Where does this Differential Equation comes from?

Im studying Stark Effect and im trying to prove that the second order correction to the ground state of hydrogen like atoms goes like

$$\begin{equation} \delta E^{(2)}_{100}= -\frac{1}{4}a_o^3 E^2(4+5Z^2) \end{equation}$$

And i tried to integrate directly using the basis $$|nlm_lm_s>$$ but the radial part is too difficult to make due to the infinite sum of laguerre polyomials, so i decided to try another aproach and started to follow Luis de la Peña procedure, he proposes a Operator F whos properties is that in the position basis it commutes with z and:

$$\begin{equation} = \end{equation}$$

so he can express the second order correction as:

$$\begin{equation} \delta E^{(2)}_{100}= e^2E^2<0|\hat{z} \hat{F} |0> \end{equation}$$

so i only have to know what is F right?, the problem i have is that in order to know that, he rewrites the second equation as this differential equation:

$$\begin{equation} \nabla^2F+2\frac{\nabla \Psi_{100}}{\Psi_{100}}\cdot \nabla F = \frac{2m}{\hbar}z \end{equation}$$

and then remarks is for our special case, so the question is, why like that? where does that differential equation comes from? how can i go from the bras and kets notation to that? i tried using the properties of bras and kets notation and operators but i can´t reach it, am i missing something important? do i have to go to position basis before? im not asking for solutions but i would really appreciate a hint in the right direction