Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

A local solution of the Schrödinger equation

Mircea Dan Grigoriu

Published Oct 30, 1998• 1 citation
Publisher

Abstract

A local method is developed for solving the Schrödinger equation. The method is local in the sense that it can determine the value of the solution of the Schrödinger equation at an arbitrary point directly rather than extracting this value from the field solution. The method is based on properties of diffusion processes, the Itô formula, and Monte Carlo simulation. Simplicity, accuracy, and generality are the main features of the proposed local solution. The extension of the proposed method to solve the stochastic version of the Schrödinger equation is elementary. Two examples with Dirichlet and Neumann boundary conditions are presented to demonstrate the application and evaluate the accuracy of the proposed local solution.

DOI
10.1088/0305-4470/31/43/010

Related papers

Monte-Carlo solution of Schrödinger’s equation

R C Grimm, R. G. Storer

Abstract With this paper Grimm and Storer introduced importance sampling to the field of quantum Monte Carlo. They had earlier used numerical quadrature for estimating the integrals required in evaluating energies in a diffusion-like path integral iteration scheme for solving the Schrödinger equation.0 This was extended to larger numbers of particles and higher dimensions with use of Monte Carlo methods. In conjunction with this they recognized they could take advantage of prior knowledge of the general form of a wavefunction to obtain more accurate estimates of the eigenvalue Eo .

•Jun 18, 2007•0 citations

About One Method of Numerical Solution of Schrodinger’s Equation

Konstantin Eduardovich Plokhotnikov

Abstract The paper considers the method of the numerical solution of the Schrodinger equation, which can partly be attributed to the class of Monte Carlo methods. The method is presented and simultaneously illustrated by the examples of solving the one-dimensional and multidimensional Schrodinger equation in the problems of a linear one-dimensional oscillator, hydrogen atom, and benzene atom.

Mathematical Models and Computer Simulations•Mar 1, 2020•13 citations

Efficient Methods for Solving the Time-Dependent Schrödinger Equation: Application to Anderson Localization

Hans A. De Raedt

Algorithms for solving the time-dependent Schrödinger equation, based on symmetrized versions of generalized Trotter-Suzuki formulae, are proposed. It is demonstrated that these stable, explicit algorithms are more accurate and efficient than the popular Cranck-Nicholson method. They are used to study the dynamic of the Anderson model for localization.

Europhysics Letters (EPL)•Jan 15, 1987•12 citations

Quantum Monte Carlo methods

Joseph Marie Thijssen

Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).

Cambridge University Press eBooks•Mar 22, 2007•26 citations

Extensions of the auxiliary field method to solve Schrödinger equations

BERNARD SILVESTRE-BRAC, Claude Semay, Fabien Buisseret

It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schrödinger equation. This technique can generate the spectrum associated with an arbitrary potential V ( r ) starting from the analytically known spectrum of a particular potential P ( r ). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P ( r ). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP ( r ) + V ( r ), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed.

arXiv:0806.2020•Journal of Physics A Mathematical and Theoretical•Sep 22, 2008•13 citations

- Approximation Methods I: Time-Independent Perturbation Theory

S. Rajasekar, R. Velusamy

The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.

CRC Press eBooks•Dec 11, 2014•0 citations
A local method is developed for solving the Schrödinger equation. The method is local in the sense that it can determine the value of the solution of the Schrödinger equation at an arbitrary point directly rather than extracting this value from the field solution. The method is based on properties of diffusion processes, the Itô formula, and Monte Carlo simulation. Simplicity, accuracy, and generality are the main features of the proposed local solution. The extension of the proposed method to solve the stochastic version of the Schrödinger equation is elementary. Two examples with Dirichlet and Neumann boundary conditions are presented to demonstrate the application and evaluate the accuracy of the proposed local solution.
Abstract With this paper Grimm and Storer introduced importance sampling to the field of quantum Monte Carlo. They had earlier used numerical quadrature for estimating the integrals required in evaluating energies in a diffusion-like path integral iteration scheme for solving the Schrödinger equation.0 This was extended to larger numbers of particles and higher dimensions with use of Monte Carlo methods. In conjunction with this they recognized they could take advantage of prior knowledge of the general form of a wavefunction to obtain more accurate estimates of the eigenvalue Eo .
Introduction In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for many-electron systems. Many of the techniques described there carry over to other quantum many-particle systems, such as liquid helium, and the protons and neutrons in a nucleus. The techniques which we discussed there were, however, all of a mean-field type and therefore correlation effects could not be taken into account without introducing approximations. In this chapter, we consider more accurate techniques, which are similar to those studied in Chapter 10 and are based on using (pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods. In Chapter 10 we applied Monte Carlo techniques to classical many-particle systems; here we use these techniques for studying quantum problems involving many particles. In the next section we shall see how we can apply Monte Carlo techniques to the problem of calculating the quantum mechanical expectation value of the ground state energy. This is used in order to optimise this expectation value by adjusting a trial wave function in a variational type of approach, hence the name variational Monte Carlo (VMC). In the following section we use the similarity between the Schrödinger equation and the diffusion equation in order to calculate the properties of a collection of interacting quantum mechanical particles by simulating a classical particle diffusion process. The resulting method is called diffusion Monte Carlo (DMC).
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schrödinger equation. This technique can generate the spectrum associated with an arbitrary potential V ( r ) starting from the analytically known spectrum of a particular potential P ( r ). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P ( r ). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP ( r ) + V ( r ), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.