I'm reading A Guide to Feynman Diagrams in the Many-Body Problem by Richard D. Mattuck (2nd edition). You can look at the relevant pages here.
On page 45, he presents a formula for $D_t c_p(t)$. However, he writes no formula for $c_p(t)$ so I'm at a loss for how he derived that equation.
I just pretended to know how he got that formula and moved on to page 46 and presents (3.27). In this case, I'm not even sure what he did. I'd appreciate a more in detail walk-through of how he turned the wave function into a laplacian then a quadratic term in the integral evaluation.
Here's my shameful attempt at reconciling what's going on in (3.26). $c_p$ is the probability amplitude of a particle being in state $\phi_p$ at time $t_0$. This is almost the same definition as the Green function (propagator), (3.10), divided by the delta function. Taking the derivative of that you get something close to (3.26). Perturbing $\epsilon$ and then taking a derivative makes the $\epsilon_a-\epsilon_b$ term believable. However, I don't even heuristically know how we get the summation or the matrix potentials though.
As for (3.27) I'm even more clueless. Conversion from the potential to the integral left me scratching my head. In addition, the manipulations inside are confusing. I don't really know what to make of (3.27).
Considering this book is about the most intuitive intro to the subject that exists, my chances don't seem to be boding well...any help is appreciated.