Delocalised Electron's Electric Field Given a delocalised region that exists in some region of space V, how would one evaluate the electric field from such an electron? Would it be some superposition of fields or would the charge of the electron just act from some expectation value of position or something altogether different?
 A: It depends.
In full-up quantum theory, that electric field is an operator-valued function (or, alternatively, a position-dependent operator),
$$
\hat{\mathbf E}(\mathbf r) = \frac{q}{4\pi\epsilon_0} \frac{1}{\|\mathbf r-\hat{\mathbf r}_e\|^2},
$$
where $\hat{\mathbf r}_e$ is the position of the electron, which is an operator, so therefore $\hat{\mathbf E}$ is also an operator.
Now, in many cases $-$ Hartree-Fock being the starting place for these $-$ it's normally fine to take the expectation value of this operator, giving you approaches known as mean-field methods where you essentially take the probability density $|\psi(\mathbf r_e)|^2$ as a charge density and you calculate the electric field it generates.
In other cases, though (the Auger effect is the first that comes to mind) you really do need to consider the electric field as an operator, and indeed it becomes an entangling operator
$$
V(\hat{\mathbf r_1},\hat{\mathbf r}_2) = \frac{q}{4\pi\epsilon_0} \frac{1}{\|\hat{\mathbf r}_2-\hat{\mathbf r}_2\|},
$$
on the coordinates of the two electrons involved, and it becomes the key ingredient in correlation-driven physics.
