I'm looking for a simple expression of the solution to the Schrödinger equation
$$ \begin{equation} i\hbar\frac{\partial}{\partial t} \left|\Psi(t)\right> = H(t) \left|\Psi(t)\right>. \end{equation} $$
For a constant Hamiltonian, the solution can be written as
$$ \begin{equation} \left|\Psi(t)\right> = e^{-\frac{i}{\hbar} Ht} \left|\Psi(0)\right>. \end{equation} $$
But how can we write the solution if $H$ is not constant in a simple way? What I'm able to come up with is
$$ \begin{equation} \left|\Psi(t)\right> = \lim_{N\to\infty} \left(1-\frac{i}{\hbar} H\left(\frac{n\cdot t}{N}\right)\frac{t}{N}\right) \left(1-\frac{i}{\hbar} H\left(\frac{(n-1)\cdot t}{N}\right)\frac{t}{N}\right) \cdots \left(1-\frac{i}{\hbar} H\left(\frac{1\cdot t}{N}\right)\frac{t}{N}\right) \left|\Psi(0)\right>, \end{equation} $$
or perhaps slightly neater,
$$ \begin{equation} \left|\Psi(t)\right> = \lim_{N\to\infty} H\left(\frac{n\cdot t}{N}\right)^{-\frac{i}{\hbar}\frac{t}{N}} H\left(\frac{(n-1)\cdot t}{N}\right)^{-\frac{i}{\hbar}\frac{t}{N}} \cdots H\left(\frac{1\cdot t}{N}\right)^{-\frac{i}{\hbar}\frac{t}{N}} \left|\Psi(0)\right>, \end{equation} $$
but none of these expression are very elegant. Looking at the Wikipedia article for the product integral, these expressions look very similar to the Volterra integral and the geometric integral, respectively, except for the fact that the factors seem to come in the reverse order in those integrals.
Is there some standard way to write product integrals in QM/QFT? If I use a geometric product integral, can I simply announce that the factors in the integral for increasing values of $t$ should be placed to the left rather than to the right in the product? Could this make the geometric product integral a suitable way to express an evolution operator $U(t_2,t_1)$ in a simple way, such that $\left|\Psi(t_2)\right> = U(t_2,t_1)\left|\Psi(t_1)\right>$?
