Schroedinger equation as the universal continuum limit of   nonrelativistic coherent hopping on a cubic spatial lattice
  The Schroedinger equation with scalar and vector potentials is the continuum limit of any coherent hopping process (where position eigenstates superpose with neighbouring eigenstates after a time step) whose hopping amplitudes are homogeneous in quadratic order of the inverse lattice spacing, inhomogeneous in first order, and satisfying a summability condition with respect to higher-than-next neighbours. 
Lutz Polley (Oldenburg University, Germany)