The phase factor depends on the conventions we choose for the statesYjℓmj. For our conventions,
the factor is−1.
~σ·~x
r
YjℓmAj=−YjℓmBjWe now have everything we need toget to the radial equations.
c1
r~σ·~x
r(
−i ̄hr∂
∂r+i~σ·~L)(
if(r)Y
mj
jℓB
g(r)YjℓmAj)
=
(
E−V(r)−mc^20
0 E−V(r) +mc^2)(
g(r)Y
mj
jℓA
if(r)YjℓmBj)
c1
r~σ·~x
r
(
−i ̄hr∂r∂ +i~σ·~L)
if(r)Y
mj
( jℓB
−i ̄hr∂r∂ +i~σ·~L)
g(r)Y
mj
jℓA
=
(
E−V(r)−mc^20
0 E−V(r) +mc^2)(
g(r)YjℓmAj
if(r)YjℓmBj)
c
1
r~σ·~x
r( (
̄hr∂r∂ −(κ−1) ̄h)
f(r)YjℓmBj
(
−i ̄hr∂r∂ +i(−κ−1) ̄h)
g(r)YjℓmAj)
=
(
E−V(r)−mc^20
0 E−V(r) +mc^2)(
g(r)YjℓmAj
if(r)YjℓmBj)
̄hc1
r~σ·~x
r( (
r∂r∂ −(κ−1))
f(r)YjℓmBj
(
−ir∂r∂ −i(1 +κ))
g(r)YjℓmAj)
=
(
E−V(r)−mc^20
0 E−V(r) +mc^2)(
g(r)YjℓmAj
if(r)YjℓmBj)
̄hc1
r((
−r∂r∂ + (κ−1))
f(r)YjℓmAj
(
ir∂r∂ +i(1 +κ))
g(r)YjℓmBj)
=
(
E−V(r)−mc^20
0 E−V(r) +mc^2)(
g(r)Y
mj
jℓA
if(r)YjℓmBj)
̄hc1
r((
−r∂r∂ + (κ−1))
( f(r)
r∂r∂ + (1 +κ))
g(r))
=
(
E−V(r)−mc^20
0 E−V(r) +mc^2)(
g(r)
f(r))
̄hc
(
−∂f∂r+(κ−r1)f)
(
∂g
∂r+(1+κ)
r g)
=
(
(E−V−mc^2 )g
(E−V+mc^2 )f)
This is now a set of two coupled radial equations. We can simplify them abit by making the
substitutionsF=rfandG=rg. The extra term from the derivative cancels the 1’s that are with
κs.
̄hc((
−^1 r∂F∂r+rF 2 +κFr 2 −rF 2)
( 1
r∂G
∂r−G
r^2 +G
r^2 +κG
r^2)
)
=
(
(E−V−mc^2 )Gr
(E−V+mc^2 )Fr)
̄hc((
−∂F∂r+κFr)
(∂G
∂r+κG
r)
)
=
(
(E−V−mc^2 )G
(E−V+mc^2 )F)
((∂F
∂r−κF
r)
(∂G
∂r+κG
r)
)
=
(mc (^2) −E+V
̄hc G
mc^2 +E−V
̄hc F
)
These equations are true for any spherically symmetric potential. Now it is time tospecialize to
the hydrogen atomfor which V ̄hc =−Zαr. We definek 1 = mc
(^2) +E
̄hc andk^2 =
mc^2 −E
̄hc and the
dimensionlessρ=
√
k 1 k 2 r. The equations then become.
((∂F
∂r−κF
r)
(∂G
∂r+κG
r)
)
=
((
k 2 −Zαr)
( G
k 1 +Zαr