Representation of electrons on symmetric electron-wave stub-filters by waves and particles

Abstract

This paper presents a method of computing sample trajectories of particle models of electrons on symmetric electron-wave stub-filters. The time-independent Schrödinger equation describing the electron-wave filters is transformed to a system of linear algebraic equations. The Fourier series expansion transforms the eigenvalues of the system to continuous and differentiable eigenfunctions. A electron wave packet is constructed from the obtained smooth eigenfunctions and a nonlinear stochastic ordinary differential equation (NSODE) which governs the behavior of the electron particle model is established by applying the stochastic quantization to the wave packet. Sample trajectories of the particles are computed by integrating the NSODE. Numerical experiments have shown that the statistical behavior of the particles is almost equal to that of the wave packets.

Wave-to-particle representation transformation for single-electrons on graphene

“…Equations (21) and (22) are the same equations as equation of continuity (12) and probability density current (13), respectively. This means that the effect of electric and magnetic fields E, B on the motion of an electron does not appear explicitly in probability density current J .…”

“…(9) and Eq. (13) or (22). Therefore, the single-electron on graphene can be represented by a probabilistic particle which is described by the Langevin and Fokker-Planck equations.…”

“…We computed probability density current J (x, y, t) given by Eq. (13). From the computed function ρ(x, y, t) and current J (x, y, t), a probabilistic particle model is built as mentioned in subsection 3.2.…”

“…The authors have asserted unification of the representation of electrons for integrated simulation of circuits built of both passive and active quantum devices. Since the circuit simulator models of the active quantum devices are available today, electrons in the passive quantum devices should be modeled as probabilistic particles [11][12][13]. In this paper, the authors insist again on transforming expression of electrons in graphene-based passive devices [14][15][16][17][18] from wave representation to particle representation.…”

Probabilistic Particle Models of Classical and Quantum Wave Systems

“…の PDF と PCD を与えると Nelson が導いた式 (38) のように モデル化し，その運動を波動関数と比較して示す (43) ． 量子系を記述する式 (23), (24) の換算プランク定数．電子質量 を h = 1，m e = 1 とする．図 11 に式 (24) のポテンシャル V (x) を図示する．数値実験では，a = 0.3, b = 14.0, c 1 = c 2 = 0.3, d 1 = d 2 = 0.23, e 1 = e 2 = 0.3 とした．電子波は V = 0 の白 色部分を伝搬し，絶縁体と仮定している灰色部分には入り込ま ないよう V = V H = 1000 に設定すると，前節の LP-ADPF の 3 本のスタブがフィルタ効果を生み出したように，この量子系は 4 本のスタブにより電子波フィルタとして機能する．図 12 は上 部に電子波の PDF を示す．図 12 (a) は ψ(x, y, ik x,init x により与えられ，標準偏差 σ x = 1, σ y = 0.06 の広がりをもつ初 期ガウス波朿を，図 12 (b), (c) は初期運動量 hk x,init = 13.0, 10.0 が与えられたときの時刻 t = 0.7 における PDF を示して いる．hk x,init に依存して電子波束はスタブ接続部分を通過， 接続部近辺で反射している．式 (40) により与えられる確率的古 典粒子モデルの軌道サンプルを図 12 の下部に示す．赤丸中の点 が t = 0, 0.7 の電子位置を示す．粒子モデルも波朿同様にスタ ブ接続部分を通過または接続部で折り返している．軌道サンプ ル 10 5 の t = 0.7 における電子位置の確率分布と波動関数から 得た PDF の差の空間積分は 10 −3 程度であった(43) ． m e = 1 とし，素電荷も e 0 = 1 とする．座標 系として円柱座標系 (r, θ, z) を採用する．スカラポテンシャルは A 0 = 0，ベクトルポテンシャルと磁界は = ∇ × A = 0 0 B 0 (51) とする．このとき，スピノール解は次式で表される． ψ n,m z ,± (r, θ, z, t) = (52) 1 n,m z (ξ(r)) exp i(m z θ + k z z − ε n,m z ,± h t ξ(r) = e 0 n,m z (x) = m z ω c πh n! (n + |m z |)!…”