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arXiv:1001.1706 · Journal of Physics A Mathematical and Theoretical

Eigenstates with the auxiliary field method

Claude Semay, BERNARD SILVESTRE-BRAC

Published Jun 2, 2010• 12 citations
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Abstract

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.

DOI
10.1088/1751-8113/43/26/265302

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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.
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We propose a new method to obtain approximate solutions for the Schrödinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows to find in many cases, analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which greatly improves the corresponding formulae that can be found in the literature.
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.
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 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.