Ulrich Mutze
The direct midpoint method as a quantum mechanical integrator
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ABSTRACT.  A computational implementation of quantum dynamics for an arbitrary 
time-independent Hamilton operator is defined and analyzed. The proposed evolution algorithm for a time step needs three additions of state vectors, three multiplications of state vectors with real numbers, and one application of the square of the Hamilton operator to a state vector. 
A trajectory starting from a unit-vector remains totally within the unit-sphere in Hilbert space if the time step is smaller than 2 divided by the norm of the Hamilton operator.If the time step is larger than this bound, the trajectory grows exponentially over all limits. The method is exemplified with a computational quantum system which models collision and inelastic scattering of two particles. Each of these particles lives in a discrete finite space which is a subset of a line. 
The two lines thus associated with the particles cross each other at right angle.