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FDTD-Q Analyses of a Mott Insulator Quantum Phase

Fumitaka Fujiwara

Published Feb 2, 2007• 0 citations
OpenAlex

Abstract

In this dissertation, the “FDTD-Q method” which is an application of the finite-difference timedomain (FDTD) method is introduced to solve the time-dependent Schrodinger equation. In this study, using the FDTD-Q method, the wave function of a two-particle two-site system is calculated and the calculation is also extended to a three-particle system in a three-site geometry. The properties of a Mott insulator phase were analysed using these results.

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In this dissertation, the “FDTD-Q method” which is an application of the finite-difference timedomain (FDTD) method is introduced to solve the time-dependent Schrodinger equation. In this study, using the FDTD-Q method, the wave function of a two-particle two-site system is calculated and the calculation is also extended to a three-particle system in a three-site geometry. The properties of a Mott insulator phase were analysed using these results.
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
A general method for numerical computation of the thermal density matrix of a single-particle confined quantum system is presented. The Schrodinger equation in imaginary time τ is solved numerically by the finite difference time domain (FDTD) method using a set of initial wavefunctions at τ = 0. By choosing this initial set appropriately, the set of wavefunctions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.
This paper introduced the FDTD method,the basic principles of the boundary and initial treatment selection of the trial wave function.On this basis,the matrix calculation to simulate the barrier change the energy of the system were given,the probability of changes in the quantum tunneling effect reproduce the important physical phenomena.Further demonstrates the use of FDTD method to solve the problem of quantum mechanics is to facilitate workable and systematic approach to cut the average energy of the border can be more satisfactory results.
A general method for numerical computation of the thermal density matrix of a single-particle quantum system is presented. The Schrodinger equation in imaginary time tau is solved numerically by the finite difference time domain (FDTD) method using a set of initial wave functions at tau=0 . By choosing this initial set appropriately, the set of wave functions generated by the FDTD method can be used to construct the thermal density matrix. The theoretical basis of the method, a numerical algorithm for its implementation, and illustrative examples in one, two and three dimensions are given in this paper. The numerical results show that the procedure is efficient and accurately determines the density matrix and thermodynamic properties of single-particle systems. Extensions of the method to more general cases are briefly indicated.
The application of the Finite Difference Time Domain(FDTD) method in quantum-mechanical problems becomes more and more interesting.The postulate,the boundary cut-off condition and the initial trial wave function are described.Based on the FDTD method,three different potential-barrier problems including rectangular potential-barrier,deltoid potential-barrier and syntonic potential-barrier are investigated.The energy variation and the probability variation are calculated and modeled,the important phenomenon of quantum tunnel effect reappeared.It is convenient to apply the FDTD method in quantum-mechanical problems and the results are acceptable while using the boundary cut-off based on the expectation energy of the system.And it is potentially further extended to other quantum-mechanical problems.
This project looks into the time evolution of a wave function within a two-dimensional quantum well. We start by solving the time-dependent Schrödinger equation for stationary states in a quantum well. Next, we express any wave function as a linear combination of stationary states, allowing us to understand their time evolution. Two methods are presented: one relies on decomposing the wave function into a basis of stationary states and the other on discretisation of the time-dependent Schrödinger equation, incorporating three-point formulas for derivatives. These approaches necessitate confronting intricate boundary conditions and require maintaining energy conservation for numerical accuracy. We further demonstrate the methods using a wave packet, revealing fundamental phenomena in quantum physics. Our results demonstrate the utility of these methods in understanding quantum systems, despite the challenges in determining stationary states for a given potential. This study enhances our comprehension of the dynamics of quantum states in constrained systems, essential for fields like quantum computing and nanotechnology.