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} <k|\hat{F}\hat{H_0} -\hat{H_0}\hat{F}|0>=<k|\hat{z}|0> \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


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