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arXiv:1712.06886 · Europhysics Letters (EPL)

Bound states of moving potential wells in discrete wave mechanics

Stefano Longhi

Published Oct 1, 2017• 5 citations
arXivHTMLPDF

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.

DOI
10.1209/0295-5075/120/20007

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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.
Abstract We explore the dynamics of matter wave scattering in finite potential wells using analytical solutions of the Schrödinger equation within the framework of a quantum shutter model. We find that the incident wave interferes with the bound states of the quantum well, resulting in time-domain oscillations. These oscillations exhibit Rabi-type frequencies, characterised by the energy differences between the incident wave and the bound states of the quantum well. We show that in systems with double-bound states, the interference pattern is characterised by quantum beats in the time-dependent probability density. The period of these beatings depends on the energy difference between the bound states, which can be tuned by controlling the potential parameters. In the general case where bound, anti-bound, and resonant states coexist in the system spectrum, complex oscillations in the probability density arise from the interactions of the incident wave with different quantum states. We demonstrate that the bound states sector can effectively describe this complex behaviour, providing a simple and reliable analytical expression for the probability density in multilevel systems. This formula highlights the significant role of bound states, whose interaction with the incident wave dominates the transient probability density. This contrasts with conventional systems with potential barriers and wells, where resonances govern the wave dynamics.
The discretized Schrödinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.
Anyonic PT\boldsymbol{\mathcal{PT}}PT symmetry, drifting potentials and non-Hermitian delocalization
We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian H satisfying the generalized (anyonic) parity-time symmetry PT H = exp ⁡ ( 2 i φ ) HPT , where P and T are the parity and time-reversal operators. For a stationary potential, the anyonic phase φ just rotates the energy spectrum of H in a complex plane, however, for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when φ ≠ 0 . In particular, in the unbroken PT phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.
In a seminal work, Horsley and collaborators [S. A. R. Horsley, et al., Nat. Photon. 9, 436 (2015)] showed that, in the framework of non-Hermitian extensions of the Schr\"odinger and Helmholtz equations, a localized complex scattering potential with spatial distributions of the real and imaginary parts related to one another by the spatial Kramers-Kronig relations are reflectionless and even invisible under certain conditions. Here we consider the scattering properties of Kramers-Kronig potentials for the discrete version of the Schr\"odinger equation, which generally describes wave transport on a lattice. Contrary to the continuous Schr\"odinger equation, on a lattice a stationary Kramers-Kronig potential is reflective. However, it is shown that a slow drift can make the potential invisible under certain conditions.
In most textbooks on quantum mechanics, it is asserted that the bound states of a localized potential have negative energies. To stress that this view should be taken with caution, we give several examples of scalar potentials that tend to zero at infinite distance and yet possess one or several discrete positive bound states for the Schrödinger single-particle wave equation. The consequence that usual tunneling may not occur, though it should according to a superficial view, is of interesting pedagogical value.