked questions

Try it live

Everything above runs in your browser — open Hydrogen Orbital, dial in n, l and m, and watch the probability cloud reshape between s, p and d as the sliders change.