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

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.

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 𝑖^𝛼ℎ_𝛼𝜕^𝛼𝜓_𝛼𝑥,𝑦,𝑧,𝑡𝜕𝑡^𝛼=−ℎ_𝛼²∇2𝑚^2𝛼𝜓_𝛼𝑥,𝑦,𝑧,𝑡+𝑉_𝛼𝑥,𝑦,𝑧𝜓_𝛼𝑥,𝑦,𝑧,𝑡, (1) where the local fractional Laplace operator is [39, 40, 42] ∇^2𝛼=𝜕^2𝛼𝜕𝑥^2𝛼+𝜕^2𝛼𝜕𝑦^2𝛼+𝜕^2𝛼𝜕𝑧^2𝛼, (2) the wave function 𝜓_𝛼(𝑥,𝑦,𝑧,𝑡) is a local fractional continuous function [39, 40], and the local fractional differential operator is given by [39, 40] 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼∣_𝑥=𝑥0=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (3)

with Δ^𝛼(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 𝑖^𝛼ℎ_𝛼𝜕^𝛼𝜓_𝛼𝑥,𝑦,𝑡𝜕𝑡^𝛼ℎ=−^𝛼2∇2𝑚^2𝛼𝜓_𝛼𝑥,𝑦,𝑡+𝑉_𝛼𝑥,𝑦𝜓_𝛼𝑥,𝑦,𝑡, (4) where the local fractional Laplace operator is given by ∇^2𝛼=𝜕^2𝛼𝜕𝑥^2𝛼+𝜕^2𝛼𝜕𝑦^2𝛼. (5) The local fractional Schrödinger equation in one-dimensional Cantorian system is presented as 𝑖^𝛼ℎ_𝛼𝜕^𝛼𝜓_𝛼(𝑥,𝑡)𝜕𝑡^𝛼ℎ=−^𝛼2𝜕2𝑚𝜕𝑥^2𝛼2𝛼𝜓_𝛼(𝑥,𝑡)+𝑉_𝛼(𝑥)𝜓_𝛼(𝑥,𝑡), (6) where the wave function 𝜓_𝛼(𝑥,𝑡) is local fractional continuous function.

With the potential energy 𝑉_𝛼=0, the local fractional Schrödinger equation in the one-dimensional Cantorian system is 𝑖^𝛼ℎ_𝛼𝜕^𝛼𝜓_𝛼(𝑥,𝑡)𝜕𝑡^𝛼ℎ=−^𝛼2𝜕2𝑚𝜕𝑥^2𝛼2𝛼𝜓_𝛼(𝑥,𝑡). (7)

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: 𝜙^𝛼_𝑡=𝐿_𝛼𝜙, (8) where 𝐿_𝛼 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: 𝜙(𝑥,𝑡)=^∞_𝑖=0𝜑_𝑖(𝑡)𝜓_𝑖(𝑥), (9) where 𝜑_𝑖(𝑡) and 𝜓_𝑖(𝑥) are the local fractional continuous function.

There are nondifferentiable terms, which are written as 𝜑_𝑖(𝑡)=𝜒_𝑖𝑡^𝑖𝛼Γ(),1+𝑖𝛼 (10) where 𝜒_𝑖 is a coefficient.

In view of (10), we get 𝜙(𝑥,𝑡)=^∞_𝑖=0𝜒_𝑖𝑡^𝑖𝛼Γ()𝜓1+𝑖𝛼^𝑖(𝑥). (11)

Therefore, 𝜙^𝛼_𝑡=^∞_𝑖=0𝜒_𝑖+1𝑡^𝑖𝛼Γ()𝜓1+𝑖𝛼^𝑖+1(𝑥),𝐿_𝛼𝜙=^∞_𝑖=0𝜒_𝑖𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝜓_𝑖(𝑥). (12)

Then, following (12), we have ^∞_𝑖=0𝜒_𝑖+1𝑡^𝑖𝛼Γ()𝜓1+𝑖𝛼^𝑖+1(𝑥)=^∞_𝑖=0𝜒_𝑖𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝜓_𝑖(𝑥). (13)

Let 𝜒_𝑖+1=𝜒_𝑖=1; then ^∞_𝑖=0𝑡^𝑖𝛼Γ()𝜓1+𝑖𝛼^𝑖+1(𝑥)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝜓_𝑖(𝑥). (14)

So, we have 𝜓_𝑖+1(𝑥)=𝐿_𝛼𝜓_𝑖, (15) where 𝐿_𝛼 is a linear local fractional operator.

