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Communications in Computational Physics

A Finite Element Method Solver for Time-Dependent and Stationary Schrödinger Equations with a Generic Potential

A. Soba

Published May 1, 2009• 1 citation
Publisher

Abstract

A general finite element solution of the Schrödinger equation for a one-dimensional problem is presented. The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term. Furthermore, it is possible to include external magnetic or electric fields, as well as spin-orbital and spin-magnetic interactions. We use analytically soluble problems to validate the solver. The predicted numerical auto-states are compared with the analytical ones, and selected mean values are used to validate the auto-functions. In order to analyze the performance of the time-dependent Schrödinger equation, a traveling wave package benchmark was reproduced. In addition, a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages. Other general problems, related to periodic potentials, are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions. Some simple cases of known periodic potential solutions are reported.

DOI
10.4208/cicp.2009.v5.p914

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A general finite element solution of the Schrödinger equation for a one-dimensional problem is presented. The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term. Furthermore, it is possible to include external magnetic or electric fields, as well as spin-orbital and spin-magnetic interactions. We use analytically soluble problems to validate the solver. The predicted numerical auto-states are compared with the analytical ones, and selected mean values are used to validate the auto-functions. In order to analyze the performance of the time-dependent Schrödinger equation, a traveling wave package benchmark was reproduced. In addition, a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages. Other general problems, related to periodic potentials, are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions. Some simple cases of known periodic potential solutions are reported.
A new method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods have been discussed critically.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
A propagator for the one dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained using the multiple-scattering theory approach. It allows for the consideration of the reflection and transmission processes as the particle scattering at the potential jump (in contrast to the conventional wave-like picture) and for accounting for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to the particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x,t)ψ(x,t)ψ(x,t)) and allows for considering the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of time. These results are related to fundamental issues such as measuring time in quantum mechanics (tunneling time, time of arrival, dwell time). For imaginary time, which represents an inverse temperature (t→−iℏ/kBT)(t\rightarrow-i{\hbar}/k_{B}T)(t→−iℏ/kB​T), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to a rectangular potential. If one shifts to imaginary time (t→−it)(t\rightarrow-it)(t→−it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals from elementary functions and thus allow for a numerical visualization of the probability density ∣ψ(x,t)∣2|ψ(x,t)|^2∣ψ(x,t)∣2, the density matrix and the solution of the diffusion-like equation. The results obtained may also be useful for spintronics applications due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.
Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.