Hydrogen wavefunctions and orbital shapes

The three labels arrive from three different demands, and each governs a different part of the picture. Single-valuedness in φ gives the integer m, which carries the whole φ dependence in the phase e^(imφ); regularity at the poles gives the integer l ≥ |m|, which with |m| fixes the angular pattern; and demanding a normalisable radial solution gives n ≥ l + 1, which fixes the energy, −13.6 eV/n², and the overall size. So n sets the scale and the total node count n − 1; l divides those nodes between radial and angular and controls how hard the electron is held off the nucleus; m changes neither the energy nor the radial factor — it moves the θ pattern between poles and equator and winds the phase about the z-axis, but never gives the density any φ dependence. That is why one level holds n² orbital states of identical energy: the sum of 2l + 1 over l = 0 to n − 1 is n².

Write Rₙₗ(r) = N (2r/na₀)^l e^(−r/na₀) L^(2l+1)_(n−l−1)(2r/na₀) and three features follow with no further work. Near the origin only the r^l survives, so ψ(0) is non-zero for s orbitals alone — which is why the Fermi contact term and the Darwin term reach l = 0 states and nothing else. Far out, the exponential of range na₀ sets the size: ⟨r⟩ = (a₀/2)[3n² − l(l+1)] makes 3s nine times the extent of 1s. In between, the Laguerre polynomial has degree n − l − 1 and exactly that many positive roots — 1s has none, 2s has one at r = 2a₀ = 106 pm, and 3s has two, at 1.90a₀ and 7.10a₀, the roots of 2ρ² − 18ρ + 27 with ρ = r/a₀. At each root the radial function passes through zero and changes sign, and the node is a whole sphere, not a point.

n − l − 1 radial nodes, l angular nodal surfaces, n − 1 in total, and that total depends on n alone. In the real orbitals the angular ones divide further: |m| planes containing the z-axis, and l − |m| cones of constant θ. A complex m ≠ 0 state keeps the cones, but its φ dependence is a pure phase, so in place of the planes it vanishes only along the z-axis itself. Two checks. The 3d_z² orbital (n = 3, l = 2, m = 0) has no radial node at all — the polynomial has degree zero — and two cones, at the roots of 3cos²θ − 1, that is θ = 54.7° and 125.3°; the doughnut round its waist is simply the region between those cones, where ψ has the opposite sign, not a separate orbital. A 4d orbital (n = 4, l = 2) adds one spherical node, at r = 12a₀ = 635 pm, to the same two cones: three nodes, matching n − 1 = 3. The count takes ten seconds and kills most wrong sketches.

Because |e^(imφ)|² = 1, the density of any ψₙₗₘ is independent of φ: it is a surface of revolution about z. For m = 0 and l = 1 that gives the familiar pair of lobes along the axis, 2p_z. For m = ±1 it gives a ring around z, and the m = +1 and m = −1 densities are identical — what distinguishes them is a circulating probability current, j_φ = (mħ/mₑ r sinθ)|ψ|², and with it the orbital magnetic moment. The lobed pₓ and p_y of every chemistry text are not m states: they are the real combinations (ψ₍₂₁₋₁₎ − ψ₍₂₁₁₎)/√2 and i(ψ₍₂₁₋₁₎ + ψ₍₂₁₁₎)/√2. Both bases are legitimate, because degenerate states can be mixed without leaving the energy eigenspace, and the question decides which to use: a field along z picks the complex ones, since those are the Zeeman eigenstates, while a bond along x picks the real ones, which point where the neighbour is and carry no current.

|ψ|² is a density spread through three dimensions, so any picture must choose something to draw, and the usual choice is an isosurface enclosing a fixed fraction of the probability — conventionally 90 per cent. For 1s, integrating 4πr²|ψ|² outwards gives P(<r) = 1 − e^(−2r/a₀)(1 + 2r/a₀ + 2r²/a₀²); setting that to 0.90 gives r = 2.66a₀ = 141 pm. Nothing physical happens at that radius: just outside it the density is still 0.5 per cent of its value at the nucleus, and it never reaches zero. Draw the 99 per cent surface instead and the same orbital looks markedly bigger. So the boundary is a reporting convention, the shading inside it is |ψ|² or sometimes ψ, and none of it is a trajectory — nothing runs round the lobes, and asking how the electron crosses a node assumes a path the model never supplies.

A node is where ψ changes sign, and that sign survives into every integral you will compute with the orbital. Parity first: ψₙₗₘ(−r) = (−1)^l ψₙₗₘ(r), since inversion leaves Rₙₗ alone and Yₗ^m supplies the (−1)^l — which is what makes the dipole matrix element vanish unless l changes by an odd number, a rule angular-momentum addition then narrows to Δl = ±1. Overlap second: a positive lobe brought against a positive lobe adds amplitude and lowers the energy, against a negative one it cancels, and that is bonding against antibonding, decided by a sign the density plot has already thrown away. The complex phase does its own work: e^(imφ) carries the current and the moment −mμ_B, so the real pₓ, an equal mixture of m = +1 and −1, has zero mean L_z and no current — yet a measurement of L_z on it still returns +ħ or −ħ, never 0.