I was reading through the proof of the Adiabatic Theorem (in Sakurai) and I realised I'm not quite sure how Schrodinger Basis kets behave when we have a time-dependent Hamiltonian. I know that with a time-independent Hamiltonian the basis kets don't change in the Schrodinger Picture.
So if $|n;t\rangle$ are the energy eigenkets of $H(t)$ at time $t$ and $|\alpha;t\rangle$ is an arbitrary state at time $t$, is the following at all true? \begin{align*} |\alpha;t\rangle = \sum_n c_n(t)|n;t\rangle = \sum_n c_n(t) e^{i\theta_n(t)}|n,t_0\rangle \end{align*} where $\theta_n(t) = -\frac{1}{\hbar}\int_{t_0}^t H(t')\,dt'$ and $e^{i\theta_n(t)}$ is a time-evolution operator
Wikipedia and Sakurai both have (each in different notation): \begin{align*} |\alpha;t\rangle = \sum_n c_n(t) e^{i\theta_n(t)}|n;t\rangle \end{align*} I feel like I'm not understanding this properly at all