Peculiarities of the quantum interchannel motion: Why the quantum lattice wave does not slide down the potential slope

Abstract

Unlike classical and quantum particles rolling down a potential slope, quantum waves on a lattice are confined in bound states on the potential hill. For the linear potential there is an equidistant spectrum of these states. The well-known Bessel functions ${\mathit{J}}_{\mathrm{\ensuremath{\alpha}}}$(kr) with the integer index \ensuremath{\alpha} as a discrete spatial coordinate (not r as usual) play the part of the bound-state wave functions---standing waves between the potential hill and the invisible boundary of the upper forbidden zone. This exactly solvable model elucidates some peculiar features of more realistic systems, particularly, the interchannel wave motion where \ensuremath{\alpha} means the discrete channel number.