Nonlocal approach to scattering in a one-dimensional problem

Grossel, Ph.; Vigoureux, Jean-Marie; Baida, Fadi

doi:10.1103/physreva.50.3627

Cited by 28 publications

(25 citation statements)

References 18 publications

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“…The scattering coefficients may be computed by considering the barrier as an open quantum system [2] outside of which the potential is constant. Typically, one may use a quantum transmitting boundary method [15], a spectral projection method [19], or a transfer matrix method [1,13,6]. In these approaches, one separates the domain into three regions, C 1 , Q, and C 2 .…”

Section: 1mentioning

confidence: 99%

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A coherent semiclassical transport model for pure-state quantum scattering

Jin

Novak²

2010

Communications in Mathematical Sciences

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Abstract. We present a time-dependent semiclassical transport model for coherent pure-state scattering with quantum barriers. The model is based on a complex-valued Liouville equation, with interface conditions at quantum barriers computed from the steady-state Schrödinger equation. By retaining the phase information at the barrier, this coherent model adequately describes quantum scattering and interference at quantum barriers, with a computational cost comparable to that of classical mechanics. We construct both Eulerian and Lagrangian numerical methods for this model, and validate it using several numerical examples, including multiple quantum barriers.Key words. multiscale method, semiclassical limit, Liouville, coherent, quantum barrier.AMS subject classifications. 65M06, 65Z05, 81Q20, 81S30, 81T80. IntroductionThe motion of electrons in a plasma or a semiconductor can be modeled with classical mechanics when the change in the applied potential is moderate. But in a region where the potential changes rapidly over a length on the order of a de Broglie wavelength, quantum mechanics is required to accurately capture wave phenomena such as tunneling, resonance, and partial transmission and reflections. Because quantumscale parameters often control the accuracy and consistency of the solution, one often must resolve the dynamics entirely at the quantum scale. But for large-scale problems, such an approach is numerically infeasible. If the quantum region is sufficiently localized, a viable approach is to solve the problem using a multiscale method that combines the large-scale classical model with the small-scale quantum model.In [7,8] the authors presented a multiscale approach which accurately models the interaction of a quantum wave packet with a thin barrier in the semiclassical regime as the scaled Planck constant ε vanishes. This thin-barrier model accurately describes the weak limit of the moments of solutions to the pure-state Schrödinger and mixedstate von Neumann equation for an isolated thin quantum barrier (a barrier of width on the order of a de Broglie wavelength). This model assumes that the dwell time of the particle in the barrier is sufficiently short so that the behavior of the wave packet may be adequately described by the steady-state Schrödinger equation. Such an assumption is realistic for O(ε) thin barriers in the semiclassical limit as ε → 0, but it is inadequate when either ε is finite or the width of the barrier is significant in comparison to the width of the wave packet.Another shortcoming of the thin-barrier model is that it can only generate a decoherent solution. Quantum mechanics is in essence wave mechanics and the Schrö-

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Section: 1mentioning

confidence: 99%

“…Initialization. Compute the complex scattering coefficients associated with each momentum incident to each quantum barrier by using transfer matrix method [1,13,6] or transmitting boundary method [15].…”

Section: Eulerian Approachmentioning

confidence: 99%

A coherent semiclassical transport model for pure-state quantum scattering

Jin

Novak²

2010

Communications in Mathematical Sciences

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“…The methodology to obtain an estimate of the density, viscosity and thickness of the soft layer is based on the principle described here after. for a potential value of the film thickness, the acoustic attenuation ∆A and the acoustic phase shift ∆φ correspond, each of them, ∆θ corresponds an unique value of the refractive index [38] that is further related to the density of the layer thanks to Eq. 8, which assumes an identical evolution of the refractive index and of the density of the composite layer.…”

Section: Theorymentioning

confidence: 99%

In Situ Evaluation of Density, Viscosity, and Thickness of Adsorbed Soft Layers by Combined Surface Acoustic Wave and Surface Plasmon Resonance

et al. 2006

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We show the theoretical and experimental combination of acoustic and optical methods for the in situ quantitative evaluation of the density, the viscosity and the thickness of soft layers adsorbed on chemically tailored metal surfaces. For the highest sensitivity and an operation in liquids, a Love mode surface acoustic wave (SAW) sensor with a hydrophobized gold coated sensing area is the acoustic method, while surface plasmon resonance (SPR) on the same gold surface as the optical method is monitored simultaneously in a single set-up for the real-time and label-free measurement of the parameters of adsorbed soft layers, which means for layers with a predominant viscous behavior. A general mathematical modeling in equivalent viscoelastic transmission lines is presented to determine the correlation between experimental SAW signal shifts and the waveguide structure including the presence of the adsorbed layer and the supporting liquid from which it segregates. A methodology is presented to identify from SAW and SPR simulations the parameters representatives of the soft layer. During the absorption of a soft layer, thickness or viscosity changes are observed in the experimental ratio of the SAW signal attenuation to the SAW signal phase and are correlated with 2 the theoretical model. As application example, the simulation method is applied to study the thermal behavior of physisorbed PNIPAAm, a polymer whose conformation is sensitive to temperature, under a cycling variation of temperature between 20 and 40• C. Under the assumption of the bulk density and the bulk refractive index of PNIPAAm, thickness and viscosity of the film are obtained from simulations; the viscosity is correlated to the solvent content of the physisorbed layer.3

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“…The square of the magnitude of the probability amplitude p(x, t) = 10(X, t) 12 gives the position density in physical space.…”

Section: From Classical To Quantum Mechanicsmentioning

confidence: 99%

“…Our idea is to solve the Schrddinger equation (either exactly if possible, or numerically via a transfer matrix method [1,18,12]) inside the quantum barrier in order to generate the transmission and reflection coefficients, and then use that information in the interface condition to solve the classical Liouville equation through the barrier, in the spirit of the Hamiltonian-preserving method of Jin and Wen. When the quantum barrier is thin (on the order of a de Broglie wavelength), solving the stationary Schrbdinger equation suffices.…”

Section: Sponsoring/monitoring Agency Name(s) and Address(es)mentioning

confidence: 99%

A Semiclassical Transport Model for Thin Quantum Barriers

Jin¹,

Novak²

2006

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'ublic reporting burden tor this collection ot intormation is estimated to average 1 hour per response, including the time tar reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Abstract. We present a one-dimensional time-dependent semiclassical transport model for mixed state scattering with thin quantum barriers. The idea is to solve a stationary Schr6dinger equation in the thin quantum barrier to obtain the scattering coefficients, and then use them to supply the interface condition that connects the two classical domains. We then build in the interface condition to the numerical flux, in the spirit of the Hamiltonian-preserving scheme introduced by Jin and Wen for a classical barrier. The overall cost is roughly the same as solving a classical barrier. We construct a numerical method based on this semiclassical approach and validate the model using various numerical examples.

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Nonlocal approach to scattering in a one-dimensional problem

Cited by 28 publications

References 18 publications

A coherent semiclassical transport model for pure-state quantum scattering

A coherent semiclassical transport model for pure-state quantum scattering

In Situ Evaluation of Density, Viscosity, and Thickness of Adsorbed Soft Layers by Combined Surface Acoustic Wave and Surface Plasmon Resonance

A Semiclassical Transport Model for Thin Quantum Barriers

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