Spherical harmonics as a basis on the sphere

Start from the operators, not a table. In spherical coordinates L̂_z = −iħ ∂_φ and L̂² = −ħ²[(1/sin θ)∂_θ(sin θ ∂_θ) + (1/sin²θ)∂_φ²], the angular part of −ħ²r²∇² from 9.5. Try Y = Θ(θ)Φ(φ). Φ must be a function on the circle, Φ(φ + 2π) = Φ(φ), which gives Φ = e^(imφ) with integer m — and 9.6 gave the honest reason no half-integer m can be rescued by a sign convention: its ladder never closes and leaves L²(S²). Θ, written in u = cos θ, obeys the associated Legendre equation (1 − u²)Θ″ − 2uΘ′ + [λ − m²/(1 − u²)]Θ = 0 with λ the eigenvalue of L̂²/ħ². The equation is singular at u = ±1, the two poles, and regularity there is the boundary condition. The solution regular at u = 1 is (1 − u²)^(|m|/2) times a power series in (1 − u); that series diverges at the other pole, u = −1, unless it terminates, and it terminates only for λ = ℓ(ℓ + 1) with ℓ an integer ≥ |m|. The regular solutions are P_ℓ^m(u) = (−1)^m (1 − u²)^(m/2) d^m P_ℓ/du^m for m ≥ 0, the (−1)^m being the Condon–Shortley sign, with negative m fixed by Y_ℓ^(−m) = (−1)^m Y_ℓ^m*, and normalising over the sphere fixes N_ℓm. The first few: Y₀⁰ = 1/√(4π), Y₁⁰ = √(3/4π) cos θ, Y₁^±1 = ∓√(3/8π) sin θ e^(±iφ), Y₂⁰ = √(5/16π)(3cos²θ − 1).

The inner product on L²(S²) is ⟨f|g⟩ = ∫ f*g dΩ with dΩ = sin θ dθ dφ, and under exactly that measure L̂_z and L̂² are Hermitian: integrating by parts in φ leaves no boundary term because everything is 2π-periodic, and in θ the boundary terms carry a factor sin θ that vanishes at both poles. Hermitian operators have orthogonal eigenfunctions for distinct eigenvalues, so ⟨ℓ m|ℓ′ m′⟩ = 0 whenever ℓ ≠ ℓ′ (L̂² separates them) or m ≠ m′ (L̂_z does). No integral has to be checked; only the length of each vector is a choice, and N_ℓm = √[(2ℓ+1)(ℓ−m)!/4π(ℓ+m)!] makes it 1. The measure is the whole story. With it, ⟨0 0|2 0⟩ ∝ ∫₋₁¹ (3u² − 1) du = 2 − 2 = 0. Without the sin θ, ∫₀^π (3cos²θ − 1) dθ = 3π/2 − π = π/2, and two functions that must be orthogonal appear not to be.

P_ℓ^m(u) has exactly ℓ − |m| zeros in −1 < u ⟨1, so |Y_ℓ^m|² vanishes on ℓ − |m| circles of latitude. |e^(imφ)|² = 1 adds none, which is why the density of a complex Y_ℓ^m is a figure of revolution about z. The real combinations (Y_ℓ^m ± Y_ℓ^(−m))/√2 ∝ cos mφ, sin mφ — the chemist's pₓ and d_xy — add |m| nodal planes through the poles, bringing the total to ℓ nodal lines. So Y₃¹ shows two polar circles and, in real form, one meridian plane. Parity is the map r̂ → −r̂, that is θ → π − θ and φ → φ + π: P_ℓ^m(−u) = (−1)^(ℓ−m) P_ℓ^m(u) and e^(im(φ+π)) = (−1)^m e^(imφ), so the product is (−1)^ℓ, odd for p, even for d, whatever m is. Degeneracy is the count of m: 2ℓ + 1 functions share L̂² = ℓ(ℓ+1)ħ², seven for ℓ = 3, and any linear combination of them is another eigenfunction with the same ℓ.

