Time and Space Fractional Schr dinger Equation with Fractional Factor∗

Pei Xiang(相培),Yong-Xin Guo(郭永新),and Jing-Li Fu(傅景礼),†

1Institute of Mathematical Physics,Zhejiang Sci-Tech University,Hangzhou 310018,China

2Key Laboratory of Optical Field Manipulation of Zhejiang Province,Hangzhou 310018,China

3School of Physics,Liaoning University,Shenyang 110036,China

Abstract In this paper,we introduce a new definition of fractional derivative which contains a fractional factor,and its physical meanings are given.When studying the fractional Schr dinger equation(FSE)with this form of fractional derivative,the result shows that under the description of time FSE with fractional factor,the probability of finding a particle in the whole space is still conserved.By using this new definition to construct space FSE,we achieve a continuous transition from standard Schr dinger equation to the fractional one.When applying this form of Schr dinger equation to a particle in an infinite symmetrical square potential well,we find that the probability density distribution loses spatial symmetry and shows a kind of attenuation property.For the situation of a one-dimensional infinite δ potential well,the first derivative of time-independent wave function Φ to space coordinate x can be continuous everywhere when the particle is at some special discrete energy levels,which is much different from the standard Schr dinger equation.

Key words:fractional derivative,fractional factor,fractional Schr dinger equation,Bessel function

1 Introduction

Since Nick Laskin put forward an extension of a fractality concept in quantum physics and constructed the fractional Schrdinger equation,[1−3]many studies have been done to apply fractional calculus to fractional quantum mechanics.[4−10]Inspired by the process of Laskin’s constructing space fractional Schrdinger equation,Mark Naber constructed time fractional Schrdinger equation by considering non-Markovian evolution. And then,Naber used Caputo fractional derivative to solve this equation.The results showed that the total probability was not conserved even for a free particle and increased over time to a limiting value which may be bigger than one.[4,11]In some extent,this phenomenon is against the normalizing condition in normal quantum mechanics and the property of probability in mathematics.This will be unavoidable if the FSE is still described by Caputo fractional derivative.[7,12]Hence,we think the FSE should be reconsidered from the aspect of redefining the fractional derivative.

Recently,a new definition of fractional derivative called conformable fractional derivative has been introduced by Khalil et al.(Eq.(6)).[13]The studies show that this form of definition is a natural one and obeys basic classical properties,which overcomes the setbacks of Riemann-Liouville definition and Caputo definition.[13−14]Then,a method called conformable fractional NU method is presented and some applications of this method are done in quantum mechanics.[6,8]In this method,the Schrdinger equation is reduced to a hypergeometric type second order differential equation and the normal differential operator is replaced by the conformable fractional derivative operator.However,when we apply the form of space fractional diffusion equation,which is produced when considering anomalous diffusion generated by non-Gaussian distributions,to constructing the space FSE,it turns out that some important physical information will be omitted if conformable fractional derivative is used,although the equation can be solved in mathematics.

In this paper,we will introduce a new kind of fractional derivative,which includes a fractional factor.[15]And its physical meanings have been discussed in Sec.2.When we use this definition to construct the time FSE,we find that the total probability is still conserved and the normalizing condition can hold all the time,which is same as normal quantum mechanics and different from the conclusions obtained by Naber.Then we apply this new fractional derivative to solving the space FSE we construct,and just like Khalil et al.used the conformable fractional derivative to realize a natural transition from normal derivative to the fractional one,we achieve a continuous transition from normal Schrdinger equation to the fractional one by constructing the space FSE with a fractional factor.

In Sec.3,we prove that under the description of the time FSE with fractional factor,the probability of finding a particle in the whole space is conserved.

In Sec.4,we present the space FSE with fractional factor and then apply it to describing the behavior of a particle in a one-dimensional infinite square potential well.It turns out that the solution is a linear combination of Bessel function and Neumann function.We also find that the energy of the particle is discrete.The results show that when the space fractional order tends to 2,the distribution of probability density tends to be consistent with the one in normal quantum mechanics,and when the order tends to 1,the distribution loses spatial symmetry and shows a property of attenuation.In the following part,we solve the space FSE with fractional factor in a one-dimensional infinite δ potential well.It turns out that when the particle is at special energy levels,the first derivative of the wave function Φ to space coordinate x can be continuous everywhere even at x=0(where the potential is infinite).This is an interesting result,which is much different from the conclusion in normal quantum mechanics.

2 The Fractional Derivative with Fractional Factor and Its Physical Meanings

The two most popular definitions of fractional derivative are Riemann-Liouville definition and Caputo definition,which are all constructed by an integral form:[16]

(i)Riemann-Liouville definition If the lower limit of definition domain of function f is a.For α ∈ [n−1,n),the α derivative offis

(ii)Caputo definition If the lower limit of definition domain of function f is a.For α ∈ [n−1,n),the α derivative offis

Two ofthem aremuch differentfrom theusual derivative and do not satisfy some operation rules of derivative.[8,13−14]For example,these two fractional derivatives do not satisfy the derivative of the product of two functions and the quotient of two functions:

in addition,due to the form of integration,the definitions above are complicated and not convenient for application.

Considering fractional derivative may be a natural extension of the usual one,Khalil et al.have given a new definition of fractional derivative called conformable fractional derivative.[13]Fractional differential equations can be solved easily by using this definition:

Definition 1Given a function f:[0,∞)→R.Then the conformable fractional derivative offof order α is defined by

where t>0,α∈(0,1).Generally,this kind of fractional derivative can be written as[13]

where α ∈ (n,n+1],and[α]is the smallest integer greater than or equal to α.Under this definition,Tα(f)(t),which refers to the fractional derivative offat t of order α,equals to the usual derivative offat t of order[α]multiplied by a fractional term t([α]−α).Khalil et al.have proved that the conformable fractional derivative satisfies all the properties of standard derivative.

