Supersymmetric variational energies of 3d confined potentials

Abstract

Within the approach of Supersymmetric Quantum Mechanics associated with the variational method a recipe to construct the superpotential of three dimensional confined potentials in general is proposed. To illustrate the construction, the energies of the Harmonic Oscillator and the Hulth\'en potential, both confined in three dimensions are evaluated. Comparison with the corresponding results of other approximative and exact numerical results is presented.Comment: 9 pages, Late

The effect of confinement on the electronic energy and polarizability of a hydrogen molecular ion

“…However, in principle, the methodology can be adapted for other excited states as it was successfully carried out for central force potentials (see, for instance, Ref. [ ]). The electronic energy eigenvalues are calculated from the trial wave function (11) and the Hamiltonian (5).…”

Oscilador Harmônico Quântico 1D Confinado e sua Energia Variacional obtida com Auxílio da Supersimetria

“…Note-se que o potencial efetivoé calculado para valores do conjunto de parâmetros {μ} que minimizam a energia. A função dada em (4.3) difere dos superpotenciais usados para confinamento em três dimensões, [10,11], pela inclusão de um termo a mais de barreira de potencial para quando x → −R. Esse termoé desnecessário no estudo de equações radiais em três dimensões, uma vez que a variável radial jamais poderá assumir valores negativos.…”

“…Observa-se que os resultados para sistemas confinados tridimensionais, [10,11], podem ser adaptados para tratamento de sistemas unidimensionais. Os resultados numéricos obtidos são consistentes com o esperado.…”

“…O elo entre potencial e função de ondaé feito através do chamado superpotencial. Esta linha de raciocínio foi primeiro usada no estudo de sistemas não confinados, [3,8,9], e, mais recentemente, para os sistemas confinados, [10,11]. Entretanto, até o momento apenas sistemas confinados tridimensionais têm sido tratados usando o método variacional aliadoà MQS.…”

Supersymmetric Quantum Mechanics Formalism in a Modeling for Protein Folding

“…In the case of problems that are not exactly solvable, however, it has been suggested that the variational method could be applied within the SQM formalism to evaluate the energy spectrum of diatomic molecules [3], [4]. The methodology has thus provided knowledge on quantum dots [5], nonlinear features of the Peyrard-Bishop model for the DNA [6], quantum confinement of H-bonds in the DNA [7], [8], and other algebraic potential problems, [9], including the analysis of the Fokker-Planck Equation (FPE) with a view to describing a diffusive process, such as the folding of proteins, [10], through a mapping in a Schrödinger type equation, SE, [11]- [12]. In this line the approach was applied to explore the kinetics of the time-dependent probability distributions over thermodynamic free energy profiles of protein folding, [13], and tri-stable free energy profiles to analyze the kinetics of intermediate conformations of the protein, [14].…”