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Physical Review C

Deterministic technique of path summation

M. Rosa-Clot, Stefano Taddei

Published Aug 1, 1994• 9 citations
Publisher

Abstract

A numerical method, based on the Euclidean path integral formulation of quantum mechanics, to evaluate the ground state energy and wave function of a quantum system is discussed. The method is illustrated in one-dimensional cases, and then applied to a two-body system interacting through central and tensor potentials. A detailed discussion of the deuteron problem with a realistic nuclear potential is given.

DOI
10.1103/physrevc.50.627

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A numerical method, based on the Euclidean path integral formulation of quantum mechanics, to evaluate the ground state energy and wave function of a quantum system is discussed. The method is illustrated in one-dimensional cases, and then applied to a two-body system interacting through central and tensor potentials. A detailed discussion of the deuteron problem with a realistic nuclear potential is given.
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The methods discussed have been developed with the nuclear many-body problem, formulated as a classical quantum mechanical many-body problem with only twoparticle interaction and without any assumption of the origin of nuclear forces in mind. Two approaches characterized by different wave functions are discussed, and the general equations for the determination of best twoparticle orbitals'' are derived. The problem of the two interacting particles are examined in a common potential. This common potential is chosen as an oscillator. (W.D.M.)
Using the discretized path integral formalism we develop a numerically convenient and accurate method for evaluating the finite time propagator (and density matrix) of a given system. The eigenfunctions and energies of large number of states of the system are also found. We demonstrate the accuracy of the method by computing eigenvalues, eigenfunctions, position and flux autocorrelation functions and complex time propagators for a large number of systems and comparing these results with those available from other sources. In all the cases we find very good agreement. The method thus provides a very simple and accurate procedure for the study of static and dynamic properties of a quantum system.
We formulate a method to study two-body correlations in a system of N identical bosons interacting via central two-body potentials. We use the adiabatic hyperspherical approach and assume a Faddeev-like decomposition of the wave function. For a fixed hyperradius we derive variationally an optimal integro-differential equation for the hyperangular eigenvalue and wave function. This equation reduces substantially by assuming the interaction range much smaller than the size of the N-body system. At most, one-dimensional integrals then remain. We view a Bose-Einstein condensate pictorially as a structure in the landscape of the potential given as a function of the one-dimensional hyperradial coordinate. The quantum states of the condensate can be located in one of the two potential minima. We derive and discuss properties of the solutions and illustrate with numerical results. The correlations lower the interaction energy substantially. The new multibody Efimov states are solutions independent of details of the two-body potential. We compare with mean-field results and available experimental data.