Adaptive Time Propagation for Time-dependent Schrödinger equations

Area of Science:

• Numerical analysis
• Quantum mechanics
• Computational physics

Background:

• Linear Schrödinger equations with time-dependent Hamiltonians are crucial in quantum mechanics.
• Accurate and efficient numerical methods are needed to solve these equations.
• Adaptive time-stepping strategies improve computational efficiency and accuracy.

Purpose of the Study:

• To compare the performance of adaptive time integrators for linear Schrödinger equations with explicit time dependence.
• To evaluate splitting methods and commutator-free Magnus-type methods.
• To determine the optimal integrator based on the structure of the Hamiltonian.

Main Methods:

• Adaptive time-stepping was employed for both splitting and Magnus-type methods.
• Local error estimators were used to adapt time-steps dynamically.
• The numerical solutions were compared based on efficiency and accuracy.

Main Results:

• Splitting methods demonstrated higher efficiency when the Hamiltonian could be naturally separated into kinetic and potential parts.
• Magnus-type integrators proved more effective when the Hamiltonian's structure did not allow for such separation.
• Adaptive time-step selection based on error estimators was effective for both integrator types.

Conclusions:

• The choice between splitting and Magnus-type methods depends on the specific structure of the time-dependent Hamiltonian.
• Adaptive time integrators offer a robust approach for solving time-dependent linear Schrödinger equations.
• These findings guide the selection of numerical methods for quantum dynamics simulations.