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Physica Scripta

A quantum mechanical model of the Bloch NMR flow equations for electron dynamics in fluids at the molecular level

Bamidele Omotayo Awojoyogbe

Published May 11, 2007• 21 citations
Publisher

Abstract

Understanding of processes such as carrier mobility, electron transfer reactions, chemical reactions in fluids, electron solvation in fluids and electron attachment and localization in clusters relies crucially on the understanding of electron dynamics in fluids at the molecular level. Because of its very small mass, an electron is a quantum object and some of its properties such as diffusion coefficient of a solvated electron in water can be explained only by resorting to quantum mechanical formulation. In this study, we have solved the Bloch NMR flow equations to describe the evolution of the wavelike properties and find the wave functions which can be useful to solve a particular fluid flow problem quantum mechanically. Based on the uncertainty principle, a wave packet is assumed to initially describe the fluid particle (electron) under study. Then, when the particle encounters a force (so its potential energy is no longer zero), the force modifies the wave packet. Finding such propagation techniques, and applying them appropriately can provide useful techniques to find solutions to biological, medical and physical problems which otherwise could not be easily solved.

DOI
10.1088/0031-8949/75/6/008

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Understanding of processes such as carrier mobility, electron transfer reactions, chemical reactions in fluids, electron solvation in fluids and electron attachment and localization in clusters relies crucially on the understanding of electron dynamics in fluids at the molecular level. Because of its very small mass, an electron is a quantum object and some of its properties such as diffusion coefficient of a solvated electron in water can be explained only by resorting to quantum mechanical formulation. In this study, we have solved the Bloch NMR flow equations to describe the evolution of the wavelike properties and find the wave functions which can be useful to solve a particular fluid flow problem quantum mechanically. Based on the uncertainty principle, a wave packet is assumed to initially describe the fluid particle (electron) under study. Then, when the particle encounters a force (so its potential energy is no longer zero), the force modifies the wave packet. Finding such propagation techniques, and applying them appropriately can provide useful techniques to find solutions to biological, medical and physical problems which otherwise could not be easily solved.
The equations of fluid mechanics, coupled with those that describe matter transportation at the molecular level must be handled effectively. Putting the fluid into equations, we model the Bloch NMR flow equations into the harmonic wave equation for the analysis of general fluid flow. We derived the solution of the modelled harmonic equation in non relativistic quantum mechanics and discuss its semi classical application to illustrate its potential wide-ranging usefulness in the search for the best possible data obtainable for general fluid flow analysis. Representing the solution of the derived harmonic wave equation by a normalized state function is quite useful in generating the properly normalized wave functions and in the efficient evaluation of expectation values of many operators that can be fundamental to the analysis of fluid flow especially at the microscopic level.
Many solutions have been propounded for Schrodinger’s equation. The application of these solutions is restricted to a few aspects of physics. One of the methods used for solving the Schrodinger equation is the Finite Difference Time Domain (FDTD) method [1]. It was used to solve numerically the Schrodinger equation for ground state and excited state eigenvalues, and eigenfunctions for a single particle in one-, two-, and three-dimensional potential wells. In the past, analytical methods which have been used to solve the Schrodinger equation are: supersymmetry [1], Nikiforov Uvarov method (NU) [3], asymptotic iteration method (AIM) [4,5], hyperviria perturbation [6], exact quantization rule (EQR) [7], perturbative formalism [8], wave function ansatz method [9], and path integral [10]. In all these methods highlighted above, none has been incorporated with the Bloch NMR to solve nuclear spin among other problems in physics and chemistry. The solution of the Schrodinger equation using the mathematical modeling of Bloch NMR has wide range of applications in many fields of physics and chemistry. It can be used for checking and improving models developed to study quantum mechanical systems, improve on the effective analysis of biological systems, analyzing spin transport, nuclear spin dynamics, improvement of the numerical methods and so on. In Sec. 2, the solution of the Schrodinger is given. In Sec. 3, the solution was validated by applying it to the harmonic solution. The energy
The quantum trajectory method (QTM) for time-dependent wave packet dynamics involves integration of the de Broglie–Bohm hydrodynamic equations for the evolving probability fluid [C. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. The equations of motion for discretized elements of the probability fluid (particles) are integrated in the Lagrangian, moving with the fluid, picture. These fluid elements move under the influence of both the usual potential energy function and the quantum potential, which involves the curvature of the quantum amplitude. The quantum potential and the quantum force are evaluated using a moving weighted least squares algorithm. As a demonstration of applicability, the QTM is applied to a model collinear reaction with an activation barrier. The reaction probabilities are in good agreement with exact quantum results, even for a relatively small number of particles in the ensemble. The QTM accurately describes tunneling using only real valued trajectories. In addition to the reaction probability, plots are presented to show the probability density and the flux distributions at several time steps during the reaction.
We explore the usefulness of a quantum fluid dynamics (QFD) approach for quantitative electronic structure calculations of many-electron systems. By combining QFD and density functional theory, a single time-dependent nonlinear QFD equation can be derived. The equation is further transformed into a diffusion-type form by an imaginary-time evolution method, whose asymptotic solution reaches a global minimum and the many-body ground-state wavefunction. The time-dependent generalized pseudospectral method is extended to solve the diffusion equation in spherical coordinates, allowing optimal and nonuniform spatial discretization and accurate and efficient solution of the diffusion function in space and time. The procedure is applied to the study of electronic energies, densities and other ground-state properties of noble gas atoms (He, Ne, Ar, Kr, Xe). The results are in good agreement with other best available values. The method offers a conceptually appealing and computationally practical procedure for the treatment of many-electron systems beyond the Hartree-Fock level.
The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.
The quantum trajectory method (QTM) is extended to the dynamics of electronic nonadiabiatic collisions. Equations of motion are first derived for the probability density, velocity, and action function for wave packets moving on each of the coupled electronic potential surfaces. These discretized equations are solved in the Lagrangian (moving with the fluid) picture to give the trajectory dynamics of fluid elements evolving on each potential surface. This trajectory method is fully quantum mechanical and does not involve “trajectory surface hopping.” The method is applied to nonadiabiatic collision models involving two coupled electronic states. The quantum trajectory results are in excellent agreement with solutions computed (using space-fixed grid methods) directly from the time-dependent Schrödinger equation.