Problem Set 10

Prob 1

Answer given sepaeately.

Prob 2

Z = 10 - 5/16

α  =  1/137.036

γ  =  Sqrt[1 - α^2 Z^2]

P1s[r_]  =  
      Sqrt[(1 + γ)/2] Sqrt[
          2 Z/Gamma[2  γ + 1]] (2  
              Z r)^γ Exp[-Z r]

Q1s[r_]  =  
      Sqrt[(1 - γ)/2] Sqrt[
          2 Z/Gamma[2  γ + 1]] (2  
              Z r)^γ Exp[-Z r]

Check that the wave functions are normalized.

Integrate[P1s[r]^2 + Q1s[r]^2, {r, 0, Infinity}]

Evaluate the inner integral

Y1[r_]  =  
      Integrate[x P1s[x] Q1s[x], {x, 0, r}, Assumptions -> r > 0]/
          r^2  +  
        r Integrate[P1s[x] Q1s[x]/x^2, {x, r, Infinity}, 
            Assumptions ->  r > 0]

(8/3) NIntegrate[
        P1s[r] Q1s[r] Y1[r], {r, 0, Infinity}]

This value agrees closely with the value 0.012110 for Z=10 given in the notes.