Evolving wave functions numerically

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Thank you to Carrie Weidner for the original guidance. All errors are mine.

When we have an Hamiltonian $H=\frac{P^2}{2m}+V(x)$, numerically evolving the system is not obvious. In finite dimensional settings, we diagonalize the Hamiltonian, and then use the time evolution operator $U=e^{tH}$ to calculate the function evolution. While our generic $H$ is not diagonal, the Potential $V(x)$ is diagonal in the position basis $|x⟩$ and our Momentum Operator $P$ is diagonal in the momentum basis $|p⟩$. We can make use of these to express our time evolution operator, with minor errors which follow the Baker-Campbell-Hausdorff formula. More specifically we get

$$\begin{array}{rll}U(t)& =e^{itH}& \\ & =e^{it\left(\frac{P^2}{2m}+V(x)\right)}& \\ & =e^{\frac{it}{2m}{P}^2+itV(x)}& \\ & =e^{\frac{it}{2m}{P}^2}{e}^{itV(x)}+\mathcal{O}(t^2)& \leftarrow \text{By taking }t\text{ small enough, trotterization provides a good approximation.}\end{array}$$

While Baker-Campbell-Hausdorff formula is not directly applicable, we might be better off using Zassenhaus formula, which expands $e^{X+Y}$ into multiple Operators.

See: https://itp.tugraz.at/~evertz/QD/Ref/Rabel_Diplomarbeit_2010.pdf