In [49], the linear local fractional operators are considered as 𝐿_𝛼=𝜕^2𝛼𝜕𝑥^2𝛼,𝐿_𝛼=𝑥^2𝛼Γ()𝜕1+2𝛼𝜕𝑥^2𝛼2𝛼,𝐿_𝛼𝜕=𝜇^2𝛼𝜕𝑥^2𝛼, (16)(17) where 𝜇 is a constant.

Here, we consider the following operator: 𝐿_𝛼𝜕=𝜂^2𝛼𝜕𝑥^2𝛼+𝛾, (18) where 𝜂 and 𝛾 are two constants.

Using the iterative formula (18), we obtain 𝜙(𝑥,𝑡)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝜓1+𝑖𝛼^𝑖(𝑥), (19) which is the solution of (8).

3. Approximation Solutions

Let us change (6) into the formula in the following form: 𝜕^𝛼𝜓_𝛼(𝑥,𝑡)𝜕𝑡^𝛼ℎ=−^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼𝜓_𝛼()+𝑉𝑥,𝑡^𝛼(𝑥)𝑖^𝛼ℎ_𝛼𝜓_𝛼(𝑥,𝑡), (20) where the linear local fractional operator is 𝐿_𝛼ℎ=−^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+𝑉_𝛼(𝑥)𝑖^𝛼ℎ_𝛼. (21)

With 𝑉_𝛼(𝑥)=1, we have 𝐿_𝛼ℎ=−^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼, (22)

so that 𝜕^𝛼𝜓_𝛼(𝑥,𝑡)𝜕𝑡^𝛼=𝐿_𝛼𝜓_𝛼(𝑥,𝑡), (23) where 𝜓_𝛼()=𝑥𝑥,0^2𝛼Γ().1+2𝛼 (24)

Using iteration relation (15), we set up 𝜓_𝛼,𝑛+1(𝑥,𝑡)=𝐿_𝛼𝜓_𝑎,𝑛(𝑥,𝑡), (25)

and an initial value is given by 𝜓_𝑎,0()=𝑥𝑥,𝑡^2𝛼Γ().1+2𝛼 (26)

Therefore, following (25), we get 𝜓_𝑎,0()=𝑥𝑥,𝑡^2𝛼Γ(),𝜓1+2𝛼_𝑎,1()=−ℎ𝑥,𝑡^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼𝜓_𝑎,0()ℎ𝑥,𝑡=−^𝛼2𝑖^𝛼𝑚+1𝑖^𝛼ℎ_𝛼𝑥^2𝛼Γ(),𝜓1+2𝛼_𝑎,2()=−ℎ𝑥,𝑡^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼𝜓_𝑎,1()=−ℎ𝑥,𝑡_𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼×−ℎ_𝛼2𝑖^𝛼𝑚+1𝑖^𝛼ℎ_𝛼𝑥^2𝛼Γ()21+2𝛼=−2𝑖^2𝛼𝑚+1𝑖^2𝛼ℎ_𝛼²𝑥^2𝛼Γ(),𝜓1+2𝛼_𝑎,3()=−ℎ𝑥,𝑡^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼𝜓_𝑎,2()=−ℎ𝑥,𝑡_𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼×−1𝑖^2𝛼𝑚+1𝑖^2𝛼ℎ_𝛼²𝑥^2𝛼Γ()31+2𝛼=−2𝑖^3𝛼𝑚ℎ_𝛼+1𝑖^3𝛼ℎ_𝛼³𝑥^2𝛼Γ(),𝜓1+2𝛼_𝑎,4()=−ℎ𝑥,𝑡^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼𝜓_𝑎,3()=−ℎ𝑥,𝑡_𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼×−32𝑖^3𝛼𝑚ℎ_𝛼+1𝑖^3𝛼ℎ_𝛼³𝑥^2𝛼Γ()41+2𝛼=−2𝑖^4𝛼𝑚ℎ_𝛼²+1𝑖^4𝛼ℎ_𝛼⁴𝑥^2𝛼Γ(),⋮𝜓1+2𝛼_𝑎,𝑛()=−ℎ𝑥,𝑡^𝛼2𝑖^𝛼𝑚𝜕^2𝛼𝜕𝑥^2𝛼+1𝑖^𝛼ℎ_𝛼𝜓_{𝑎,𝑛−1}()ℎ𝑥,𝑡=−^𝛼𝑛−2𝑛2𝑖^𝑛𝛼𝑚+1𝑖^𝑛𝛼ℎ_𝛼^𝑛𝑥^2𝛼Γ(),1+2𝛼 (27)(28)(29)(30)(31)(32)

