<P> Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects . </P> <Dl> <Dd> E n (1) = ⟨ ψ 0 H ′ ψ 0 ⟩ = − 1 8 m 3 c 2 ⟨ ψ 0 p 4 ψ 0 ⟩ = − 1 8 m 3 c 2 ⟨ ψ 0 p 2 p 2 ψ 0 ⟩ (\ displaystyle E_ (n) ^ ((1)) = \ left \ langle \ psi ^ (0) \ right \ vert H' \ left \ vert \ psi ^ (0) \ right \ rangle = - (\ frac (1) (8m ^ (3) c ^ (2))) \ left \ langle \ psi ^ (0) \ right \ vert p ^ (4) \ left \ vert \ psi ^ (0) \ right \ rangle = - (\ frac (1) (8m ^ (3) c ^ (2))) \ left \ langle \ psi ^ (0) \ right \ vert p ^ (2) p ^ (2) \ left \ vert \ psi ^ (0) \ right \ rangle) </Dd> </Dl> <Dd> E n (1) = ⟨ ψ 0 H ′ ψ 0 ⟩ = − 1 8 m 3 c 2 ⟨ ψ 0 p 4 ψ 0 ⟩ = − 1 8 m 3 c 2 ⟨ ψ 0 p 2 p 2 ψ 0 ⟩ (\ displaystyle E_ (n) ^ ((1)) = \ left \ langle \ psi ^ (0) \ right \ vert H' \ left \ vert \ psi ^ (0) \ right \ rangle = - (\ frac (1) (8m ^ (3) c ^ (2))) \ left \ langle \ psi ^ (0) \ right \ vert p ^ (4) \ left \ vert \ psi ^ (0) \ right \ rangle = - (\ frac (1) (8m ^ (3) c ^ (2))) \ left \ langle \ psi ^ (0) \ right \ vert p ^ (2) p ^ (2) \ left \ vert \ psi ^ (0) \ right \ rangle) </Dd> <P> where ψ 0 (\ displaystyle \ psi ^ (0)) is the unperturbed wave function . Recalling the unperturbed Hamiltonian, we see </P>

Fine structure splitting of spectral lines in hydrogen