5.5. Real time evolution method

In HPhi++, real time evolution calculation is done by using the following relation

\[|\Phi (t_n)\rangle = \exp^{-i {\cal H} \Delta t_n}|\Phi (t_{n-1})\rangle,\]

where \(|\Phi(t_0)\rangle\) is an initial wave function and \(t_n = \sum_{j=1}^n \Delta t_j\). In calculation, we approximate \(\exp^{-i {\cal H} \Delta t_n}\) as

\[\exp^{-i {\cal H} \Delta t_n} =\sum_{l=0}^m \frac{1}{l!}(-i {\cal H} \Delta t_n)^l .\]

Here, the cut-off integer \(m\) can be set by ExpandCoef in ModPara. We can judge whether the expansion order is enough or not by checking the norm conservation \(\langle \Phi (t_n)|\Phi (t_n)\rangle=1\) and energy conservation \(\langle \Phi (t_n)|\hat{\cal H}|\Phi (t_n)\rangle=E\).