The spin-orbit interaction arises from relativistic effects: an electron moving through an electric field E experiences a magnetic field B that couples to its spin. The energy shift is proportional to S·L = (J² − L² − S²)/2, causing levels with the same (n,ℓ) but different j = ℓ ± 1/2 to split. The hydrogen 2p level splits into 2P₁/₂ and 2P₃/₂ states, confirming relativistic corrections.
Derive the spin-orbit coupling energy from the relativistic interaction of spin magnetic moment with the electric field of the nucleus. Calculate splittings for hydrogen low-n states and compare with observed spectral line splitting.
Spin-orbit coupling is a relativistic effect, not explained by a classical spinning charge in a magnetic field. The coupling depends on how fast the electron orbits (higher ℓ means smaller effect in hydrogen). The term 'Thomas precession' refers to an essential relativistic correction in deriving the factor of 1/2.
You know that an electron has spin angular momentum S and an associated magnetic moment μ_s = −g_s μ_B S/ℏ. You also know that the hydrogen 2p level has orbital angular momentum L with quantum number ℓ = 1. The question is: do S and L interact? The answer is yes, and the mechanism is relativistic.
In the electron's rest frame, the proton is moving — and a moving charge produces not just an electric field but also a magnetic field. This magnetic field B felt by the electron is proportional to the electric field E of the nucleus crossed with the electron's velocity: B ~ v × E/c². The electron's spin magnetic moment sits in this field with energy −μ_s · B. Expanding this out, v × E is proportional to the orbital angular momentum L (since L = m r × v and E points radially), so the interaction energy is proportional to S · L. This is the spin-orbit coupling term.
There is a subtlety: a naive derivation gives H_SO = (e²/2m²c²r³) S·L, but the correct relativistic treatment introduces a factor of 1/2 from Thomas precession — a purely relativistic kinematic effect that arises because the electron's rest frame is non-inertial (it accelerates in a circular orbit). Without the Thomas factor the prediction would be twice the observed splitting. With it, the spin-orbit Hamiltonian is H_SO = (1/2)(e²/2m²c²r³) S·L.
To find the energy eigenvalues, use the identity S·L = (J² − L² − S²)/2, where J = L + S is the total angular momentum. The quantum numbers j, ℓ, s are good quantum numbers for the perturbed Hamiltonian, replacing mℓ and ms. For a 2p electron (ℓ=1, s=1/2), j can be 3/2 or 1/2. The expectation value of S·L = ℏ²[j(j+1) − ℓ(ℓ+1) − s(s+1)]/2 differs for the two j values, so the single 2p level splits into 2P₃/₂ and 2P₁/₂ states separated by a small energy. This doublet splitting is directly observed as the closely spaced pair of lines in the hydrogen spectrum, and its measured magnitude matches the relativistic spin-orbit prediction — one of the early confirmations that special relativity and quantum mechanics must be unified.