Numerical Solutionand Structural ReconstructionoftheTime-Dependent SchrödingerEquation

This work proposes a structural reconstruction of the time-dependent Schrödinger equation (TDSE) by incorporating non- Hermitian dissipation terms inspired by the Information Dynamical System(IDS)framework. Weimplementanumericalsolverforafree particlesystemusingfinitedifferencediscretizationandRunge-Kutta 45 (RK45) integration, with a Gaussian wave packet as the initial condition. The model includes dynamic external potentials and a gradient-based dissipation term, Γ(𝑥,𝑡)∝𝜕𝑥|𝜓(𝑥,𝑡)|, enabling irre- versible dynamics. Compared to traditional Hermitian solvers, our approachenhancesnumericalstabilityviaadaptiveabsorbingbound- aries and sparse matrix acceleration, avoiding costly Crank-Nicolson schemes. Robustness tests across spatial step size (𝑑𝑥), wave packet width (𝜎), wave number (𝑘0), and time span confirm stability and accuracy. Results show normalization errors below 10−6 for Hermi- tian cases and controlled norm decay in non-Hermitian cases, with energy expectations aligning with theory. This implementation high- lightsabreakthroughinmodelingirreversiblequantumdynamicswith IDS-inspiredstructuraldissipation.