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Multi-point quasi-rational approximants for the energy eigenvalues of two-power potentials

Martin,P.; Castro,E.; Paz,J.L.
Fonte: Sociedad Mexicana de Física Publicador: Sociedad Mexicana de Física
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/08/2012 EN
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Analytic approximants for the eigenvalues of the one-dimensional Schrödinger equation with potentials of the form V(x) = xª + λx b are found using a multi-point quasi-rational approximation technique. This technique is based on the use of the power series and asymptotic expansion of the eigenvalues in λ, as well as the expansion at intermediate points. These expansions are found through a system of differential equations. The approximants found are valid and accurate for any values of λ > 0 (with b > a). As an example, the technique is applied to the quartic anharmonic oscillator.

Energy eigenvalues for free and confined triple-well potentials

Aquino,N.; Garza,J.; Campoy,G.; Vela,A
Fonte: Sociedad Mexicana de Física Publicador: Sociedad Mexicana de Física
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/02/2011 EN
Relevância na Pesquisa
36.6%
Some confined and unconfined (free) one-dimensional triple-well potentials are analyzed with two different numerical approaches. Confinement is achieved by enclosing the potential between two impenetrable walls. The unconfined (free) system is recovered as the positions of the walls move to infinity. The numerical solutions of the Schrodinger equation for the symmetric and asymmetric potentials without confinement, are comparable in precision with those obtained anaylitically. For the symmetric triple-well potentials, V (x) = αx2 - βx4 + x6, it is found that there are sets of two or three quasi-degenerate eigenvalues depending on the parameters a and ¡3. A heuristic analysis shows that if the conditions α= (β2 /4) ± 1 (with α > 0 and β > 0) are satisfied, then there are sets of three eigenvalues with similar energy. An interesting behavior is found when one impenetrable wall is fixed and the other is moved to different positions. In summary, the number of local minima that the potential has in the confined region determines a two- or three-fold degeneracy.