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.
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.
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.
In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.
We present quantum numerical methods for the typical initial boundary value problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVP is discretized into a series of linear systems via finite difference methods and explicit time marching schemes. To solve these discrete systems in quantum computers, we design a series of quantum circuits, including four stages of encoding, amplification, adding source terms, and incorporating boundary conditions. In the encoding stage, the initial condition is encoded in the amplitudes of quantum registers as a state vector to take advantage of quantum algorithms in space complexity. In the following three stages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmetic calculations in quantum amplitudes are also realized to sum up the increments from these stages. The proposed quantum algorithm is implemented within the open-source quantum computing framework Qiskit [2]. By simulating one-dimensional transient problems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokes equations, we demonstrate the capability of quantum computers in fluid dynamics.
The quantum hydrodynamic equations associated with the de Broglie–Bohm formulation of quantum mechanics are solved using a meshless method based on a moving least squares approach. An arbitrary Lagrangian–Eulerian frame of reference is used which significantly improves the accuracy and stability of the method when compared to an approach based on a purely Lagrangian frame of reference. A regridding algorithm is implemented which adds and deletes points when necessary in order to maintain accurate and stable calculations. It is shown that unitarity in the time evolution of the quantum wave packet is significantly improved by propagating using averaged fields. As nodes in the reflected wave packet start to form, the quantum potential and force become very large and numerical instabilities occur. By introducing artificial viscosity into the equations of motion, these instabilities can be avoided and the stable propagation of the wave packet for very long times becomes possible. Results are presented for the scattering of a wave packet from a repulsive Eckart barrier.
The quantum hydrodynamics dynamics (QHD) is introduced to study the probability of electron through the tunneling junction. The complex form of the wave function was transformed into polar form, the time evolution equations of C and S were given in the Lagrange framework, and the explicit central differencing scheme was adopted for time integration of the QHD equations. In the Lagrangian framework, the grid points of the system will inevitable become scattered and unstructured during the evolution, the spatially smooth functions C and S are especially suitable for rational function approximation method to enable the implementation of a robust numerical scheme. As an example, the dynamics of a Gaussian wave packet is calculated and results are presented in the figures. In these figures, wave packet penetration of the barrier can be clearly observed, showing the quantum effect obviously.
Recently, in an effort to solve more realistic problems in quantum dynamics, much attention has been directed into numerically integrating the quantum hydrodynamic equations of motions (QHEM), as opposed to directly solving the time-dependent Schrödinger equation (TDSE). Such efforts have been provoked by the many numerical drawbacks encountered when solving the TDSE on a fixed-grid. In this dissertation, one trajectory method for integrating the QHEM is reviewed, and two novel trajectories methods are described. The first of these, the quantum trajectory method (QTM), was introduced in 1999 and has been used to solve many problems in quantum dynamics since then. However, severe numerical problems are encountered when this method is applied to problems that form wave function nodes. To get around this problem, new methods for numerically integrating the QHEM are needed. In the first novel method described, the arbitrary Lagrangian-Eulerian (ALE) method, particle trajectories are governed by a predetermined equation of motion that is user-supplied. The ALE method remedies inflation and compression problems encountered in the pure Lagrangian QTM. In the second new method discussed, the derivative propagating method (DPM), single quantum trajectories can be calculated one at a time, as opposed to the ensemble propagation of the QTM and ALE method. Using these two methods, new solutions to the QHEM are obtained where the QTM fails. In addition to solving the QHEM, the DPM is also used to solve the classical Klein-Kramers equation in this dissertation. This equation governs the Markovian phase space evolution of a system coupled to an environment such as a heat bath. This marks the first time single trajectories have been used to solve both the QHEM and the Klein-Kramers equations.