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Journal of Physics Condensed Matter

A path-integration calculation method based on the real-space finite-difference scheme

Hidekazu Goto, Tomoya Ono, Kikuji Hirose

Published Aug 24, 2007• 2 citations
Publisher

Abstract

We propose a new path-integration calculation method to treat the time evolution of a wavefunction within the framework of the real-space finite-difference formalism, and also develop an effective scheme to compute the scattering wavefunction for an incident electron with arbitrary energy, in which an impulse wavefunction is adopted as an initial state of the time evolution. In this method, once the time evolution of the initial impulse wavefunction is calculated, all of the solutions in the scattering problem can be derived by means of Fourier analysis of the time-evolved wavefunction, which leads to a reduction of the calculation time. In order to test the applicability of our newly developed simulation procedures, we implemented simulations for the one-dimensional scattering problem. Each simulation showed the usefulness of the present scheme by yielding the steady scattering states in agreement with exact ones.

DOI
10.1088/0953-8984/19/36/365205

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We propose a new path-integration calculation method to treat the time evolution of a wavefunction within the framework of the real-space finite-difference formalism, and also develop an effective scheme to compute the scattering wavefunction for an incident electron with arbitrary energy, in which an impulse wavefunction is adopted as an initial state of the time evolution. In this method, once the time evolution of the initial impulse wavefunction is calculated, all of the solutions in the scattering problem can be derived by means of Fourier analysis of the time-evolved wavefunction, which leads to a reduction of the calculation time. In order to test the applicability of our newly developed simulation procedures, we implemented simulations for the one-dimensional scattering problem. Each simulation showed the usefulness of the present scheme by yielding the steady scattering states in agreement with exact ones.
Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution.Purpose: We test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables.Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials.Results: The method is applied to compute matrix elements of M\o{}ller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation.Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.
The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.
A nonperturbative technique is developed to solve for the quantum dynamics of an electron interacting strongly with its environment. Real-time path-integral methods are used to solve numerically a quantum dissipation problem that simulates nonequilibrium electron transport in solids. A linearly coupled electron-phonon interaction is used to model the dissipation. This method is nonperturbative in both the electron-phonon coupling and the external electric field, making it useful for electron-transport calculations in cases of both high fields and large phonon-scattering rates. Monte Carlo sampling over the electron paths is done using the Feynman-Vernon influence functional as a weight. Importance-sampling methods are developed for the numerical solution of the quantum propagation. Results of the calculations are shown which include the motion of the electron, the wave-packet spread, and temperature effects.
In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, in the second representation this trajectory is complex and evolves according to an effective, time dependent potential. Using a cumulant expansion, the first correction to the variational approximation is also evaluated. The high energy behavior of the approximation is investigated, and is shown to contain exactly the leading and next-to-leading order of the eikonal expansion, and parts of higher terms. Our results are then numerically tested in two particular situations where others approximations turned out to be unsatisfactory. Substantial improvements are found.
While scattering phenomena are discussed in most quantum mechanics textbooks, the analysis is usually performed using time-independent techniques. These techniques have certainly been successful in predicting the results of scattering events but fail to present an intuitive picture of what happens to the physical system during a collision. A remedy to this situation could be affected by the introduction of time-dependent techniques, utilizing the conceptually simple idea of a wavefunction evolving over time, but these methods have traditionally been computationally unmanageable. This paper describes a novel numerical technique, the split-operator Fourier transform method, which greatly reduces the amount of computational effort required to accurately solve the time-dependent Schrödinger equation. This method is applied to the Rutherford scattering problem in two dimensions and it is shown that the method is not only efficient enough to yield results in less than 15 min when implemented on a standard personal computer, but is also simple enough to be employed by an advanced undergraduate. The method also has the advantage of reinforcing the quantum-mechanical formalism in the student’s mind; both the evolution operator and the complementary relationship between operators and wavefunctions in coordinate and momentum space are exploited. The results of the calculations are presented as color graphs portraying the magnitude and phase of wave packets as they evolve through time.