We have only kept terms to orderα^4. Now we mustsimplify two of the terms in the equation
which contain the momentum operator acting on the field.
p^2 A 0 ψ = − ̄h^2 ∇^2 A 0 ψ=−h ̄^2 ∇·~ ((∇~A 0 )ψ+A 0 ∇~ψ) =−h ̄^2 ((∇^2 A 0 )ψ+ 2(∇~A 0 )·(∇~ψ) +A 0 ∇^2 ψ)
= − ̄h^2 (∇^2 A 0 )ψ+ 2i ̄hE~·~pψ+A 0 p^2 ψ~σ·~pA 0 ~σ·~p =
̄h
i~σ·(∇~A 0 )~σ·~p+A 0 ~σ·~p~σ·~p=−̄h
i~σ·E~σ~ ·~p+A 0 p^2 =i ̄h(σiEiσjpj) +A 0 p^2= ̄h(σiσjEipj) +A 0 p^2 =ihiǫ ̄ ijkσkEipj+i ̄hE~·~p+A 0 p^2 =−h~σ ̄ ·E~×~p+i ̄hE~·~p+A 0 p^2Plugging this back into the equation, we can cancel several terms.
(
p^2
2 m
−eA 0 −p^4
8 m^3 c^2+
e ̄h~σ·E~×~p−ie ̄hE~·~p−eA 0 p^2
4 m^2 c^2)
ψ=
(
E(NR)−e− ̄h^2 (∇^2 A 0 ) + 2i ̄hE~·~p+A 0 p^2
8 m^2 c^2−eA 0p^2
8 m^2 c^2)
ψ
(
p^2
2 m−eA 0 −p^4
8 m^2 c^2+
e ̄h~σ·E~×~p−ie ̄hE~·~p
4 m^2 c^2)
ψ=(
E(NR)−e− ̄h^2 (∇^2 A 0 ) + 2i ̄hE~·~p
8 m^2 c^2)
ψ
(
p^2
2 m−eA 0 −p^4
8 m^3 c^2+
e ̄h~σ·E~×~p
4 m^2 c^2+
e ̄h^2 ∇·~ E~
8 m^2 c^2)
ψ=E(NR)ψNow we canexplicitly put in the potentialdue to the nucleus 4 Zeπrin our new units. We identify
~r×~pas the orbital angular momentum. Note that∇·~ E~=ρ=Zeδ^3 (~r). The equation can now be
cast in a more familiar form.
−eA 0 = −eφ=−eZe
4 πr
e ̄h~σ·E~×~p
4 m^2 c^2=
Ze^2 S~·~r×~p
8 πm^2 c^2 r^3=
Ze^2 ~L·S~
8 πm^2 c^2 r^3
e ̄h^2 ∇·~ E~
8 m^2 c^2=
e ̄h^2 Ze
8 m^2 c^2δ^3 (~r) =Ze^2 ̄h^2
8 m^2 c^2δ^3 (~r)(
p^2
2 m−
Ze^2
4 πr−
p^4
8 m^3 c^2+
Ze^2 L~·S~
8 πm^2 c^2 r^3+
Ze^2 ̄h^2
8 m^2 c^2δ^3 (~r))
ψ=E(NR)ψThis“Schr ̈odinger equation”, derived from the Dirac equation, agrees well with the one
we used to understand the fine structure of Hydrogen. The firsttwo terms are the kinetic and
potential energy terms for the unperturbed Hydrogen Hamiltonian. Note that our units now put a
4 πin the denominator here. (The 4πwill be absorbed into the new formula forα.) The third term
is the relativisticcorrection to the kinetic energy. The fourth term is the correctspin-orbit
interaction, including theThomas Precessioneffect that we did not take the time to understand
when we did the NR fine structure. The fifth term is the so calledDarwin termwhich we said
would come from the Dirac equation; and now it has.