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We present a new numerical method for accurate computations of solutions to(linear) one dimensional Schr\"odinger equations with periodic potentials. Thisis a prominent model in solid state physics where we also allow forperturbations by non-periodic potentials describing external electric fields.Our approach is based on the classical Bloch decomposition method which allowsto diagonalize the periodic part of the Hamiltonian operator. Hence, thedominant effects from dispersion and periodic lattice potential are computedtogether, while the non-periodic potential acts only as a perturbation. Becausethe split-step communicator error between the periodic and non-periodic partsis relatively small, the step size can be chosen substantially larger than forthe traditional splitting of the dispersion and potential operators. Indeed itis shown by the given examples, that our method is unconditionally stable andmore efficient than the traditional split-step pseudo spectral schemes. To thisend a particular focus is on the semiclassical regime, where the new algorithmnaturally incorporates the adiabatic splitting of slow and fast degrees offreedom.

A Bloch decomposition based split-step pseudo spectral method for quantum dynamics with periodic potentials