Many-Body Forces with the Envelope Theory

Abstract

Many-body forces are sometimes a relevant ingredient in various fields, such as atomic, nuclear or hadronic physics. Their precise structure is generally difficult to uncover. So, phenomenological effective forces are often used in practice. Nevertheless, they are always very heavy to treat numerically. The envelope theory, also known as the auxiliary field method, is a very efficient technique to obtain approximate, but reliable, solutions of many-body systems interacting via one-or two-body forces. It is adapted here to allow the treatment of a special form of many-body forces. In the most favourable cases, the approximate eigenvalues are analytical lower or upper bounds. Otherwise, numerical approximation can always be computed. Two examples of many-body forces are presented, and the critical coupling constants for generic attractive many-body potentials are computed. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues. *

Quantum support to BoHua Sun’s conjecture

“…In this case, the necessity to (anti)symmetrize the wave-function puts strong constraints on the solution, which can help the computation. The envelope theory is an efficient and convenient tool to obtain approximate solutions for quantum systems made of N identical particles [7][8][9]. With this method, the quantum energy of N self-gravitating particles with a mass m in D-dimensional space is given by [10]…”

Classical and Quantum Kepler's Third Law of N-Body System

“…where the quantum period T q and quantum energy E q of N self-gravitating particles with a mass m in Ddimensional space [1,8,9]. For further discussion, we call Eq.…”

“…where Q is a global quantum number to be determined for unequal bodies. Semay obtained the global quantum number for identical bodies [8].…”

Tests of the Envelope Theory in One Dimension

“…Computations for D = 1 show that formulas obtained in [4,6] are still valid, but with the momentums and the positions of particles which are now scalar quantities, and the global quantum number Q with the centre of mass removed which is defined by ( = 1)…”

“…The envelope theory (ET) [1][2][3] is a simple technique to compute approximate solutions, eigenvalues and eigenvectors, of N -body systems with arbitrary kinematics in D (> 2) dimensions for identical particles [4][5][6]. In the most favourable cases, the approximate eigenvalues are analytical lower or upper bounds.…”