Atomic Structure Theory

Solutions 273
2.11
Below are mathematica statements used to verify the reduction.
First, express the radial wave functions P and Q in terms of F1 and F2.
In the equations below, we set E/c 2 = ɛ for simplicity:
x =2λr
P [x ]:=Sqrt[1 + ɛ]Exp[−x/2](F1[x]+F2[x])
Q[x ]:=Sqrt[1 − ɛ]Exp[−x/2](F1[x] − F2[x])
Next, write down the radial Dirac equations:
eq1 = (−Z/r + c ∧ 2)P [x]+c(D[Q[x],r] − κ/rQ[x]) ==c ∧ 2ɛP [x]
eq2 = (−Z/r − c ∧ 2)Q[x] − c(D[P [x],r]+κ/rP [x]) ==c ∧ 2ɛQ[x]
Now we solve the equations for dF1/dx and dF2/dx:
sol = Solve[{eq1, eq2}, {F1 ′ [2λr], F2 ′ [2λr]}]
Simplify the result; dF1/dx → f1p dF2/dx >→ f2p:
f1p = Expand[FullSimplify[F1 ′ [2λr]/.sol]]
f2p = Expand[FullSimplify[F2 ′ [2λr]/.sol]]
Find coefficients of F1 and F2 on RHS of equations:
f11 = Coefficient[f1p, F1[2λr]]
f12 = Coefficient[f1p, F2[2λr]]
f21 = Coefficient[f2p, F1[2λr]]
f22 = Coefficient[f2p, F2[2λr]]
Combine 1st, 2nd, and 4th terms of f11 and of f22 to further simplify:
Simplify[f11[[1, 1]] + f11[[1, 2]] + f11[[1, 4]]/. λ->cSqrt[1 − ɛ ∧ 2]]
0
Simplify[f22[[1, 1]] + f22[[1, 2]] + f22[[1, 4]]/. λ->cSqrt[1 − ɛ ∧ 2]]
1
Recombine the simplified coefficients and write out the resulting RHS terms:
g11 = f11[[1, 3]]F1[x]
g12 = f12[[1]]F2[x]
g21 = f21[[1]]F1[x]
g22 = (1 + f22[[1, 3]])F2[x]
Clear[x]
r = x/(2λ)
g11 + g12
ZɛF1[x]
cx √ +
1−ɛ2 g21 + g22
− Z
cx √ κ −
1−ɛ2 x
Z
cx √ κ −
1−ɛ2 x
F1[x]+
F2[x]
1 − Zɛ
cx √ 1−ɛ2
F2[x]
Therefore, noting that λ =c √ 1 − ɛ 2 , we find:
dF1
dx
dF2
dx
= g11 + g12 = Zɛ
λx
= g21 + g22 =(-Z
λx
F1[x] +(Z
λx
- κ
x
2.12 Start with the identity:
- κ
x
) F1[x]+ (1- Zɛ
λx
) F2[x]
) F2[x]
(Eb − Ea) 〈φb|rk|φa〉 = 〈φb|[H, rk]|φa〉 .