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arXiv:1010.1404

Solution of three-dimensional Faddeev equations: ultracold Helium trimer calculations with a public quantum three-body code

Elena A. Kolganova, Vladimir Roudnev, M. J. Cavagnero

Published Oct 7, 2010• 2 citations
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Abstract

We present an illustration of using a quantum three-body code being prepared for public release. The code is based on iterative solving of the three-dimensional Faddeev equations. The code is easy to use and allows users to perform highly-accurate calculations of quantum three-body systems. The previously known results for He$_3$ ground state are well reproduced by the code.

DOI
10.48550/arxiv.1010.1404

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We present an illustration of using a quantum three-body code being prepared for public release. The code is based on iterative solving of the three-dimensional Faddeev equations. The code is easy to use and allows users to perform highly-accurate calculations of quantum three-body systems. The previously known results for He3_33​ ground state are well reproduced by the code.
We present detailed calculations of bound and scattering states of dimers and trimers of He to produce highly accurate data and to test a non-relativistic three-body code currently in development for public distribution. For these systems, uncertainties and inaccuracies in the fundamental constants frequently used in published works can substantially exceed numerical errors. Our benchmark calculations include specific estimates of the numerical accuracy of the calculations and also explore sensitivity to fundamental constants and their uncertainties. The use of an inexact coupling constant in the previous calculations leads to 0.08% error for the ground state energy, 0.3% error for the excited state energy and up to 0.15% error for the atom–dimer scattering length in the system of three 4 He atoms. The corresponding errors for the asymmetric 4 He 2 3 He system are 0.3% for the bound state energy and 0.03% for the atom–dimer scattering length.
Faddeev-Yakubovski equations are solved numerically for 4He^{4}\mathrm{He}4He tetramer and trimer states using realistic helium-helium interaction models. We describe the properties of ground and excited states, and we discuss with a special emphasis the $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}_{3}$ low energy scattering.
The Faddeev equations for the atomic helium-trimer systems are solved numerically with high accuracy both for the most sophisticated realistic potentials available and for simple phenomenological potentials. An efficient numerical procedure is described. The large-distance asymptotic behaviour, crucial for weakly bound three-body systems, is described almost analytically for arbitrary potentials. The Efimov effect is especially considered. The geometric structures of the bound states are quantitatively investigated. The accuracy of the schematic models and previous computations is comparable, i.e. within 20% for the spatially extended states and within 40% for the smaller -trimer ground state.