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arXiv:0705.3581 · Journal of Physics A Mathematical and Theoretical

The bends on a quantum waveguide and cross-products of Bessel functions

Martin Horvat, Tomaž Prosen

Published May 30, 2007• 7 citations
arXivHTMLPDF

Abstract

A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical calculations of scattering across the bend. The reflection is a purely quantum phenomenon and is discussed in more detail over a larger energy interval. The behaviour of the reflection is explained partially by a one-dimensional scattering model and heuristic calculations of the scattering matrix for narrow bends. In the same spirit, we explain the numerical results for the Wigner–Smith delay time in the bend.

DOI
10.1088/1751-8113/40/24/006

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A detailed analysis of the wave-mode structure in a bend and its incorporation into a stable algorithm for calculation of the scattering matrix of the bend is presented. The calculations are based on the modal approach. The stability and precision of the algorithm is numerically and analytically analysed. The algorithm enables precise numerical calculations of scattering across the bend. The reflection is a purely quantum phenomenon and is discussed in more detail over a larger energy interval. The behaviour of the reflection is explained partially by a one-dimensional scattering model and heuristic calculations of the scattering matrix for narrow bends. In the same spirit, we explain the numerical results for the Wigner–Smith delay time in the bend.
The properties of noninteracting, low-energy atom waves propagating through circular bends are investigated. Time-independent quantum mechanical calculations using various simple harmonic oscillator based confining potentials explore the transmission, reflection, and mode-transfer probabilities over a range of energies. It is shown that at low energies single-mode wave propagation dominates. At higher energies, however, excitation and transmission of other modes become significant. Notably, reflections generally remain negligible even for sharply curved bends. The behavior near the mode thresholds and their associated resonances is emphasized.
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A new numerical algorithm for tunneling is presented. This algorithm can be used to analyse an onedimensional\npotential barrier with an arbitrary profile. It is able to calculate not only the transmission coefficient\naccurately, but also the phases of the transmitted and reflected waves. Using the Schrodinger equation and plane waves,\nwe introduce a tunneling matrix to relate the transmitted and incident wave functions. Thus the transmission coefficient\nand phase can be calculated from the tunneling matrix.To test the accuracy of this algorithm,we calculate the transmission\ncoefficients and phases for a double -barrier (resonant tunneling) and a triangular barrier (field emission), and\nmake comparisous with corresponding analytical values. It shows that this algorithm is fast, accurate and efficient.
A reverse algorithm is presented for computation of one- dimensional quantum scattering. It is assumed that there is only the transmitted wave in transmission side. The Schrdinger equation is solved from the transmission side to the incident side according to the initial value problem,the real part and the imaginary part of the wave function,probability density function and then in the incident side,the maximum and minimum value of the function are obtained,the transmittance and reflectance arealso calculated simply. Variation of the quantum phase in scattering process is studied by using this method,an interesting phenomenon of π- jump's in the reflection wave phase at transparent points is found.
Eisenbud-Wigner-Smith delay and the Larmor time give different estimates for the duration of a quantum scattering event. The difference is most pronounced in the case where the de Broglie wavelength is large compared to the size of the scatterer. We use the methods of quantum measurement theory to analyse both approaches and to decide which one of them, if any, describes the duration a particle spends in the region that contains the scattering potential. The cases of transmission, reflection, and three-dimensional elastic scattering are discussed in some detail.