# Approximation Solutions for Local Fractional Schrödinger Equation in the One-Dimensional Cantorian System

Authors: Yang Zhao; De-Fu Cheng; Xiao-Jun Yang
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

The local fractional Schrödinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative.


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## Body

## 1. Introduction

As it is known, in classical mechanics, the equations of motions are described as Newton’s second law, and the equivalent formulations become the Euler-Lagrange equations and Hamilton’s equations. In quantum mechanics, Schrödinger's equation for a dynamic system like Newton's law plays an important role in Newton's mechanics and conservation of energy. Mathematically, it is a partial differential equation, which is applied to describe how the quantum state of a physical system changes in time [1, 2]. In this work, the solutions of Schrödinger equations were investigated within the various methods [3–12] and other references therein.Recently, the fractional calculus [13–30], which is different from the classical calculus, is now applied to practical techniques in many branches of applied sciences and engineering. Fractional Schrödinger's equation was proposed by Laskin [31] via the space fractional quantum mechanics, which is based on the Feynman path integrals, and some properties of fractional Schrödinger's equation are investigated by Naber [32]. In present works, the solutions of fractional Schrödinger equations were considered in [33–38].Classical and fractional calculus cannot deal with nondifferentiable functions. However, the local fractional calculus (also called fractal calculus) [39–56] is best candidate and has been applied to model the practical problems in engineering, which are nondifferentiable functions. For example, the systems of Navier-Stokes equations on Cantor sets with local fractional derivative were discussed in [42]. The local fractional Fokker-Planck equation was investigated in [43]. The basic theory of elastic problems was considered in [44]. The anomalous diffusion with local fractional derivative was researched in [48–50]. Newtonian mechanics with local fractional derivative was proposed in [51]. The fractal heat transfer in silk cocoon hierarchy and heat conduction in a semi-infinite fractal bar were presented in [53–55] and other references therein.More recently, the local fractional Schrödinger equation in three-dimensional Cantorian system was considered in [56] as
(1)iαhα∂αψα(x,y,z,t)∂tα=-hα22m∇2αψα(x,y,z,t)+Vα(x,y,z)ψα(x,y,z,t),
where the local fractional Laplace operator is [39, 40, 42]
(2)∇2α=∂2α∂x2α+∂2α∂y2α+∂2α∂z2α,
the wave function ψα(x,y,z,t) is a local fractional continuous function [39, 40], and the local fractional differential operator is given by [39, 40]
(3)f(α)(x0)=dαf(x)dxα∣x=x0=limx→x0Δα(f(x)-f(x0))(x-x0)α,withΔα(f(x)-f(x0))≅Γ(1+α)Δ(f(x)-f(x0)).The local fractional Schrödinger equation in two-dimensional Cantorian system can be written as(4)iαhα∂αψα(x,y,t)∂tα=-hα22m∇2αψα(x,y,t)+Vα(x,y)ψα(x,y,t),
where the local fractional Laplace operator is given by
(5)∇2α=∂2α∂x2α+∂2α∂y2α.
The local fractional Schrödinger equation in one-dimensional Cantorian system is presented as
(6)iαhα∂αψα(x,t)∂tα=-hα22m∂2α∂x2αψα(x,t)+Vα(x)ψα(x,t),
where the wave function ψα(x,t) is local fractional continuous function.With the potential energyVα=0, the local fractional Schrödinger equation in the one-dimensional Cantorian system is
(7)iαhα∂αψα(x,t)∂tα=-hα22m∂2α∂x2αψα(x,t).In this paper our aim is to investigate the nondifferentiable solutions for local fractional Schrödinger equations in the one-dimensional Cantorian system by using the local fractional series expansion method [49]. The organization of the paper is organized as follows. In Section 2, we introduce the local fractional series expansion method. Section 3 is devoted to the solutions for local fractional Schrödinger equations. Finally, conclusions are given in Section 4.


