Operator representation of p in x basis.

[qubits]

For the Schrodinger equation for a hydrogen atom, we need to write out:

p^2/2m for the electron.

If we define our basis states to be a linear discrete array of points, let's say 4 points. 0,d,2d, and 3d, where is some distance, on the order of a Bohr radius. How do I write p as an operator in that positional basis? I kind of have an idea of how to do it, but I don't want to say anything to bias your answers. I'll hear your ideas then tell you what I did.

[qubits]

Basically I want to write this in operator notation where I have defined the basis states.

\hat{H} = \frac{1}{2m} \hat{p}_{1}^2 - \frac{1}{4 \pi \epsilon_0} \frac{Z e}{\hat{r}_{1} - R}