I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow this course, though I have no university physics education.)
In general, I find these lectures focus a bit too much on the math and not really on the physical motivation behind it, but so be it (if there are other courses aimed at those with reasonable math skills that focus more on physical meaning, let me know!). However, that's only indirectly related to what my question is about.
In Lecture 4, just after the 40-minute mark, Susskind sets out to derive an expression for what he earlier labelled the time-development operator $U$:
$$|\psi(t)\rangle = U(t)|\psi(0)\rangle$$
He starts out as such:
$$U(\epsilon) = I + \epsilon H$$
which makes sense because the change in time will have to be small, i.e. on the order of a small $\epsilon$. However, he then goes ahead and changes this into:
$$U(\epsilon) = I - i\epsilon H$$
which of course is still fine, because we still don't know what $H$ is supposed to be. Now my problem lies with the fact that Susskind then proceeds to derive an expression for $H$ and, subsequently, the Schrödinger equation in which it figures, from the above equation. The $i$ never gets lost and ends up in that equation.
Could we not just as easily have left the $i$ out, or put a 6 or whatever there? Why put $i$ there? I finished the entire lecture hoping Susskind would get back to this, but he never does, unfortunately. (Which is, I guess, another symptom of this course, with which I'm otherwise quite happy, occasionally lacking in physical motivation.)
For those of you who know this lecture, or similar styles of teaching: am I missing something here?
Alternatively, a general answer as to why there is an $i$ in the Hamiltonian and the Schrödinger equation?