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arXiv:1501.01383 · Few-Body Systems

Numerical Tests of the Envelope Theory for Few-Boson Systems

Claude Semay

Published Feb 11, 2015• 19 citations
arXivHTMLPDF

Abstract

The envelope theory, also known as the auxiliary field method, is a simple technique to compute approximate solutions of Hamiltonians for N identical particles in D -dimension. The accuracy of this method is tested by computing the ground state of N identical bosons for various systems. A method is proposed to improve the quality of the approximations by modifying the characteristic global quantum number of the method.

DOI
10.1007/s00601-015-0960-5

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The envelope theory, also known as the auxiliary field method, is a simple technique to compute approximate solutions of Hamiltonians for N identical particles in D -dimension. The accuracy of this method is tested by computing the ground state of N identical bosons for various systems. A method is proposed to improve the quality of the approximations by modifying the characteristic global quantum number of the method.
The envelope theory, also known as the auxiliary field method, is a technique to compute approximate solutions of a quantum system with N identical particles. It has been generalized for various systems with arbitrary kinematics and one- or two-body potentials. The basic idea is to replace the Hamiltonian H under study by an auxiliary Hamiltonian ˜H which is solvable, the eigenvalues of H˜ being optimized to be as close as possible to those of H. The method is easy to implement since it reduces to find the solution of a transcendental equation. Recently, its accuracy has been tested for eigenvalues and eigenvectors by computing the ground state of various systems containing up to 10 bosons. We show here that the envelope theory can be extended to compute the eigensolutions of a system of identical particles with a type of many-body forces often used in phenomenological models.
The envelope theory is a method to easily obtain approximate, but reliable, solutions for some quantum many-body problems. Quite general Hamiltonians can be considered for systems composed of an arbitrary number of different particles in D dimensions. In the case of identical particles, a compact set of three equations can be written to find the eigensolutions. This set provides also a nice interpretation and a starting point to improve the method. It is shown here that a similar set of seven equations can be determined for a system containing an arbitrary number of two different particles.
A method based on the envelope theory is presented to compute approximate solutions for N-body Hamiltonians with identical particles in D dimensions (D ⩾ 2). In some favorable cases, the approximate eigenvalues can be analytically determined and can be lower or upper bounds. The accuracy of the method is tested with several examples, and an application to a N-body system with a minimal length is studied. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.
The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum p0p_0p0​ and of the potential energy computed at a mean distance r0r_0r0​. The quantities p0p_0p0​ and r0r_0r0​ are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states.
The auxiliary field method is a powerful technique to obtain approximate closed-form energy formulas for eigenequations in quantum mechanics. Very good results can be obtained for Schrödinger and semirelativistic Hamiltonians with various potentials, even in the case of many-body problems. This method can also provide approximate eigenstates in terms of well-known wavefunctions, for instance harmonic oscillator or hydrogen-like states, but with a characteristic size which depends on quantum numbers. In this paper, we consider two-body Schrödinger equations with linear, logarithmic and exponential potentials and show that analytical approximations of the corresponding eigenstates can be obtained with the auxiliary field method, with very good accuracy in some cases.