Central potentials and the 3D Schrödinger equation

An isolated atom is two particles, so its wavefunction lives on six coordinates: Ψ(r₁, r₂), with H = −(ℏ²/2m₁)∇₁² − (ℏ²/2m₂)∇₂² + V(r₁ − r₂). Nothing separates in these variables, because V couples them. Change to the centre of mass R = (m₁r₁ + m₂r₂)/(m₁ + m₂) and the separation r = r₁ − r₂. The kinetic energy — the one piece of algebra worth doing by hand once — becomes −(ℏ²/2M)∇²_R − (ℏ²/2μ)∇²ᵣ, with M = m₁ + m₂ and 1/μ = 1/m₁ + 1/m₂, and the cross terms cancel identically. Now Ψ = e^(iK⋅R)ψ(r) works: the centre of mass is a free particle carrying kinetic energy ℏ²K²/2M, and every level, every orbital and every spectral line of the atom belongs to the second factor. The only thing used was that V depends on the difference of the positions, which is translation invariance — so this reduction is exact, not a model.

μ = m₁m₂/(m₁ + m₂) is smaller than either mass and is dominated by the lighter one: for hydrogen μ = mₑ × 1836.15/1837.15 = 0.9994557 mₑ. Let the nucleus grow without limit and μ → mₑ, which is the clamped-nucleus picture — so nailing the proton down is the approximation, and μ is the exact answer, not the other way round. The correction is small but never invisible. Every level of a hydrogenic system carries one factor of μ and every length one factor of 1/μ, so hydrogen's Rydberg energy is 13.5983 eV against the infinite-mass 13.6057 eV, and Lyman-α sits at 121.568 nm rather than 121.502 nm. Deuterium, with a nucleus twice as heavy, has μ = 0.9997276 mₑ, and the 0.18 nm gap between the two Balmer-α lines near 656 nm is how Urey identified deuterium in 1932 — a new isotope found in a mass ratio. Positronium, where the two masses are equal, gives μ = mₑ/2 exactly: half the binding, twice the radius.

In spherical coordinates ∇² = (1/r²)∂ᵣ(r²∂ᵣ) + (1/r²sinθ)∂_θ(sinθ ∂_θ) + (1/r²sin²θ)∂²_φ. The two angular terms are not a mess to be endured: together they are exactly −L²/ℏ², an operator built only from θ and φ, and they sit divided by exactly r². So H = −(ℏ²/2μ)(1/r²)∂ᵣ(r²∂ᵣ) + L²/(2μr²) + V(r). If V depends on r alone, then L² commutes with every term — it cannot see V(r), and the radial derivatives cannot see θ or φ. Put ψ = R(r)Y(θ, φ), divide by RY and multiply by −2μr²/ℏ². The r² cancels the 1/r² carried by the angular block, and what is left is a function of r alone on one side and a function of θ and φ alone on the other. Two such functions can be equal only if both are the same constant, and that constant is written l(l + 1). That step, and only that step, is where separability comes from.

Separation manufactures constants; boundary conditions choose their values, and each of the three comes from a different demand. Take φ first: d²Φ/dφ² = −m²Φ gives Φ = e^(imφ), and going once round the axis must return the same wavefunction, Φ(φ + 2π) = Φ(φ), so m is an integer. The θ equation then has solutions that diverge at cosθ = ±1 unless its series terminates; demanding a finite ψ on the polar axis forces l to be a non-negative integer with l ≥ |m|, so each l carries 2l + 1 values of m. Only the third constant, E, comes from the radial equation, where R must stay finite at the origin and normalisable at infinity. Notice what that ordering means: two of the three quantum numbers are fixed by geometry before the potential has even been named, and they come out the same for a nuclear well as for hydrogen. E is the only one that ever knows what V is.

The saving is large. Yₗₘ(θ, φ) is the answer for every central potential there is: the three-dimensional isotropic oscillator, the finite spherical well, the Woods-Saxon potential of a nucleus, the Yukawa and Thomas-Fermi screened forms, the confinement of a spherical quantum dot, and the Coulomb 1/r. That is why the s, p, d labels and the 2l + 1 counting carry across from atoms to nuclei to nanocrystals with no rederivation. Rotational symmetry also hands you a degeneracy for free: because H contains no preferred direction, the 2l + 1 states of a given l share one energy in any V(r), which is exactly why something with a direction in it — an applied field — is what splits them. What the potential controls is the radial equation, and through it E(n, l) and the node count. The extra l-degeneracy of hydrogen is not generic: move the exponent off −1 and states of different l separate at once, while every Yₗₘ stays untouched.

Three limits are worth naming in advance. First, two bodies only. Remove the centre of mass from a three-body atom and you are left not merely with reduced masses but with a mass-polarisation term, −(ℏ²/M)∇₁⋅∇₂ with M the nuclear mass, coupling the electrons through the recoiling nucleus; helium has no exact one-particle reduction even before the repulsion between its electrons is counted. Second, central means central. An external electric field, the r₁₂ repulsion between two electrons, a quadrupole-deformed nucleus — each puts a direction into V and kills the factorisation on the spot. Third, spherical symmetry is a far weaker requirement than a point charge. A proton of radius 0.84 fm gives a potential that is not −e²/4πε₀r inside the nucleus, yet it is still a function of r alone, so ψ = R Yₗₘ survives intact and only the radial equation changes. Keep the two ideas apart: separation is a statement about symmetry, and 1/r is a statement about the source.