A semiclassical transport model for two-dimensional thin quantum barriers

Jin, Shi; Novak, Kyle A.

doi:10.1016/j.jcp.2007.06.006

Cited by 14 publications

(9 citation statements)

References 24 publications

(27 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This model extends the thin-barrier models introduced in [7,8] by including phase information so that the model can treat problems in which quantum coherence is critical. This model uses the complex-valued Liouville equation with an interface condition determined using complex-valued quantum scattering coefficients.…”

Section: Resultsmentioning

confidence: 98%

“…We begin by presenting an overview of the thin-barrier model which was developed in [7,8] for quantum barriers or wells that are sufficiently thin in comparison to the support of the wavepacket. The thin-barrier model fails to capture important phenomena such as interference.…”

Section: Semiclassical Modelsmentioning

confidence: 99%

“…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).…”

Section: Introductionmentioning

confidence: 99%

“…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).…”

mentioning

confidence: 99%

See 3 more Smart Citations

A coherent semiclassical transport model for pure-state quantum scattering

Jin

Novak²

2010

Communications in Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

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-

show abstract

Section: Resultsmentioning

confidence: 98%

Section: Semiclassical Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A coherent semiclassical transport model for pure-state quantum scattering

Jin

Novak²

2010

Communications in Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…A notion of the solution to the Hamiltonian system (4.2)-(4.3), using a probability interpretation, was introduced in [11], which was the basis for a Monte-Carlo particle method in [15]. Basically, one solves the system using a standard ODE or Hamiltonian solver, but at the interface, we introduce the following Monte-Carlo solution (we give the solution in the case of p − > 0; the other case is similar):…”

Section: 1)mentioning

confidence: 99%

Numerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond

Jin¹

2009

Hyperbolic Problems: Theory, Numerics and Applications

Self Cite

View full text Add to dashboard Cite

Abstract. This paper reviews some recent numerical methods for hyperbolic equations with singular (discontinuous or measure-valued) coefficients. Such problems arise in wave propagation through interfaces or barriers, or nonlinear waves through singular geometries. The connection between the well-balanced schemes for shallow-water equations with discontinuous bottom topography and the Hamiltonian preserving schemes for Liouville equations with discontinuous Hamiltonians is illustrated. Various developments of numerical methods with applications in high frequency waves through interfaces, and in multiscale coupling between classical and quantum mechanics, are discussed..

show abstract

A High Order Numerical Method for Computing Physical Observables in the Semiclassical Limit of the One-Dimensional Linear Schrödinger Equation with Discontinuous Potentials

Wen

2009

J Sci Comput

View full text Add to dashboard Cite

A semiclassical transport model for two-dimensional thin quantum barriers

Cited by 14 publications

References 24 publications

A coherent semiclassical transport model for pure-state quantum scattering

A coherent semiclassical transport model for pure-state quantum scattering

Numerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond

A High Order Numerical Method for Computing Physical Observables in the Semiclassical Limit of the One-Dimensional Linear Schrödinger Equation with Discontinuous Potentials

Contact Info

Product

Resources

About