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arXiv:1102.1321 · Journal of Mathematical Physics

Duality relations in the auxiliary field method

BERNARD SILVESTRE-BRAC, Claude Semay

Published May 1, 2011• 13 citations
arXivHTMLPDF

Abstract

The eigenenergies ε(N)(m; {ni, li}) of a system of N identical particles with a mass m are functions of the various radial quantum numbers ni and orbital quantum numbers li. Approximations E(N)(m; Q) of these eigenenergies, depending on a principal quantum number Q({ni, li}), can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities E(N)(m; Q) and E(p)(m′; Q′) for various forms of the potentials (independent of m and N) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies ε(N)(m; {ni, li}).

DOI
10.1063/1.3589959

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The eigenenergies ε(N)(m; {ni, li}) of a system of N identical particles with a mass m are functions of the various radial quantum numbers ni and orbital quantum numbers li. Approximations E(N)(m; Q) of these eigenenergies, depending on a principal quantum number Q({ni, li}), can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities E(N)(m; Q) and E(p)(m′; Q′) for various forms of the potentials (independent of m and N) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies ε(N)(m; {ni, li}).
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.
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schrödinger equation. This technique can generate the spectrum associated with an arbitrary potential V ( r ) starting from the analytically known spectrum of a particular potential P ( r ). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P ( r ). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP ( r ) + V ( r ), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed.
This chapter contains a brief compilation of information about various fields and contractions. The field formalism is fully applicable also to the ordinary quantum mechanics of systems with a finite number of degrees of freedom. The simplest example is that of a particle in a given external field. Preliminary separation of the δ function is necessary, for example, when calculating the amplitude by the stationary-phase method (the semiclassical approximation), since the conservation laws expressed by the δ functions automatically make the kernel of the quadratic form of the second variation of the action degenerate in the neighborhood of a classical trajectory. Nonrelativistic field theory is the quantum mechanics of systems with an arbitrary number of identical particles in the second-quantized formalism. In the classical theory such systems are described by a complex field ψ, ψ + which is an ordinary or anticommuting function.
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.