Analytical and numerical estimates of nonlocal effects at low atomic scales: periodic structures, Landau levels, and quantum box

A generalized nonlocal uncertainty relation is constructed based on the notion of quantum acceleratum operator obtained in the framework of nonlocal-in-time kinetic energy approach for the case of reversible motion. The new uncertainty relation modified all quantum Hamiltonians and predicts nonlocal corrections to various phenomena at low-dimensional and nanoscales. We evaluate analytically and numerically nonlocal corrections for various quantum phenomena, mainly the Landau levels, the periodic structures, and the quantum box, and we constrain our results with the scanning tunneling microscope experiment. Several features were discussed accordingly.