arXiv:physics/9909005 · The Journal of Chemical Physics

Comparison of numerical methods for the calculation of cold atom collisions

George H. Rawitscher, Brett D. Esry, Eite Tiesinga, J. P. Burke, I. Koltracht

Published Dec 15, 1999• 28 citations

Abstract

Comparison between three different numerical techniques for solving a coupled channel Schrödinger equation is presented. The benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are (i) a recently developed spectral type integral equation method based on Chebyshev expansions, (ii) a finite element expansion, and (iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods (i) and (ii), achieving an accuracy of ten significant figures with a double precision calculation. Method (iii) achieves seven significant figures. The scattering length and effective range are also obtained.

DOI: 10.1063/1.480431

Related papers

Ultracold atom-atom collisions by renormalized Numerov method

Jing Bai, Ting Xie

The renormalized Numerov algorithm is applied to solving time-independent Schrödinger equation relating to atom-atom collisions at ultralow temperature. The proprieties of Feshbach resonance in 39K-133Cs collisions are investigated as an example. The results show that the renormalized Numerov method can give excellent results for ultracold colliding process. In contrast to improved log derivative method, the renormalized Numerov method displays a certain weakness in computational efficiency under the same condition. However, it is much stable in a wide range of grid step size. Hence a new propagating method is proposed by combining renormalized Numerov and logarithmic derivative method which can save computational time with a better accuracy. This algorithm can be used to solve close-coupling Schrödinger equation at arbitrary temperature for two-body collisions.

Acta Physica SinicaJan 1, 20220 citations

Efficiency and accuracy of numerical solutions to the time-dependent Schrödinger equation

W. van Dijk, James Corgan Brown, Kyle Spyksma

Recent significant improvements of the numerical solutions of the time-dependent Schrödinger equation beg the question as to whether these recent methods are comparable in efficacy (in terms of accuracy and computational time) to the current "method of choice," i.e., the Chebyshev expansion of the time-evolution operator and the fast-Fourier-transform method of determining the kinetic energy. In this paper we review the methods in question and, by studying the time development of a coherent wave packet in an oscillator well, we are able to assess the effectiveness of the various methods. It turns out that the new generalizations come close (to within an order of magnitude) to being able to generate solutions as precisely and efficiently as the Chebyshev-fast-Fourier-transform method. The strict unitarity of the generalized methods may be an advantage. We also show that the fast-Fourier-transform approach to calculating the kinetic energy can be replaced by straightforward numerical differentiation to obtain the same precision.

Physical Review ENov 17, 201119 citations

Numerical determination of the scattering length for atomic collisions

Ondřej Basler

The goal of this bachelor thesis is to compare numerical methods designed for the determination of scattering length for atomic collisions. We compare nume- rical solutions calculated by the method of direct integration of the Schrödinger equation, the variable phase equation, the Riccati equation for the scattering length and the method of exterior complex scaling. Methods are tested on the exponential potential and the empirical potential of the Cs-Cs dimer. Calculated scattering lengths are further compared with results from other autors. 1

Jan 1, 20170 citations

The Faddeev and Schrödinger approaches to Efimov states—a numerical comparison

Kajsa-My Tempest, S. Jonsell

Abstract We compare effective hyperradial three-body potentials calculated using the S-wave part of the Faddeev equations to calculations using the full Schrödinger equation. As two-body model potential we test both a short-range potential and a Lennard-Jones potential with van der Waals tail. In the former case we find excellent agreement between the two methods for the lowest adiabatic state, indicating that the Faddeev method can be a useful tool also for numerical computations. For excited states the two methods show important differences, but agree for hyperradii larger than about five times the range of the potential (independent of the value of the scattering length and of the number of bound states). For the van der Waals potential, we focus on how well the Faddeev method reproduces the so-called van der Waals universality. We find that indeed the universality is manifest also using this method, but at a slightly different value of the universal parameter κ ∗ . We further derive an efficient method to solve the integro-differential equation arising in the Faddeev method.

Journal of Physics B Atomic Molecular and Optical PhysicsSep 8, 20231 citation

Elastic Scattering of Ultracold 133 Cs and 85 Rb Atoms in Triplet State

Sun Jin-Feng, Hu Qiu-Bo, Zunlue Zhu +2 more

Elastic scattering properties of the ultracold interaction for the triplet state of 133 Cs and 85 Rb atoms are studied using two kinds of potentials by the same phase . One is the interpolation potential, and another is Lennard-Jones potential (LJ12,6). The radial Schrodinger equation is also solved using two computational methods, the semiclassical method (WKB), and the Numerov method. Our results are found to be in an excellent agreement with the more recent theoretical values. It shows that the two potentials and methods are applicable for studying ultracold collisions between the mixing alkali atoms.

Communications in Theoretical PhysicsJun 1, 20073 citations

A Comparison of Different Numerical Propagation Schemes for Solving the Time-Dependent Schrödinger Equation in the Position Representation in One Dimension for Mixed Quantum-and Molecular Dynamics Simulations

Salomon R. Billeter, Wilfred F. Van Gunsteren A

Various numerical integration schemes to calculate the propagation of a state following the time-dependent Schrodinger equation in the one dimensional position representation are presented and compared to each other. Three potentials have been used: a harmonic, a double-well and a zero potential. Eigenstates and a coherent state have been chosen as initial states. Special attention has been given to the long-time stability of the algorithms. These are: kinetic referenced split operator (KRSO), kinetic referenced Cayley (KRC), distributed approximating functions (DAF), Chebysheff expansion (CH), residuum minimization (RES), second-order differencing (SOD), an eigenstate expansion (EE) and a corrected kinetic referenced split operator (CKRSO). In addition, a speedup of the KRC and KRSO methods is presented which is specially suited when very few grid points are used. Numerical results are compared to analytically calculated values. Mixed classical/quantum mechanical simulations require a representation of the quantum state on a limited number of grid points, classical integration time steps of about one femtosecond and compatibility with methods to solve the time-ordering problem. For the considered potentials which differ quite essentially from the potentials used for scattering problems in particle physics, the EE method has been found to be faster, more accurate and more stable than the other methods if only a few grid points are required. Otherwise, good results have been obtained with KRC, KRSO, CH, DAF and RES. SOD has been found to be too slow, and CKRSO is not stable enough for long simulation times.

Molecular SimulationNov 1, 199517 citations