Suppose there is a time-dependent Hamiltonian and the Schrodinger equation is solved. $$ i\hbar \partial_t U(t) = H(t) U(t) $$
Now, how easy is it to solve a scaled version of the Hamiltonian (e.g., $H'(t) = 2 H(t)$)? For the time-independent Hamiltonian, we can take $U'(t) = U(2t)$. Does a global scaling of the Hamiltonian cause the evolution to change in a non-trivial way?
We can write this with a time-ordered exponential/Dyson series for $U(t)$, which comes from $H(t)$, and $U^\prime(t)$, which comes from $\lambda H(t)$, where $\lambda$ is a positive real constant. \begin{align} U(t) &= T \exp[-i \int_0^t d\tau H(\tau)]\\ U^\prime(t) &= T \exp[-i \int_0^t d\tau \, \lambda H(\tau)]\\ &= T \exp[-i \int_0^{\lambda t} d(\lambda \tau) H(\tau)]\\ &= T \exp[-i \int_0^{\lambda t} d\tau H(\tau/\lambda)]\\ \end{align} You can get the $\lambda$ to multiply the time coordinate with a change of variables, but it's trickier than the time-independent case because of the $H(\tau / \lambda)$. The action of the Hamiltonian is "spread out" or "bunched up" over time, so if you have nontrivial time dependence things could get weird. Intuitively, the magnitude of the Hamiltonian sets one timescale which governs how the $\exp(i\omega t)$ coherence factors evolve over time, and a time-dependent Hamiltonian $H(t)$ sets another, separate timescale, and they don't have to scale together in the trivial way that you want. I don't have a minimal working example of this but I'm sure you could come up with one as a problem in Sakurai or a slight modification thereof.
