We will be attempting to get the correct Schr ̈odinger equation to orderα^4 , like the one we used to
calculate the fine structure in Hydrogen. Since this energy term weare expanding is multiplied in
the equation byp^2 , we only need the first two terms in the expansion (order 1 and orderα^2 ).
~σ·~p1
2 m(
1 −
E(NR)+eA 0
2 mc^2)
~σ·~pψA = (E(NR)+eA 0 )ψAThenormalization conditionwe derive from the Dirac equation is
j 0 =ψγ ̄ 4 ψ=ψ†γ 4 γ 4 ψ=ψ†ψ=ψ†AψA+ψ†BψB= 1.
ψB≈
pc
2 mc^2ψAψA†ψA+ψB†ψB≈(
1 +
( pc
2 mc^2) 2 )
ψA†ψA= 1
(
1 +p^2
4 m^2 c^2)
ψA†ψA= 1ψ≡(
1 +
p^2
8 m^2 c^2)
ψAψA≡(
1 −
p^2
8 m^2 c^2)
ψWe’ve definedψ, the 2 component wavefunction we will use, in terms ofψAso that it is properly
normalized, at least to orderα^4. We cannow replaceψAin the equation.
~σ·~p1
2 m(
1 −
E(NR)+eA 0
2 mc^2)
~σ·~p(
1 −
p^2
8 m^2 c^2)
ψ = (E(NR)+eA 0 )(
1 −
p^2
8 m^2 c^2)
ψThis equation is correct, but not exactly what we want for the Schr ̈odinger equation. In particular,
we want toisolate the non-relativistic energy on the right of the equationwithout other
operators. We can solve the problem by multiplying both sides of the equation by
(
1 − p2
8 m^2 c^2)
.
(
1 −
p^2
8 m^2 c^2)
~σ·~p1
2 m(
1 −
E(NR)+eA 0
2 mc^2)
~σ·~p(
1 −
p^2
8 m^2 c^2)
ψ=
(
1 −
p^2
8 m^2 c^2)
(E(NR)+eA 0 )(
1 −
p^2
8 m^2 c^2)
ψ
(
~σ·~p~σ·~p
2 m−
p^2
8 m^2 c^2~σ·~p~σ·~p
2 m−
~σ·~p
2 mE(NR)+eA 0
2 mc^2~σ·~p−~σ·~p~σ·~p
2 mp^2
8 m^2 c^2)
ψ=
(
(E(NR)+eA 0 )−
p^2
4 m^2 c^2E(NR)−
p^2
8 m^2 c^2eA 0 −eA 0
p^2
8 m^2 c^2)
ψ(
p^2
2 m
−
p^2
8 m^2 c^2p^2
2 m−
p^2
2 mE(NR)
2 mc^2−
e~σ·~pA 0 ~σ·~p
4 m^2 c^2−
p^2
2 mp^2
8 m^2 c^2)
ψ=
(
(E(NR)+eA 0 )−p^2
4 m^2 c^2E(NR)−
p^2
8 m^2 c^2
eA 0 −eA 0p^2
8 m^2 c^2)
ψ
(
p^2
2 m−
p^4
8 m^3 c^2−eA 0 −e~σ·~pA 0 ~σ·~p
4 m^2 c^2)
ψ=(
E(NR)−
p^2
8 m^2 c^2eA 0 −eA 0p^2
8 m^2 c^2)
ψ