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Applied Mathematical Sciences

Analytical solutions of a three dimensional time-dependent Schrodinger equation using the Bloch NMR approach for NMR studies

Moses Eterigho Emetere

Published Jan 1, 2014• 9 citations
Publisher

Abstract

The three dimensional time dependent Schrödinger equation was solved using the Bloch NMR approach with the help of the hermit differential polynomial. Its solution shed more light on the nature of the transverse magnetization. It was discovered that the transverse and longitudinal magnetization is the same mathematical but may vary with respect of the speed of the particles. This gives the theory wide range application in semiconductors, superconductivity and spintronics.

DOI
10.12988/ams.2014.4012

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The three dimensional time dependent Schrödinger equation was solved using the Bloch NMR approach with the help of the hermit differential polynomial. Its solution shed more light on the nature of the transverse magnetization. It was discovered that the transverse and longitudinal magnetization is the same mathematical but may vary with respect of the speed of the particles. This gives the theory wide range application in semiconductors, superconductivity and spintronics.
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 derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau-Lifshitz-Bloch equation, has been described. Starting from the classical Landau-Lifshitz-Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation.
Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood. Since the inception of Magnus’ expansion in 1954, no fundamentally novel mathematical method for solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems and is invariant under scale transformations. We solve for the dynamics of all two-level systems as well as of many-body Hamiltonians with a particular emphasis on NMR applications (Bloch-Siegert effect and N-spin systems involving the dipolar Hamiltonian and spin diffusion).