Point-like nature of particle interaction and wave function non-locality Let us consider the Hamiltonian for the hydrogen atom
$$
\hat{\mathcal{H}}_{\mathrm{H}}=\hat{\mathcal{T}}_{\mathrm{N}}+\mathrm{\hat{\mathcal{T}}}_{\mathrm{e}}+\hat{\mathcal{V}}_{\mathrm{Ne}}=-\dfrac{\hbar^{2}}{2m_{\mathrm{N}}}\nabla_{\mathbf{R}}^{2}-\dfrac{\hbar^{2}}{2m_{\mathrm{N}}}\nabla_{\mathbf{r}}^{2}-\dfrac{1}{4\pi\epsilon_{0}}\dfrac{e^{2}}{\left|\mathbf{R}-\mathbf{r}\right|}
$$
The following discussion can be of course generalized for other systems with point-like interactions.
How the point-like nature of term $\hat{\mathcal{V}}_{\mathrm{Ne}}$ can be related to the non-local nature of the wave function describing the system?
In other words, how can a point-like interaction generate a description for the system in which the particles that interacts are described in a non-local way (by a wave function)?
 A: It's easy: the argument of the interaction potential, although it depends "locally" on $\mathbf{R}$ and $\mathbf{r}$, is uncertain due to the fact that these variables can take and do take any values in a bound state. Nothing fixes the relative and absolute distances in QM, unlike in CM.
In the Heisenberg picture, these variables are operators (matrices), not just functions of time.
A: This answer is essentially Vladimis'a answer, but is goes a bit deeper into what the wave-function means. When we say that a quantum object is point-like, we don't say that it has a fixed position. A point-like object of a fixed position is described by a delta-Dirac function. It's not the case of the electron, and proton that you ask about. 
Now, the wave-function in your case tell us that the electron may be at the position r_1, or r_2, or r_3, etc., any position where the wave-function is not zero. So, let's consider such a position, and let's give it the name r. Again, r can be r_1, r_2, etc. The same for the nucleus, it can be at R_1, R_2, etc., s.t. let's consider for it an arbitrary position where the wave-function says that the nucleus can be, and let's name it R. Well, for the electron at r and the nucleus at R, we write the Hamiltonian that you mentioned.
In other words, the Hamiltonian is valid for whatever positions r, respectively R, the two particles can take. Now it remains to find the above wave-function.
For that, we solve the Schrodinger equation containing this Hamiltonian. The wave-function we get will tell us that that r and R cannot be ANY positions in the space. For some R, the value of r will fall in some allowed range.
I hope that it is clear.
Good luck !
