Rapidly converging bound state eigenenergies for the two dimensional quantum dipole

ISSN： 0953-4075

Source： Journal of Physics B: Atomic, Molecular and Optical Physics, Vol.46, Iss.11, 2013-06, pp. : 115002-115008

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Abstract

We examine the effectiveness of a new spectral method in solving the two dimensional dipole problem (DP), as originally formulated by Dasbiswas et al (2010 Phys. Rev. B: At. Mol. Opt. Phys. 81 064516), and recently analysed by Amore and Fernandez (AF, 2012 Phys. Rev. B: At. Mol. Opt. Phys. 45 235004), through a large, non-orthogonal basis, Rayleigh-Ritz (RR) analysis. This deceptively simple problem has a long history of poorly approximated energy values, particularly for the ground state, until the recent work by AF. In contrast to their approach, we implement an orthogonal polynomial projection quantization (OPPQ) analysis (Handy and Vrinceanu 2013 J. Phys. A: Math. Theor. 46 135202), involving expanding the wavefunction in terms of a complete basis, &$Psi ({overrightarrow{r}}) = sum _n Omega _n mathcal{P}_n( {overrightarrow{r}}) mathcal{R}({overrightarrow{r}})$;, where &$mathcal{P}_n$; are the orthogonal polynomials relative to the weight &$mathcal{R}$;. For systems transformable into a moment equation, such as DP, the projection coefficients are determinable in closed form, yielding an efficient quantization procedure, particularly when the weight assumes the asymptotic form of the physical solutions. There are several theoretical reasons why the OPPQ should be more effective than the above RR approach. Indeed, comparable results are achieved with significantly fewer OPPQ variational parameters as compared to RR-variational parameters. For instance, with regards to the delicate ground state energy, 130 OPPQ variables are required to achieve E_gr = −0.137 7614 (E_gr = −0.137 7514 after a Shanks transform) as opposed to the 821 required within the RR formulation: E_gr = −0.137 7478. Despite this, the relative slow convergence for low lying even parity states, within both the OPPQ and RR formulations, suggests that significant logarithmic contributions to the wavefunction, at the origin, have been ignored by all previous investigators. Modifying the RR variational analysis to include log-dependent basis, affirms this through an accelerated RR convergence for the ground state: E_gr = −0.137 748 utilizing 68 variational parameters. This result suggests further improvements within either the OPPQ or RR formulation.