As a former physicist, I never understood the full math behind Schrodinger's equation. Since then I ventured into CS, so I wonder if this book will be a good refreshment.
Look at it again with that understanding in your head: https://en.wikipedia.org/w/index.php?title=Special:MathWikib...
H acting on a state gives you the time translation of the state. That's the crux of it.