Given any angular function, its coordinates in the basis are c_ℓm = ⟨ℓ m|f⟩ = ∫ Y_ℓ^m* f dΩ. Take f = cos²θ. There is no φ-dependence, so only m = 0 survives, and cos²θ = 1/3 + (2/3)(3cos²θ − 1)/2 = (1/3)P₀ + (2/3)P₂. Converting with P₀ = √(4π) Y₀⁰ and P₂ = √(4π/5) Y₂⁰ gives c₀₀ = √(4π)/3 = 1.182 and c₂₀ = (4/3)√(π/5) = 1.057, every other c_ℓm zero. Parseval checks it: c₀₀² + c₂₀² = 4π/9 + 16π/45 = 4π/5 = 2.513, which is ∫cos⁴θ dΩ = 2π × 2/5 computed directly. Read as a state, ψ = √(5/4π) cos²θ is normalised and equals (√5/3)Y₀⁰ + (2/3)Y₂⁰, so a measurement of L̂² returns 0 with probability 5/9 and 6ħ² with probability 4/9, L̂_z returns 0 with certainty, and ⟨L̂²⟩ = 8ħ²/3, a value no single ℓ can give.

Completeness, Σ_ℓm Y_ℓ^m(Ω) Y_ℓ^m*(Ω′) = δ(cos θ − cos θ′) δ(φ − φ′), says every f in L²(S²) has an expansion, and it says how the truncation f_L = Σ_(ℓ≤L) fails: by Parseval, ‖f − f_L‖² = Σ_(ℓ>L)|c_ℓm|², the tail of a convergent series, so the error in the norm goes to zero. That is convergence in the mean. Make f the step that is 1 on the northern hemisphere and 0 on the southern. Only m = 0 enters, c_ℓ0 = √((2ℓ+1)π) ∫₀¹ P_ℓ du, and the even ℓ ≥ 2 vanish: c₀₀ = √π = 1.772, c₁₀ = √(3π)/2 = 1.535, c₃₀ = −√(7π)/8 = −0.586, c₅₀ = √(11π)/16 = 0.367. The ℓ ≤ 5 sum holds 5.976 of ‖f‖² = 2π = 6.283, so 95.1% of the norm and rising. At the equator, though, every P_ℓ(0) with odd ℓ is zero, so f_L(π/2) = c₀₀Y₀⁰ = 1/2 at every order: the series lands on the midpoint of the jump and never leaves it, and just beside the jump the partial sums overshoot to about 1.09 — at L = 19 and still at L = 159: the ripple narrows toward the equator as L grows but never flattens. The figure below draws exactly this.

For any central potential, H = p̂ᵣ²/2μ + L̂²/(2μr²) + V(r) commutes with L̂² and L̂_z, so the stationary states factor as R(r) Y_ℓ^m(θ, φ) with the same Y_ℓ^m whatever V is — hydrogen, the three-dimensional oscillator, a nucleon in a Woods–Saxon well. The angular basis therefore carries no energy of its own; E comes from the radial equation, in which ℓ appears only through the centrifugal term ħ²ℓ(ℓ+1)/(2μr²). Hydrogen shows one extreme: E = −13.6 eV/n² is blind to ℓ, so 2s and 2p are degenerate. The rigid rotor shows the other: freeze r at r₀, the radial equation disappears, and E_ℓ = ħ²ℓ(ℓ+1)/(2I). For HCl, I = μr₀² = 1.627 × 10⁻²⁷ kg × (1.275 × 10⁻¹⁰ m)² = 2.64 × 10⁻⁴⁷ kg m², so ħ²/2I = 2.10 × 10⁻²² J = 10.6 cm⁻¹ and the ladder reads 0, 21.2 and 63.5 cm⁻¹ for ℓ = 0, 1, 2. Same Y_ℓ^m, two utterly different spectra: the harmonics tell you the shape and the quantum numbers, and the radial physics tells you the price.