Solutions to ProblemSet 3

Prob 1

Plot  (Subscript[P, nl])(r) for n=4 and all values of l

coul[n_, l_, r_]  =  
      Sqrt[Z (n + l)!/(n - l - 1)!]/(n (2 l + 
                    1)!)  (2  Z r/n)^(l + 1) Exp[-Z r/
              n] Hypergeometric1F1[l + 1 - n, 2  l + 2, 
          2 Z r/n]

Z = 2

Plot[coul[4, 0, r], {r, 0, 40}]

Plot[coul[4, 1, r], {r, 0, 40}]

Plot[coul[4, 2, r], {r, 0, 40}]

Plot[coul[4, 3, r], {r, 0, 40}]

Clear[Z]

Prob 2

Verify orthonormality of (Subscript[P, nl]) and (Subscript[P, ml]) for l=1 and {m,n} <= 5

Table[
        Integrate[coul[n, 1, r] coul[m, 1, r], {r, 0, Infinity}, 
          Assumptions ->  Z > 0], {n, 2, 5}, {m, 2, 5}] // 
      MatrixForm

Prob 3

Finite nuclear size correction to energies for states with l=0

f0[r_]  = 
        Normal[ Series[
            coul[n, 0, 
                  r]^2 (-(Z/R) (3/
                          2 - (1/2) (r^2/R^2)) + Z/r), {r, 0, 
              4}]];

g0[n_]  =  Expand[Integrate[f0[r], {r, 0, R}]]

de0[n_] =  Normal[Series[g0[n], {R, 0, 2}]]

Finite nuclear size correction for states with l=1

f1[r_]  = 
        Normal[ Series[
            coul[n, 1, 
                  r]^2 (-(Z/R) (3/
                          2 - (1/2) (r^2/R^2)) + Z/r), {r, 0, 
              6}]];

g1[n_]  =  Expand[Integrate[f1[r], {r, 0, R}]]

de1[n_]  =  Simplify[Normal[Series[g1[n], {R, 0, 4}]]]

Corrections for H in (cm^(-1)):   R = 0.85 fm,  1 au = 219474.6 (cm^(-1))

Z = 1

R =  
      0.85 10^(-13) /(0.529177 10^(-8))

e2s  = 219474.6  de0[2]

e2p  =  219474.6 de1[2]

Corrections for U in eV :   R = 0.836 (A^(1/3))+ 0.570 = 5.75 fm,  1 au = 27.211 eV

Z = 92

R =  (0.836 (238)^(1/3)  +  
            0.570) 10^(-13) / (0.529177 10^(-8))

e2s  =  27.211  de0[2]

e2s  =  27.211  de1[2]

Clear[Z, R]