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Direct solution to the few-body integrodifferential equation

Morris Ramantswana

Published Oct 1, 2019• 0 citations
OpenAlex

Abstract

In this work, a direct solution to the few-body Integrodifferential equation using the Lagrange-mesh method is presented.With the Lagrange-mesh method, the few-body Integrodifferential equation is converted into a matrix eigenvalue problem for numerical treatment.The accuracy and stability of the solution is tested by calculating groundstate binding energies for few-boson systems interacting via the Ali-Bodmer, Volkov and Malfliet-Tjon (MTV) nuclear potentials for A-body systems (where A = 4, 5, 6, 7).A rapid convergence of the results is achieved and calculated ground-state energies are in good agreement with those available in the literature, where different approaches were used.The method is promising and could be extended to larger number of particles such as those involved in the Bose-Einstein condensates.

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•Jan 31, 2019•0 citations
In this work, a direct solution to the few-body Integrodifferential equation using the Lagrange-mesh method is presented.With the Lagrange-mesh method, the few-body Integrodifferential equation is converted into a matrix eigenvalue problem for numerical treatment.The accuracy and stability of the solution is tested by calculating groundstate binding energies for few-boson systems interacting via the Ali-Bodmer, Volkov and Malfliet-Tjon (MTV) nuclear potentials for A-body systems (where A = 4, 5, 6, 7).A rapid convergence of the results is achieved and calculated ground-state energies are in good agreement with those available in the literature, where different approaches were used.The method is promising and could be extended to larger number of particles such as those involved in the Bose-Einstein condensates.
In this work, the traditional two-variable few-body integrodifferential equations approach is reviewed. Boundary conditions are implemented in both the hyperradial and hyperangular domains. A novel procedure of including effects of higher partial waves of the interaction potential is introduced. The new approach reproduces results obtained by an exact method for boson systems. These results suggest that many-body correlations are negligibly small in systems with short-range interactions and that effect of higher partial waves are adequately accounted for by the revised hypercentral potential.
We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with FFF the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of FFF and in this range the deviation of WWW from the Bogoliubov value is small. There is, however, a transition point FT{F}_{T}FT​, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT{F}_{T}FT​. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the 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.
We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.