## 2. The Local Fractional Series Expansion Method

According to local fractional series expansion method [49], we consider the following local fractional differentiable equation:
(8)ϕtα=Lαϕ,
where Lα is the linear local fractional operator and ϕ is a local fractional continuous function.In view of (8), the multiterm separated functions with respect to x,t are expressed as follows:
(9)ϕ(x,t)=∑i=0∞φi(t)ψi(x),
where φi(t) and ψi(x) are the local fractional continuous function.There are nondifferentiable terms, which are written as(10)φi(t)=χitiαΓ(1+iα),
where χi is a coefficient.In view of (10), we get
(11)ϕ(x,t)=∑i=0∞χitiαΓ(1+iα)ψi(x).Therefore,(12)ϕtα=∑i=0∞χi+1tiαΓ(1+iα)ψi+1(x),Lαϕ=∑i=0∞χitiαΓ(1+iα)(Lαψi)(x).Then, following (12), we have
(13)∑i=0∞χi+1tiαΓ(1+iα)ψi+1(x)=∑i=0∞χitiαΓ(1+iα)(Lαψi)(x).Letχi+1=χi=1; then
(14)∑i=0∞tiαΓ(1+iα)ψi+1(x)=∑i=0∞tiαΓ(1+iα)(Lαψi)(x).So, we have(15)ψi+1(x)=Lαψi,
where Lα is a linear local fractional operator.In [49], the linear local fractional operators are considered as
(16)Lα=∂2α∂x2α,Lα=x2αΓ(1+2α)∂2α∂x2α,(17)Lα=μ∂2α∂x2α,
where μ is a constant.Here, we consider the following operator:(18)Lα=η∂2α∂x2α+γ,
where η and γ are two constants.Using the iterative formula (18), we obtain
(19)ϕ(x,t)=∑i=0∞tiαΓ(1+iα)ψi(x),
which is the solution of (8).


## 3. Approximation Solutions

Let us change (6) into the formula in the following form:
(20)∂αψα(x,t)∂tα=-hα2iαm∂2α∂x2αψα(x,t)+Vα(x)iαhαψα(x,t),
where the linear local fractional operator is
(21)Lα=-hα2iαm∂2α∂x2α+Vα(x)iαhα.WithVα(x)=1, we have
(22)Lα=-hα2iαm∂2α∂x2α+1iαhα,so that(23)∂αψα(x,t)∂tα=Lαψα(x,t),
where
(24)ψα(x,0)=x2αΓ(1+2α).Using iteration relation (15), we set up
(25)ψα,n+1(x,t)=Lαψa,n(x,t),and an initial value is given by(26)ψa,0(x,t)=x2αΓ(1+2α).Therefore, following (25), we get
(27)ψa,0(x,t)=x2αΓ(1+2α),(28)ψa,1(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)ψa,0(x,t)=-hα2iαm+1iαhαx2αΓ(1+2α),(29)ψa,2(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)ψa,1(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)×(-hα2iαm+1iαhαx2αΓ(1+2α))=-22i2αm+1i2αhα2x2αΓ(1+2α),(30)ψa,3(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)ψa,2(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)×(-1i2αm+1i2αhα2x2αΓ(1+2α))=-32i3αmhα+1i3αhα3x2αΓ(1+2α),(31)ψa,4(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)ψa,3(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)×(-32i3αmhα+1i3αhα3x2αΓ(1+2α))=-42i4αmhα2+1i4αhα4x2αΓ(1+2α),⋮(32)ψa,n(x,t)=(-hα2iαm∂2α∂x2α+1iαhα)ψa,n-1(x,t)=-hαn-2n2inαm+1inαhαnx2αΓ(1+2α),and so on.Hence, from (32) we obtain the solution of (23) as
(33)ψα(x,t)=∑n=0∞ψa,n(x,t)=∑n=0∞tnαΓ(1+nα)(-hαn-2n2inαm+1inαhαnx2αΓ(1+2α)).We transform (7) into the following equation:
(34)∂αψα(x,t)∂tα=-hα2miα∂2α∂x2αψα(x,t).The initial condition is presented as(35)ψα(x,0)=x2αΓ(1+2α).Applying (17), we can write the iterative relations as follows:
(36)ψα,n+1(x,t)=Lαψa,n(x,t),ψa,0(x,t)=x2αΓ(1+2α),
where
(37)Lα=-hα2miα∂2α∂x2α.
From (36)–(38), we give the local fractional series terms as follows:
(38)ψa,0(x,t)=x2αΓ(1+2α),(39)ψa,1(x,t)=-hα2miα∂2α∂x2αψa,0(x,t)=-hα2iαm,(40)ψa,2(x,t)=-hα2miα∂2α∂x2αψa,1(x,t)=0,(41)ψa,3(x,t)=-hα2miα∂2α∂x2αψa,2(x,t)=0,⋮(42)ψa,n(x,t)=-hα2miα∂2α∂x2αψa,n-1(x,t)=0,and so forth.Hence, we have the nondifferentiable solution of (34) as follows:
(43)ψα(x,t)=∑n=0∞ψa,n(x,t)=x2αΓ(1+2α)+hα2miα.


## 4. Conclusions

In the work, we have obtained the nondifferentiable solutions for the local fractional Schrödinger equations in the one-dimensional Cantorian system by using the local fractional series expansion method. The present method is shown that is an effective method to obtain the local fractional series solutions for the partial differential equations within local fractional differentiable operator.



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*Source: 291386-2013-09-11.xml*