\(\(\( (*Here\ we\ solve\ the\ Thomas - 
        Fermi\ equation . \ \ We\ try\ different\ initial\ condiations\ as\ discussed\ by\ Feynman\ et\ al, \ 
      PR75\ \((1561)\)\ 1949*) \)\(
\)\(sol2\  = 
      NDSolve[{x^{1/2}\ \(tf''\)[x]\ \  - tf[x]^{3/2}\  == 0, 
	
          tf[0.00000000002] == 
            1, \(tf'\)[
              0.00000000002] == \ \(-1.58878\)}, {tf}, 
\ {x,  .00000000002, 
          7.2}, MaxSteps\  -> 100000, 
        WorkingPrecision\  -> \ 18]

    tfs[x_] = tf[x] /. sol2

    Evaluate[tf[1/2] /. sol2]

    Plot[{Evaluate[tfs[x]]}, {x, 0.002, 7.2}, 
      AxesLabel -> {"x", "chi"}]\)\)\)

\(sol2\  = 
      NDSolve[{x^{1/2}\ \(tf''\)[x]\ \  - tf[x]^{3/2}\  == 0, 
	
          tf[0.00000000002] == 
            1, \(tf'\)[
              0.00000000002] == \ \(-1.55\)}, {tf}, 
\ {x,  .00000000002, 
          7}, MaxSteps\  -> 100000, 
        WorkingPrecision\  -> \ 18]\)
Evaluate[tf[1/2] /. sol2]
Plot[{Evaluate[tf[x] /. sol2]}, {x, 0.002, 7}, 
      AxesLabel -> {"x", "chi"}]

b =  
      0.885 92^{-1/3}

    V[r_] =  -1/r (92)/137 tfs[r/b]

    Plot[{Evaluate[V[r]]}, {r, 0.04,  .7}, 
      AxesLabel -> {"r", "V"}]