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The Journal of Chemical Physics

Finite-difference approach to solving Heisenberg’s operator equations of motion: Application to one-dimensional time dependent Hamiltonians

Omar A. Sharafeddin

Published Jul 8, 1995• 2 citations
Publisher

Abstract

We reviewed and expanded on the finite difference approach introduced by Moncrief [Phys. Rev. D 28, 2485 (1983)] and Bender [Phys. Rev. Lett. 55, 901 (1985)] for solving Heisenberg’s operator equations of motion. In this approach, finite-difference recurrence relations are used to evolve the matrix representation of the operators in time. The advantages and disadvantages of this approach for the study of quantum processes in real time are discussed. The approach performed very well as illustrated by examples of harmonic and Morse oscillators interacting with continuous wave and pulsed laser fields.

DOI
10.1063/1.470097

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We reviewed and expanded on the finite difference approach introduced by Moncrief [Phys. Rev. D 28, 2485 (1983)] and Bender [Phys. Rev. Lett. 55, 901 (1985)] for solving Heisenberg’s operator equations of motion. In this approach, finite-difference recurrence relations are used to evolve the matrix representation of the operators in time. The advantages and disadvantages of this approach for the study of quantum processes in real time are discussed. The approach performed very well as illustrated by examples of harmonic and Morse oscillators interacting with continuous wave and pulsed laser fields.
We discuss the Schr\"odinger equation with a time-dependent Hamiltonian that can be written as a linear combination of operators which span a finite-dimensional Lie algebra. The equations of motion for the operators in the Heisenberg representation are shown to be useful in calculating matrix elements and transition probabilities, as well as in obtaining the time-evolution operator. A general time-dependent one-dimensional bilinear Hamiltonian is considered as an illustrative example, and the product form for the time-evolution operator is shown to be global.
A method of integration of the Heisenberg equation of motion is proposed in which the time evolution of the basis set {Tm,{\mathit{T}}_{\mathit{m},}Tm,​n(t)} of the Weyl-ordered operators introduced by Bender and Dunne [Phys. Rev. D 40, 3504 (1989)] is obtained from the Taylor expansion and is expressible in terms of the initial {Tm,{\mathit{T}}_{\mathit{m},}Tm,​n(0)}'s. In the absence of damping forces, the constant values of the energy and the position-momentum commutation relation are used to check the accuracy of the integration. This method is applied to obtain the mean position and velocity of the particle as a function of time as well as the dwell time of the particle inside the barrier. In the example that is considered here, the potential is assumed to be the sum of a harmonic and a cubic term, and the calculation is done with and without dissipative coupling.
We present an algorithm for solving the time-dependent Schrödinger equation that is based on the finite-difference expression of the kinetic energy operator. Students who have some knowledge of linear algebra can understand the theory used to derive the algorithm. This is because the finite-difference kinetic energy matrix and the Hückel matrix for linear conjugated hydrocarbons have similar forms. We offer 3 modules that can be used to solve 1-dimensional time-dependent Schrödinger equations, in real time. Module 1 contains two worksheets designed to show quantum dynamics in bound potentials. Module 2 contains 2 worksheets designed to illustrate tunneling. Module 3 contains 2 worksheets set up to illustrate the effect of time-dependent external fields. Further, modules 1 and 3 also include the Verlet algorithm so students can compare the quantum dynamical evolution of the systems to their classical counterpart in real time.
The matrix U' contains the full dynamics of a bilinear oscillator, because any observable of the system is an analytical function of the operators that determine the matrix representation. One cannot solve the time evolution equation in the general case, but there are many exactly solvable problems that simulate and enable one to understand the dynamics of more elaborate and realistic quantum-mechanical models. In the case of the solvable model just discussed it is easy to derive analytical expressions for operators in the Heisenberg picture as well as for invariants or constants of the motion. The mathematical treatment of the driven oscillator just discussed is trivial because the Hamiltonian operator belongs to a solvable Lie algebra. This model is a convenient starting point to illustrate the use of the Lie algebraic method, but it is also an oversimplification of most oscillatory phenomena. Nonlinear Hamiltonians are useful in many physical applications such as the simulation of frictional forces and dissipation effects.
A method based off of operator consideration for solving the time evolution of a wave function is developed. The method is applied to free space, constant force and harmonic oscillator potentials where general solutions are derived for the wave function time evolutions for these potentials. Solving the Heisenberg equations of motion is discussed. A method to obtain exact solutions to Heisenberg's equations of motion is developed based off of solutions of classical equations of motion. Approximation methods for the time evolution of operators and wave functions are developed. A connection between classical and quantum chaos is made where it is realized that whenever a classical system behaves chaotically, the quantum system will as well.
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.