I suspect the 2/x term, which is singular at the origin, is what's causing problems.

Thanks for your reply. it should be something like the wavefunction corresponding to n=1 in the attached figure.

Attachments:

The plotted wave function is not the correct one

You did not tell the readers what the correct one should be? But this eliminates the warnings you were getting, may be it will get you started. I have no idea what the plot should look like, or if your equation and parameters are correct or not.

ClearAll[x, psi, v];
v[x_] := If[0 < x < 0.5, 0, 50];
energy = 6; xMax = 1;

solution = NDSolve[{psi''[x] + (2/x) psi'[x] == -2 (energy - v[x]) psi[x],psi[xMax] == 0, psi'[xMax] == 0.001},
  psi, {x, 0.00001, xMax}, Method -> "StiffnessSwitching"]
Plot[psi[x] /. solution, {x, 0.00001, xMax}]