Acceptable solutions of the radial Schrodinger equation for a particle in a central potential

We revisit the discussion about the boundary condition at the origin in the Schrodinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function psi does not satisfy the full Schrodinger equation. If R(r) is singular, we show that the corresponding psi follows an equation similar to Schrodinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R proportional to 1/r near r = 0, which gives rise to a normalizable psi, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students. (c) 2023 Published under an exclusive license by American Association of Physics Teachers.