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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Numerical solution of the Schrödinger equation using discrete kinetic theory

Sauro Succi A

Published Feb 1, 1996• 89 citations
Publisher

Abstract

The quantum lattice Boltzmann equation a new variant of the lattice Boltzmann equation specifically designed to describe quantum mechanical motion, is checked against analytical results for the case of a free particle and the harmonic oscillator in (1+1) space-time dimensions. The proper parameter regime under which the method needs to be operated in order to reproduce faithfully nonrelativistic quantum motion is discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.53.1969

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The quantum lattice Boltzmann equation a new variant of the lattice Boltzmann equation specifically designed to describe quantum mechanical motion, is checked against analytical results for the case of a free particle and the harmonic oscillator in (1+1) space-time dimensions. The proper parameter regime under which the method needs to be operated in order to reproduce faithfully nonrelativistic quantum motion is discussed. \textcopyright{} 1996 The American Physical Society.
The multidimensional formulation of the quantum lattice Boltzmann (QLB) scheme is validated against analytical solutions of the time-dependent nonrelativistic Schrödinger equation in two and three spatial dimensions. As a result, we demonstrate the viability of the quantum lattice Boltzmann scheme for the numerical solution of the time-dependent Schrödinger equation in multiple spatial dimensions. In addition, the links between the QLB scheme and sequential splitting methods for partial differential equations are also clarified.
Lattice Boltzmann (LB) methods have proven to be powerful tools for simulating classical physical systems such as waves, diffusion and fluid dynamics. For quantum mechanics in one dimension there are two main schemes: one proposed by Meyer, where the Schrödinger equation is interpreted as a diffusion equation with imaginary diffusion constant; the other, proposed by Succi, is based on the Dirac equation, which in the mesoscopic limit reproduces the non-relativistic quantum behaviour in a similar way to lattice Boltzmann models for fluids. In the present work we compare these models by testing the quantum time evolution of a free Gaussian wavepacket. It is found to be in close correspondence with the results of the two models. Time oscillations are found both for the mean and variance of the wavepacket. Also we found their dependences from factors not present in the predicted values. These elements could provide clues for developing a good adaptation of lattice Boltzmann method for the quantum world.
By modifying the collision operator in the quantum lattice gas (QLG) algorithm one can develop an imaginary time (IT) integration to determine the ground state solutions of the Schrödinger equation and its variants. These solutions are compared to those found by other methods (in particular the backward-Euler finite-difference scheme and the quantum lattice Boltzmann). In particular, the ground state of the quantum harmonic oscillator is considered as well as bright solitons in the one-dimensional (1D) non-linear Schrödinger equation. The dark solitons in an external potential are then determined. An advantage of the QLG IT algorithm is the avoidance of any real/complex matrix inversion and that its extension to arbitrary dimensions is straightforward.
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.