Spinor spherical harmonics and the action of σ⋅r̂

Units are SI with ħ kept, and the metric signature is (+, −, −, −) wherever a four-vector appears; none is needed here. Quoted from the previous topic: in a central field H commutes with J², J_z, the parity operator βP and K = β(Σ⋅L + ħ), whose eigenvalue −κħ has κ = −(j + ½) when l = j − ½ and κ = +(j + ½) when l = j + ½. Write ψ as an upper pair over a lower pair. Since β = diag(1, −1) and Σ = diag(σ, σ), K acts as +(σ⋅L + ħ) on the upper pair and as −(σ⋅L + ħ) on the lower. One eigenvalue −κħ for the whole spinor therefore needs σ⋅L + ħ to give −κħ above and +κħ below: the lower pair is the harmonic labelled −κ. Same |κ|, so the same j; opposite sign, so the other orbital value, l′ = 2j − l. For 2P_(3/2), κ = −2, l = 1 and l′ = 2. Parity agrees: βP gives the upper pair (−1)ˡ and the lower −(−1)^(l′) = (−1)ˡ, one parity for the state, read from the upper l.

χ_κᵐ is the coupled state |l, ½, j, m⟩ of Quantum Mechanics II, quoted with Condon-Shortley phases and l coupled before s, read as a two-component function of direction. For j = l + ½, κ < 0, it is √((l + m + ½)/(2l + 1)) Yₗ^(m−1/2) ↑ + √((l − m + ½)/(2l + 1)) Yₗ^(m+1/2) ↓. For j = l − ½, κ > 0, the coefficients become −√((l − m + ½)/(2l + 1)) and +√((l + m + ½)/(2l + 1)). So χ₋₁¹⁄² = Y₀⁰ ↑ = ↑/√(4π) is the s_(1/2) harmonic, and χ₁¹⁄² = −√(1/3) Y₁⁰ ↑ + √(2/3) Y₁¹ ↓ = −(cos θ ↑ + sin θ e^(iφ) ↓)/√(4π) is p_(1/2). Each is dimensionless and orthonormal over the sphere, so |χ_κᵐ|² is a density per steradian. On each, σ⋅L = 2L⋅S/ħ takes the value [j(j + 1) − l(l + 1) − ¾]ħ = −(κ + 1)ħ: 0 for s_(1/2), +ħ for p_(3/2), −2ħ for p_(1/2).

σ⋅r̂ = σₓ sin θ cos φ + σ_y sin θ sin φ + σ_z cos θ depends on direction only. It is Hermitian, and (σ⋅a)(σ⋅b) = a⋅b + iσ⋅(a×b) gives (σ⋅r̂)² = r̂⋅r̂ = 1, so it is unitary as well. It is the dot product of two vectors, r̂ turned by L and σ turned by S, so it commutes with J = L + (ħ/2)σ and cannot change j or m. Under parity r̂ reverses and σ, an axial vector, does not: σ⋅r̂ is a pseudoscalar and reverses orbital parity. Of the harmonics with given j and m exactly one has the other parity, so σ⋅r̂χ_κᵐ = cχ_(−κ)ᵐ. The same follows algebraically from the identity (σ⋅L + ħ)(σ⋅r̂) = −(σ⋅r̂)(σ⋅L + ħ), derived from r̂⋅L = 0 and L×r̂ = −r̂×L + 2iħr̂: σ⋅r̂ turns the eigenvalue −κħ into +κħ. Hermiticity and (σ⋅r̂)² = 1 force |c| = 1, and because σ⋅r̂ is a rotational scalar, c is the same for every m. A single evaluation fixes it.

Look along +z, where θ = 0. Every Yₗ^(mₗ) vanishes there except Yₗ⁰ = √((2l + 1)/(4π)), so for m = ½ only the spin-up term with mₗ = 0 survives. For κ < 0, with l = |κ| − 1, the coefficient √((l + 1)/(2l + 1)) gives χ_κ¹⁄²(ẑ) = +√(|κ|/(4π)) ↑; for κ > 0, with l = |κ|, the coefficient −√(l/(2l + 1)) gives −√(|κ|/(4π)) ↑. Partners of equal |κ| are equal and opposite on the axis. There σ⋅r̂ = σ_z and σ_z ↑ = ↑, so σ⋅r̂ leaves χ_κ¹⁄²(ẑ) unchanged, which is −χ_(−κ)¹⁄²(ẑ): c = −1, derived. For 1s, 0.2821 ↑ against χ₁¹⁄²(ẑ) = −0.2821 ↑. One c serves every m and direction, so σ⋅r̂χ_κᵐ = −χ_(−κ)ᵐ everywhere, and |χ_(−κ)ᵐ|² = |χ_κᵐ|² point by point because σ⋅r̂ is unitary in each direction. The −1 is a convention: coupling s before l flips it, F changes sign with it, and no observable moves.

Put (σ⋅r̂)² = 1 in front of σ⋅p and multiply out the right-hand pair: σ⋅p = (σ⋅r̂)[r̂⋅p + iσ⋅(r̂×p)]. Here r̂⋅p = −iħ∂/∂r, and r̂×p = L/r with 1/r on the left, so σ⋅p = (σ⋅r̂)[−iħ∂/∂r + iσ⋅L/r], derived. Act on (G/r)χ_κᵐ. The derivative gives −iħ(G′/r − G/r²)χ_κᵐ and the angular term gives −iħ(κ + 1)(G/r²)χ_κᵐ, so the bracket is −(iħ/r)(G′ + κG/r)χ_κᵐ: the G/r² from differentiating G/r has cancelled the 1 in κ + 1. Then σ⋅r̂ supplies −χ_(−κ)ᵐ, and σ⋅p[(G/r)χ_κᵐ] = (iħ/r)(dG/dr + κG/r)χ_(−κ)ᵐ, derived. The lower pair follows from κ → −κ: σ⋅p[(F/r)χ_(−κ)ᵐ] = (iħ/r)(dF/dr − κF/r)χ_κᵐ. With ψ = (1/r)(Gχ_κᵐ, iFχ_(−κ)ᵐ), every term of the Dirac equation now carries one of the two harmonics, which divide out; the next topic writes the two radial equations that remain.

G is in m^(−1/2), so (G/r)χ_κᵐ is in m^(−3/2) like ψ, and (ħ/r)(dG/dr) and (ħ/r)(κG/r) both carry kg m s⁻¹ times m^(−3/2), as σ⋅p acting on ψ must. Applied twice, the result must give p², since (σ⋅p)² = p²: it gives −(ħ²/r)[G″ − κ(κ + 1)G/r²]χ_κᵐ, and κ(κ + 1) = l(l + 1) for either sign, 2 for κ = −2 and for κ = +1, which is Quantum Mechanics II's radial kinetic operator. For a free spherical wave, G equal to r times the spherical Bessel function of order l at kr, the Bessel recurrences turn dG/dr + κG/r into −kr (κ < 0) or +kr (κ > 0) times the Bessel function of order l′, as a free particle of orbital l′ requires. The measured check runs through σ⋅L = −(κ + 1)ħ: Unit 6's spin-orbit term, quoted, weights 2P_(3/2) by +1 and 2P_(1/2) by −2, a splitting of mc²α⁴/32 = 4.53 × 10⁻⁵ eV, or 10.95 GHz. Hydrogen's measured interval is 10.97 GHz; the 0.2% rest is mostly the Lamb shifts.