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arXiv:0811.0748 · Journal of Mathematical Physics

Some equivalences between the auxiliary field method and envelope theory

Fabien Buisseret, Claude Semay, BERNARD SILVESTRE-BRAC

Published Mar 1, 2009• 15 citations
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Abstract

The auxiliary field method has been recently proposed as an efficient technique to compute analytical approximate solutions of eigenequations in quantum mechanics. We show that the auxiliary field method is completely equivalent to the envelope theory, which is another well-known procedure to analytically solve eigenequations, although relying on different principles a priori. This equivalence leads to a deeper understanding of both frameworks.

DOI
10.1063/1.3087424

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The auxiliary field method has been recently proposed as an efficient technique to compute analytical approximate solutions of eigenequations in quantum mechanics. We show that the auxiliary field method is completely equivalent to the envelope theory, which is another well-known procedure to analytically solve eigenequations, although relying on different principles a priori. This equivalence leads to a deeper understanding of both frameworks.
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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.
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.
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.
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.