Info

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

When we have an Hamiltonian , numerically evolving the system is not obvious. In finite dimensional settings, we diagonalize the Hamiltonian, and then use the time evolution operator to calculate the function evolution. While our generic is not diagonal, the Potential is diagonal in the position basis and our Momentum Operator is diagonal in the momentum basis . 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 p

While Baker-Campbell-Hausdorff formula is not directly applicable, we might be better off using Zassenhaus formula, which expands into multiple Operators.