Bound states of moving potential wells in discrete wave mechanics

Abstract

-Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schrödinger equation is retrieved, spatial discretization and lattice effects can deeply modify wave dynamics. Here we discuss implications of breakdown of exact Galilean invariance of the discrete Schrödinger equation on the bound states sustained by a smooth potential well which is uniformly moving on the lattice with a drift velocity v. While in the continuous limit the number of bound states does not depend on the drift velocity v, as one expects from the covariance of ordinary Schrödinger equation for a Galilean boost, lattice effects can lead to a larger number of bound states for the moving potential well as compared to the potential well at rest. Moreover, for a moving potential bound states on a lattice become rather generally quasi-bound (resonance) states.Introduction. -One of the cornerstones of nonrelativistic quantum mechanics is the Schrödinger equation, which describes the temporal evolution of a particle wave function based on its initial state. Traditionally, in wave mechanics space and time are considered continuous. However, on several occasions authors have debated about the nature of space-time manifold and the possibility of considering variant forms of the Schrödinger equation in which the wave function is defined on discrete lattice sites of space, time, or space-time, instead of on the space-time continuum [1][2][3][4][5][6][7][8][9][10][11][12]. Fundamental limits to a minimum measurable length were suggested in the early days of quantum physics, notably by Heisenberg, and appear, for example, in modern theories of loop quantum gravity, where spacetime looks granular [13]. A phenomenological approach to account for a minimum length scale is to consider an extension of the uncertainty principle by deforming the canonical commutation relations of position and momentum operators [14], leading to an extended form of the Schödinger equation [15,16]. Other simple models consider the non-relativistic Schrödinger equation defined on a discrete lattice [7][8][9][10][11][12][17][18][19][20][21][22], leading to so-called discrete wave mechanics [7] or discrete quantum mechanics [20]. While earlier models of discrete wave mechanics [7][8][9][10][11][12] did not find great relevance as foundational theories, in

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