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arXiv:0806.2020 · Journal of Physics A Mathematical and Theoretical

Extensions of the auxiliary field method to solve Schrödinger equations

BERNARD SILVESTRE-BRAC, Claude Semay, Fabien Buisseret

Published Sep 22, 2008• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/41/42/425301

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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.
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.
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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The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.