I'm trying to evaluate the expectation value of some perturbation Hamiltonian $$H=\alpha \delta^3(\vec{r}),$$ where $\alpha$ is a positive constant, for the ground state wavefunction of the hydrogen atom $$\psi_{100}~\propto~ \exp[-r/a]$$ (I want to calculate the shift in the energy in first order perturbation theory). Why is it wrong to write:
$$\langle\psi_{100}|H|\psi_{100}\rangle ~\propto~ \int_0^\infty dr~ r^2 \exp[-2r/a] \alpha \delta(r) ~=~ 0$$
Does it have to do with either:
I can't just write $\delta^3(\vec{r}) = \delta(r)$ in the integral over $r$, or
I can't evaluate the delta function at zero, since it is at the endpoint of the integration limits [not inside the interval $(0,\infty)$]?