A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

Area of Science:

• Quantum mechanics
• Computational chemistry
• Theoretical physics

Background:

• Solving the time-dependent Schrödinger equation is crucial for quantum dynamics.
• Existing methods like dynamic Fourier with split-operator are accurate but grid-point intensive.
• Moving grids with wavepackets can improve efficiency but may break time-reversibility.

Purpose of the Study:

• To develop a time-reversible algorithm for quantum dynamics.
• To enhance computational efficiency by using an adaptive moving grid.
• To preserve geometric properties like stability and norm conservation.

Main Methods:

• Simultaneous evolution of wavefunction and grid using Ehrenfest theorem and splitting method.
• Development of a conditionally stable, symmetric, and time-reversible algorithm.
• Analytical and numerical validation on harmonic and He-H2 scattering models.

Main Results:

• The proposed algorithm recovers time-reversibility, unlike naive methods.
• Significant speedups (10,000x) and reduced grid points (64x) were achieved with higher-order and adaptive grids.
• Demonstrated applicability to high-dimensional quantum dynamics (Hénon-Heiles model).

Conclusions:

• The novel algorithm offers a time-reversible, stable, and norm-conserving approach for quantum dynamics.
• Adaptive grids and higher-order methods drastically improve computational efficiency.
• The method is applicable to complex, high-dimensional quantum systems.