Application of the Variational Method to the particle in the box problem : the problem of the choice of base set

David D. Keeports

It is the purpose of this paper to consider a more general application of the variational method to the particle-in-the-box problem with polynomial trial functions.

Hagen M. Kleinert, A. M. Chervyakov, B. Hamprecht

Vı́ctor Luaña, Alberto Otero‐de‐la‐Roza, M. A. Blanco +1 more

The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The x n ( a − x ) n and x n ( a /2 − x )( a − x ) n polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a ] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.

Akira Inomata, Vijay A. Singh

S. P. Merkur′ev

Leonard Rosenberg, Stephen G. Tolchin

A variational approach to the approximate solution of a wide variety of self-consistency problems is described. Attention is focused on the variational formulation of the nonlinear integral equation, which determines the single-particle Green's function of a many-particle system; such an equation arises from a particular truncation of the hierarchy of equations which dynamically couple the complete set of Green's functions. To illustrate the method, we have applied it, with encouraging results, to the numerical determination of the self-energy function for the helium atom in the Hartree-Fock approximation. The applicability of this approach to the general problem of electron-atom scattering is pointed out.