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Abstract
We solve the space-fractional Schrödinger equation for a quadrupolar triple Dirac-δ (QTD-δ) potential for all energies using the momentum-space approach. For the E < 0 solution, we consider two cases, i.e., when the strengths of the potential are V0 > 0 (QTD-δ potential with central Dirac-δ well) and V0 < 0 (QTD-δ potential with central Dirac-δ barrier) and derive expressions satisfied by the bound-state energy. For all fractional orders α considered, we find that there is one eigenenergy when V0 > 0, and there are two eigenenergies when V0 < 0. We also obtain both bound- and scattering-state (E > 0) wave functions and express them in terms of Fox's H-function.
Highlights
Applications of fractional quantum mechanics (FQM) developed by Laskin[1, 2] via constructing fractional path integral over paths of Levy flights have gained interest over the past 13 years
The formulation offers generalization of some results obtained in the standard quantum mechanics (SQM)
One of its interesting applications is to delta potentials.[3]. We present another application of FQM by considering a quadrupolar triple Dirac-δ (QTD-δ) potential in one dimension, which was first analyzed by Patil[4] in the framework of SQM
Summary
Applications of fractional quantum mechanics (FQM) developed by Laskin[1, 2] via constructing fractional path integral over paths of Levy flights have gained interest over the past 13 years. Where Dα is the generalized quantum diffusion coefficient [D2 = 1/(2m) with m being the mass of the particle], ψ(x) is the wave function, V (x) is the potential, E Its Fourier transform and convolution with ψ(p) are given, respectively, by V (p) = V0(eiap/ − 2 + e−iap/ )
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