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I think some things to remember would be that the wavefunction gives us the energy of the electron, Ψ represents the height of a wave at a position, and (Ψ)^2 represents the probability of finding an electron. Also, the H (hamiltonian) gives us a double derivative of the sin/cos wave function.
I think the most important conceptual aspect of Schrodinger's equation is just how it led to the wave functions and orbitals that we use for electron configuration and other electronic structure. We don't need to know the complicated math stuff!
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