Finite Spherical Potential Well

[dsr39]

This is more a qualitative question than a specific homework question, but a homework problem got me wondering about it.

I was solving the finite potential well.

$$V(r) = 0 \hspace{1cm} r \geq a$$
$$V(r) = -V_0\hspace{1cm} r < a$$

I am trying to solve for the ground state energy. When I find the forms of the solution in the interior of the well, I find that I get

$$\frac{c_1 \sin{(kr)} + c_2 \cos{(kr)}}{r}$$

I know from doing other reading that I should end up throwing away the cosine term, but I do not understand why.

I can see that it blows up at $$r=0$$, but it still looks like it will be normalizable to me since a volume integral in spherical coordinates provides an extra factor of $$r^2$$

 

[Avodyne]

It doesn't satisfy the Schrödinger equation, because $\nabla^2(1/r) \propto \delta^3(\vec x)$.

 

[dsr39]

I don't understand that reply. I got it by solving the Schrödinger equation, so it must satisfy it, no?