A fundamental principle of quantum mechanics, as far as I can tell, states that the Hamiltonian generates time evolution. A common result about generators are the following: let $\mathrm T$ be the Hermitian generator of $\mathrm U(\tau)$, then we have
$$i\frac{\partial}{\partial\tau}\mathrm U\vert\psi\rangle = \mathrm T\vert\psi\rangle$$
If $\mathrm H$ generates $\mathrm U(t)$ (specifically time evolution this time) then, writing for the evolved state $\vert\psi'\rangle=\mathrm U\vert\psi\rangle$:
$$i\frac{\partial\vert\psi'\rangle}{\partial t} = \mathrm H\vert\psi\rangle$$ and we've lost an $\hbar$ from the real TDSE. Does this then actually imply that time evolution is generated by $\frac{\mathrm H}{\hbar}$? I ask only because I've heard from multiple places the exact statement "the Hamiltonian generates time evolution".