Prob 1

cnr  =  (Sqrt[3] ThreeJSymbol[{1, 0}, {1, 0}, {0, 0}])^2/
        3

norm[n_, l_, 
        Z_]  :=  Sqrt[Z (n + l)!/(n - l - 1)!]/
          n/(2  l + 1)!

Pnl[n_, l_, Z_, r_]  :=  
      norm[n, l, 
          Z] (2 Z r/n)^(l + 1) Exp[- Z r/
              n] Hypergeometric1F1[-n + l + 1, 2 l  +  2,  
          2 Z r/n]

P1s  =  Pnl[1, 0, 3 - 5/16, r]

P2p  =  Pnl[2, 1, 1.25, r]

S1c =  2 NIntegrate[r^2 P1s^2, {r, 0, Infinity}] // N

S1v  =  NIntegrate[
          r^2 P2p^2, {r, 0, Infinity}]//N

S1v + S1c

T1v  =  -(1/3) (NIntegrate[
                P1s P2p r, {r, 0, Infinity}])^2 // N

R  =  Sqrt[S1v + S1c + 2  T1v]

Prob 3

P3d  =  Pnl[3, 2, 1, r]

Integrate[P3d^2, {r, 0, Infinity}]

P4f  =  Pnl[4, 3, 1, r]

Integrate[P4f^2, {r, 0, Infinity}]

w[r]  = - α/2/r^4

Integrate[P4f^2 w[r], {r, 0, Infinity}] /.  α ->  
        0.189

Integrate[P3d^2 w[r], {r, 0, Infinity}] /.  α ->  
        0.189