This is what the equation is that they found pinned to the wall in his house. This appears to be a set of equations from time-dependent perturbation theory in quantum mechanics. The notes describe how the wavefunction \psi(t)\ evolves over time when subjected to a perturbing Hamiltonian \V(t). The key idea here is that the wavefunction is expressed as a sum over energy eigenstates \\phi_k\, with time-dependent coefficients \c_k(t). The exponential factor \e^{-iE_k t / \hbar}\ represents the time evolution of each eigenstate under the unperturbed Hamiltonian \H_0.
The equation \sum (H_0 + V(t)) c_k(t) e^{-iE_k t/\hbar} \phi_k = i\hbar \sum \frac{\partial c_k(t)}{\partial t} e^{-iE_k t/\hbar} \phi_k \ suggests a way to derive the equation of motion for the coefficients \c_k(t)\, which govern how the perturbation causes transitions between states.
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