Time independent fractional Schrödinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative

Abstract

In this paper we obtain approximate bound state solutions of N -dimensional fractional time independent Schrödinger equation for generalised Mie-type potential, namely V (r α ) = A r 2α + B r α +C. Here α(0 < α < 1) acts like a fractional parameter for the space variable r. When α = 1 the potential converts into the original form of Mie-type of potential that is generally studied in molecular and chemical physics. The entire study is composed with Jumarie type fractional derivative approach. The solution is expressed via Mittag-Leffler function and fractionally defined confluent hypergeometric function. To ensure the validity of the present work, obtained results are verified with the previous works for different potential parameter configurations, specially for α = 1.At the end, few numerical calculations for energy eigenvalue and bound states eigenfunctions are furnished for a typical diatomic molecule.

Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential

“…If we choose the natural unit = c = 1 then for large-N expansion [33], N-dimensional fractional time independent Schrödinger equation for a diatomic molecule in centre of mass coordinate is written as [27],…”

“…When α = 1, all these units are well familiar within the natural unit scheme. The term Ω α N within the argument of ψ denotes angular variables θ α 1 , θ α 2 , θ α 3 · · · θ α N −2 , φ α [27]. The term ∇ 2α N is called fractional Laplacian operator in N dimension.…”

“…where ℓ is orbital angular momentum quantum number (can take quantized values 0, 1, 2, 3 · · · only), we have the fractional order hyperradial or in short 'radial' equation [27] d 2α dr 2α +…”

“…(3.15) is very complicated in fractional domain. We have approximately solved this type of equation in our previous work [27] for α ≈ 1.00 with the help of 'α-logarithmic' function [31] in Jumarie sense i.e,…”

Study of memory effects in an inventory model using fractional calculus

“…In classical calculus, a system at each time t depends only on the input at that time because the derivatives of integer orders are determined by the property of differentiable functions of time only, and they are differentiable infinitely small neighborhood of the measured point of the time. In the last thirty years, fractional calculus exhibited a remarkable progress in several fields of science such as mechanics, chemistry, biology [29], economics [21,31], control theory, physics [27][28], signal and image processing [4][5][6][7][8] etc. But it is less explored in the field of inventory management in operations research.…”