The first thing that comes to mind is the fact that energy eigenstates have a periodic phase factor, $e^{iEt/\hbar}$. In general we usually say that states are defined up to a phase factor, so for a single particle in an energy eigenstate we can ignore it, but this phase factor can manifest physically when you have a state which is a superposition of energy eigenstates, each with its own oscillating phase. A simple example of this which actually results in periodic behavior would be a spin-$1/2$ particle in a magnetic field. Such a system will have a Hamiltonian proportional to the Pauli spin matrix $\sigma_z$. 789
So our time-evolution operator is $U(t) = e^{i\omega \sigma_z t}$. Recall that the Pauli matrices are idempotent, so when we Taylor expand this we can group our terms together and we will find
\begin{equation} U(t) = \cos\left(\omega t\right)+ i\sigma_z \sin\left(\omega t\right) \end{equation}
For eigenstates of $\sigma_z$ this isn't particularly interesting. The $\sigma_z$ will give us just a + for $\vert \uparrow \rangle$ or a - for $\vert \downarrow \rangle$ and of course the modulus of the state won't change but it will have a time-oscillating phase factor. Instead let's think about a superposition of such eigenstates, like $\vert \psi \rangle \equiv\frac{1}{2}\left(\vert \uparrow \rangle + \vert \downarrow \rangle\right)$.
\begin{equation} \begin{split} U(t) \vert \psi \rangle = \frac{1}{2}\cos\left(\omega t\right)\vert \left(\vert \uparrow \rangle + \vert \downarrow \rangle\right) + \frac{i}{2} \sin\left(\omega t\right) \left(\vert \uparrow \rangle - \vert \downarrow \rangle\right) \ = \frac{1}{2} e^{i\omega t} \vert \uparrow \rangle + \frac{1}{2} e^{-i \omega t} \vert \downarrow \rangle \end{split} \end{equation}
Not incidentally, this state was chosen to have a spin of $+\hbar/2$ along the $x$ direction, but we can see in the above equation that when we evolve it in time we reach a point where it will have spin $-\hbar/2$ along the $x$ direction, when $\cos(\omega t) = 0$ and $\sin(\omega t) = 1$, so the expectation value of the $x$ spin of this particle is periodic.
Because the Hamiltonian operator is Hermitian by assumption, it can be diagonalized and the operator exponential can be rewritten as a manageable <PERSON> series, but this nice periodic form only works because the square of our Hamiltonian is proportional to the identity matrix, so we could group terms in the <PERSON> expansion to give us a cosine multiplied by the identity and a sine multiplied by the $\sigma_z$ operator. In general this procedure does not work out quite as nicely, but you can always think about the interference of the phase factors I mentioned in the first paragraph. 66