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Nonlinear Theory and Its Applications IEICE

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

Shota Hayakawa, Nobuyuki Hirami, Hisato Fujisaka

Published Jan 1, 2019• 3 citations
Publisher

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.

DOI
10.1587/nolta.10.414

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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.
The physical model of a nonrelativistic quantized Schrodinger's electron (SE) is offered. The behaviour of the SE well spread elementary electric charge had been understood by means of two independent and different in a frequency and size motions. The description of this resultant motion may be done by substitution of the classical Wiener continuous integral with the quantized Feynmam continuous integral. There are possibility to show by means of the existent not only formal but substantial analogy between the quadratic differential wave equation in partial derivatives of Schrodinger and quadratic differential particle equation in partial derivatives of Hamilton-Jacoby that the addition of a kinetic energy of the stochastic harmonic oscillation of some quantized micro particles to the kinetic energy of classical motion of the same micro particles formally determines their wave behaviour.It turns out the stochastic motion of the quantized micro particles powerfully to break up the smooth thin line of the classical motion of the same micro particle in many broad cylindrically spread path. The SE participate in stochastically roughly determined circumferences within different flats and with different radii, with centres which are successively arranjed over short and very disorderly orientated lines. Therefore the quantized motion of some micro particle cannot be descripted by smooth thin well contured (focused) line, describing the classical motion of the macro particle.
We make an initial exploration of a spectral filtering approach to numerically approximating solutions to the time-independent Schrödinger equation for a model system of two spinless fermions with harmonic interactions. Fourier transformation of the time evolution of an arbitrary antisymmetrized wave packet results in resonance at frequencies proportional to eigenenergies of the system Hamiltonian, while the Fourier coefficient approximates the corresponding eigenfunction. Spectral filtering is explored using the two-particle model through both direct numerical integration of the Schrödinger equation and a semiclassical parametrisation. In the former, a discrete position basis describes the time-evolution operator, which is then repeatedly applied to an antisymmetrized wave packet to obtain the packet's time evolution. In the semiclassical parametrisation, equations of motion for the parameters are obtained from the Dirac-Frenkel-McLachlan functional. Integration of the parameter equations of motion describes the packet's time evolution. The semiclassical approach requires less memory and accurately obtains the eigenspectrum. In a step towards application of spectral filtering to realistic systems, a possible ansatz for use with Coulomb potentials is explored, using a simple model Hamiltonian.
The physical model (PhsMdl) of a Schrodinger nonrelativistic quantized electron (ShEl) is built by means of a transition of the quadratic differential particle equation of Hamilton-Jacoby into the quadratic differential wave equation of Schrodinger in this work, which interprets the physical reason of its quantum (wave and stochastic) behaviour by explanation of the physical reason, which forces the classical Lorentz electron (LrEl) to participate in Furthian quantized stochastic oscillation motion, which turn it into quantum ShEl. It is performed that this transition is realized by my consideration the Bohm's quantum potential as a kinetic energy of the forced Furthian quantized stochastic oscillation motion of the ShEl's well spread elementary electric charge close to a smooth thin trajectory of a classical LrEl. There exist as an essential analogy between the Furthian quantum stochastic trembling oscillation motion and the Brownian classical stochastic trembling motion so and between the description of their behaviours.
Quantum particles possess both wave and particle natures. When quantum effect devices exploiting the wave nature and the devices described from the view point of particle nature coexist in a system, the following advantages are gained by representing former devices in the same way as we describe the latter devices: First, it is not necessary to compute probability distribution from wave functions and sample device states in accordance with the probability distribution. Secondly, the particle models of the devices are suitable models for conventional circuit simulators. In this paper, simple particle models of two quantum systems operating as wave filters are constructed by applying Nelson's stochastic quantization theory. The filter characteristics obtained from the wave functions and from the behavior of the particle models were almost equal.
This paper proposes a probabilistic particle model of single-electrons on graphene. The behavior of a single-electron on graphene is described approximately by the massless Dirac equation. The electron is non-relativistic and given pseudo-spin. The Dirac equation originally describes the motion of a relativistic quantum particle with actual spin. Then, it has seemed difficult that the Nelson's stochastic quantization theory could build a particle model of the electron on graphene since the theory can deal with only non-relativistic quantum particles with spin not being taken into account. In this paper, Nelson's theory is interpreted by using probability density function and probability density current so that it can build a particle model of a single-electron on graphene. Single-electrons on a graphene nano-ribbon and on a graphene sheet in constant magnetic field were modeled as probabilistic particles. The models were described by nonlinear stochastic ordinary differential equations. It has been numerically confirmed that probability distributions of the electron models coincide with distributions derived from the wave functions.
A brief presentation is given of the basic concepts in quantum probability theory in analogy to the classical ones, as well as the necessary information about noncommutative stochastic integration and its explicit representation in the Fock space. Algebraic differential relations that define quantum diffusion motion with nondemolition observation are derived. In the Markov case we obtain the stochastic equation of quantum diffusion filtering which is analogous to the Zakai equation of classical nonlinear filtering. A quantum linear stochastic model with continuous observation is given for which this equation is reduced to a linear stochastic quantum filtering equation of Kalman-Busi type and to the operator Riccati equation. We also consider an example of observing the coordinate of a free quantum particle, for which the solution of a stationary quantum filtering problem is obtained.