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arXiv:1008.5181 · Modern Physics Letters B

TRANSMISSION PROPERTIES OF THE ONE-DIMENSIONAL ARRAY OF DELTA POTENTIALS

G. Cordourier-Maruri, ROMEO DE COSS, VIRENDRA GUPTA

Published Mar 31, 2011• 11 citations
arXivHTMLPDF

Abstract

The problem of one-dimensional quantum wire along which a moving particle interacts with a linear array of N delta-function potentials is studied. Using a quantum waveguide approach, the transfer matrix is calculated to obtain the transmission probability of the particle. Results for arbitrary N and for specific regular arrays are presented. Some particular symmetries and invariances of the delta-function potential array for the N = 2 case are analyzed in detail. It is shown that perfect transmission can take place in a variety of situations.

DOI
10.1142/s0217984911026322

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The problem of one-dimensional quantum wire along which a moving particle interacts with a linear array of N delta-function potentials is studied. Using a quantum waveguide approach, the transfer matrix is calculated to obtain the transmission probability of the particle. Results for arbitrary N and for specific regular arrays are presented. Some particular symmetries and invariances of the delta-function potential array for the N = 2 case are analyzed in detail. It is shown that perfect transmission can take place in a variety of situations.
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The relationship between transmission resonances and Bloch states for a periodic array consisting of N delta function potentials is discussed. The transmission resonances are derived for matter waves incident on the periodic array, while the Bloch states are calculated using periodic boundary conditions for the array. It is shown that approximately half of the transmission resonances map into pairs of degenerate Bloch states. Wave functions are shown for both the transmission resonances and the Bloch states for arrays of five and six delta function potentials. The origin of the band structure of the Bloch states is interpreted in terms of the wave functions and eigenenergies for a particle confined to move on a ring, subjected to a periodic array of delta function potentials on the ring.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.