Problem Set 4

Prob. 2

First, express the radial wave functions P and Q in terms of F1 and F2

x  =  2 λ r

P[x_]  :=  
      Sqrt[1 + ϵ] Exp[-x/2] (F1[x] + F2[x])

Q[x_]  :=  
      Sqrt[1 - ϵ] 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/r Q[x])  == 
        c^2 ϵ P[x]

eq2  =  (-Z/r - c^2) Q[x]  -  
          c (D[P[x], r] + κ/r P[x])  == 
        c^2 ϵ Q[x]

Now solve the equations for dF1/dx and dF2/dx

Solve[{eq1, 
        eq2}, {F1'[2 λ r], F2'[
          2 λ r]}]

Simplify the result;  dF1/dx -> f1p  dF2/dx -> f2p

f1p  =  
      Expand[FullSimplify[F1'[
              2 λ r]  /.  %6]]

f2p  =  
      Expand[FullSimplify[F2'[2 λ r]  /.  %6]]

Find coefficients of F1 and F2 on RHS of equations

f11  =  Coefficient[f1p,  F1[2 λ r]]

f12  =    Coefficient[f1p,  
        F2[2 λ r]]

f21  =  Coefficient[f2p,  F1[2 λ r]]

f22  =  Coefficient[f2p,  F2[2 λ r]]

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  ->  
          c Sqrt[1 - ϵ^2]]

Simplify[
      f22[[1, 1]] + f22[[1, 2]] +  
          f22[[1, 4]] /.  λ  ->  
          c Sqrt[1 - ϵ^2]]

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/(2 λ)

g11 + g12

g21 + g22

Therefore, noting that   λ = c (Sqrt[1 - (ϵ^2)]), we find

(dF1/dx)= g11 + g12 =  Z(ϵ/(λ x)) F1[x] +((Z/( λ x)) - (κ/x)) F2[x]
(dF2/dx)=  g21 + g22  = (-(Z/( λx)) - (κ/x)) F1[x]+ (1 - ((Z ϵ)/( λ x))) F2[x]