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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

One-dimensional solutions of quantum hydrodynamic equations for an electron packet in free space and electric field

Andrey L. Sanin, Vera G. Ulianova

Published Jun 17, 2003• 0 citations
Publisher

Abstract

Quantum hydrodynamic equations are used to describe motion of single electron. The initial and boundary conditions are formulated from the analytic formula for the electron packet in a free space. The numerical integration of equations for the free packet is performed. Then, we investigated the wave packet propagation through the back-ground positive charge, when the electric field is self-consistent. Moreover, we studied the influence of the external harmonic electric fields on the properties of the motion. The purpose of the present work consists in study of the spatial-temporal processes including packets when the quantum wave effects are essential.

DOI
10.1117/12.501359

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Quantum hydrodynamic equations are used to describe motion of single electron. The initial and boundary conditions are formulated from the analytic formula for the electron packet in a free space. The numerical integration of equations for the free packet is performed. Then, we investigated the wave packet propagation through the back-ground positive charge, when the electric field is self-consistent. Moreover, we studied the influence of the external harmonic electric fields on the properties of the motion. The purpose of the present work consists in study of the spatial-temporal processes including packets when the quantum wave effects are essential.
Quantum hydrodynamic equations jointly with Maxwell's equation for the electric field are applied to describe motion of single electron including self-field one. The set of equations listed here and including the self-field of an electron is fundamental and represents the alternative quantum theory. Solutions of these equations are self-consistent and determine corresponding motion. In this paper we investigated one- dimensional stationary motion under variable distributions of background positive charge. Numerical integration of equations was carried out with Runge-Kutta method of 4th order approximation. Fast Fourier-transformation was used to analyze the spatially inhomogeneous distributions of electron density. Self-consistent solutions are compared with that of quantum hydrodynamic equations without considering the self-field of an electron (as in ordinary quantum mechanics). It is shown that difference takes place between them.
Oscillatory motion of quantum packet at the specific initial and zeroth boundary conditions with arbitrary initial velocity is investigated analytically and numerically. Influence of integer and half-integer initial velocities on the spatially temporal properties of solutions has been studied in detail. Hydrodynamical variables such as the probability density, mean values of coordinate and current, phase trajectories for coordinate and current, and Fourier spectra of these quantities were calculated. The behavior of quantum potential and kinetic energy near the nodal point was also analyzed.
Quantum hydrodynamics equations describing the dynamics of quantum fluid are a subject of this report (QFD).These equations can be used to decide the wide class of problem. But there are the calculated difficulties for the equations, which take place for nonlinear hyperbolic systems. In this connection, It is necessary to impose the additional restrictions which assure the existence and unique of solutions. As test sample, we use the free wave packet and study its behavior at the different initial and boundary conditions. The calculations of wave packet propagation cause in numerical algorithm the division. In numerical algorithm at the calculations of wave packet propagation, there arises the problem of division by zero. To overcome this problem we have to sew together discrete numerical and analytical continuous solutions on the boundary. We demonstrate here for the free wave packet that the numerical solution corresponds to the analytical solution.
A motion of a Gaussian wave packet of a single electron in a harmonic potential is obtained by solving the time dependent Schrödinger equation analytically. The analytic formula of the phase of the wave function which depends on both time and space is also presented explicitly. The probability density of the wave packet moves changing its width and central position periodically. The period of the change of the width is half of that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.
Energy eigenstates are superimposed in order to form the wave packet for an electron propagating in one dimension under the influence of a uniform, time-independent electric field. Flux velocity, phase velocity, and envelope velocity are considered. Asymptotic expressions are derived which describe the motion when the electric field is small and/or the wave packet is far away from the classical turning point. A graphical method is presented by which one can obtain both the position and shape of the envelope at a time t if the position and shape are known at some initial time.
We investigate the dynamics of a charged particle interacting with a multimode quantized electromagnetic field and obtain an analytic solution for the full electron--field system. This framework enables the calculation of position expectation values and uncertainties for arbitrary wave packets and field states, allowing us to identify quantum corrections to the corresponding classical motion. While the corrections to the position expectation value are weak and largely insensitive to the quantum state of the field, the wave packet broadening exhibits a pronounced dependence on the field state. In particular, the quantum uncertainty of the radiation is directly imprinted onto the spatial uncertainty of the particle. We illustrate these effects for Gaussian wave packets interacting with coherent, Fock, and squeezed states, including bright squeezed vacuum. The interaction with a finite-duration laser pulse is also analyzed as a multimode example. Our results provide a transparent analytic route toward understanding how quantum fluctuations of light influence electron dynamics in strong-field settings.