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We compared adaptive time integrators for solving time-dependent linear Schrödinger equations. Splitting methods are efficient for separable Hamiltonians, while Magnus-type methods are better for non-separable problems.

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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.