and so on.

Hence, from (32) we obtain the solution of (23) as 𝜓_𝛼(𝑥,𝑡)=^∞_𝑛=0𝜓_𝑎,𝑛()=𝑥,𝑡_∞𝑛=0𝑡^𝑛𝛼Γ()−ℎ1+𝑛𝛼^𝛼𝑛−2𝑛2𝑖^𝑛𝛼𝑚+1𝑖^𝑛𝛼ℎ_𝛼^𝑛𝑥^2𝛼Γ().1+2𝛼 (33)

We transform (7) into the following equation: 𝜕^𝛼𝜓_𝛼(𝑥,𝑡)𝜕𝑡^𝛼ℎ=−^𝛼2𝑚𝑖^𝛼𝜕^2𝛼𝜕𝑥^2𝛼𝜓_𝛼(𝑥,𝑡). (34)

The initial condition is presented as 𝜓_𝛼()=𝑥𝑥,0^2𝛼Γ().1+2𝛼 (35)

Applying (17), we can write the iterative relations as follows: 𝜓_𝛼,𝑛+1(𝑥,𝑡)=𝐿_𝛼𝜓_𝑎,𝑛(),𝜓𝑥,𝑡_𝑎,0()=𝑥𝑥,𝑡^2𝛼Γ(),1+2𝛼 (36) where 𝐿_𝛼ℎ=−^𝛼2𝑚𝑖^𝛼𝜕^2𝛼𝜕𝑥^2𝛼. (37) From (36)–(38), we give the local fractional series terms as follows: 𝜓_𝑎,0()=𝑥𝑥,𝑡^2𝛼Γ(),𝜓1+2𝛼_𝑎,1()ℎ𝑥,𝑡=−^𝛼2𝑚𝑖^𝛼𝜕^2𝛼𝜕𝑥^2𝛼𝜓_𝑎,0()ℎ𝑥,𝑡=−^𝛼2𝑖^𝛼𝑚,𝜓_𝑎,2()ℎ𝑥,𝑡=−^𝛼2𝑚𝑖^𝛼𝜕^2𝛼𝜕𝑥^2𝛼𝜓_𝑎,1()𝜓𝑥,𝑡=0,_𝑎,3()ℎ𝑥,𝑡=−^𝛼2𝑚𝑖^𝛼𝜕^2𝛼𝜕𝑥^2𝛼𝜓_𝑎,2()⋮𝜓𝑥,𝑡=0,_𝑎,𝑛()ℎ𝑥,𝑡=−^𝛼2𝑚𝑖^𝛼𝜕^2𝛼𝜕𝑥^2𝛼𝜓_{𝑎,𝑛−1}(𝑥,𝑡)=0, (38)(39)(40)(41)(42)

and so forth.

Hence, we have the nondifferentiable solution of (34) as follows: 𝜓_𝛼(𝑥,𝑡)=^∞_𝑛=0𝜓_𝑎,𝑛()=𝑥𝑥,𝑡_2𝛼Γ()+ℎ1+2𝛼^𝛼𝑖2𝑚^𝛼. (43)

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.

Acknowledgment

This work was supported by Natural Science Foundation of Hebei Province (no. F2010001322).