However,some fractional problems of physics cannot be described very well by this definition,which will be shown in this study.So we revise this definition by presenting a fractional factor and the problem above will be solved successfully.

The definition of fractional derivative with a fractional factor is the following:[15]

Definition 2Suppose y=f(x)is defined on an open interval(a,b).We introduce fractional factor e−(1−α)x(α ∈(0,1])and fractional increment∆αx of the independent variable,which satisfies∆x=x − x0= ∆αxe−(1−α)x.Hence,increment of the function is

Then,we can obtain

We can prove that it can be simplified as

For α∈(n,n+1],and f is an(n+1)-differentiable function at point x,where x>0.Then the high order fractional derivative offof order α is the product of fractional factor e−(n+1−α)xand function f(n+1)(x):

where Dα(f)denotes the fractional derivative offof order α.

In this paper,we use FF to denote the fractional factor e−(n+1−α)xand FT to denote the fractional term t([α]−α).

Next,we will discuss the physical meanings of the fractional derivative with fractional factor.Firstly,we consider the fractional generalization of the ordinary differential equations,which govern the relaxation and the oscillation phenomena.The fractional differential equation can be written as[17]

where ω is a positive constant as a frequency.In the cases 0<α≤1 and 1<α≤2,Eq.(10)can be referred to as the fractional relaxation or the fractional oscillation equation,respectively.For the former one,the initial condition is u(0+)=u0.And for the latter one,the initial conditions are u(0+)=u0and u′(0+)=v0.To ensure the continuous dependence of the solution of Eq.(10)on the parameter α in the transition from α =1−to α =1+,we assume v0≡0.

Combine our definition of fractional derivative with fractional factor with Eq.(10),the fractional relaxation equation and the fractional oscillation equation read,respectively,

Considering the respective initial conditions,we can obtain the solutions to Eqs.(11)and(12):

where the constants A,B,C satisfy,respectively,

J0(x)and J1(x)are zeroth-order and first-order Bessel functions.Y0(x)and Y1(x)are zeroth-order and firstorder Neumann functions.

Fig.1 The curves of fractional relaxation when the fractional order is 0.1,0.3,0.5,0.7,0.9,respectively.(a)The situation of low-frequency,ω=0.01;(b)The situation of high-frequency,ω=100.

The analyses of these solutions indicate that for the fractional relaxation equation with fractional factor,there is a boundary frequency ωc,which equals 1.When ω ≪ 1,the relaxation time will increase with fractional order α tending from 0 to 1.In contrast,the relaxation time decreases with fractional order increasing when ω ≫ 1(see Fig.1).In addition,for the fractional oscillation equation with fractional factor,no matter what the frequency is,the equilibration time increases with the increase of fractional order.However,the relationship between the vibration intensity and fractional order is different for the situation of high-frequency and low-frequency.When the frequency is high,the vibration gets intenser with the increase of fractional order.But for the situation of low-frequency,the result is opposite(see Fig.2).In conclusion,the relaxation(oscillation)equation will transform from the fractional one to the ordinary one when fractional order tends to 1(2).With the fractional factor,the fractional relaxation equation describes a system whose law of relaxation time is opposite for high-frequency and low-frequency.And described by the fractional oscillation equation,a process of damped oscillation is produced.

When considering non-Gaussian distribution,three types of fractional diffusion are produced.The space fractional,the time fractional,the time-space fractional diffusion equation can read,respectively,

Thus,the fractional diffusion equations with fractional factor can be produced

The solutions to these equations can be obtained by the method of separating variables.Ignore the source terms and initial conditions,the solutions are

Here,function Z0represents the linear superposition of zero-order Bessel function and zero-order Neumann function,which means

here c1,c2,c3and c4are arbitrary constants.

Fig.2 The curves of fractional oscillation when α is 1.1,1.5,1.7,1.99.(a)The situation of low-frequency,and ω=0.01.(b)The situation of high-frequency,ω=50.

The main difference between the time fractional diffusion equation and the ordinary one is the time term,and for the space fractional diffusion equation,the difference is the space term.The two functions of these two terms have been discussed before.

3 Time Fractional Schr dinger Equation with Fractional Factor

The standard form of the Schrdinger equation is

When the time derivative is fractional,we can obtain a new form of Schrdinger equation:[4]

Mark Naber has solved the time FSE for a free particle by using Caputo definition of the fractional derivative,which is a kind of frequently used definition now.The results show that the total probability increases over time,and finally tends to a finite value 1/ν2depending on the order ν.[4]It means that the normalizing of the wave function ψ is not invariant as time goes on any more,which is quite different from the conclusions in the ordinary quantum mechanics.In addition,since the order ν is smaller than one,the finite value 1/ν2is bigger than one.That is to say,the total probability that a free particle exists in the whole space may be bigger than one,which is not only against the basic physical law,but also against the mathematical definition of probability.However,for the fractional Schrdinger equation with fractional factor like Eq.(30)or Eq.(31),we can prove that the total probability has nothing to do with time and the wave function normalization will be kept.

For Eq.(30),it can be written as

Here,ψ∗is the conjugate complex of ψ.Combine Eq.(30)with Eq.(32),we can obtain

As the same with the probabilistic interpretation in ordinary quantum mechanics,we can define probability density ρ as

Then,

For a free particle,V=0.Then we can obtain

Define the probability current J as

Then,

Compute the integral for the spatial coordinates in the closed region D:

where S is the surface of the region D.When D represents the whole space,let VD→∞(VDrefers to the volume of D).Take account of finiteness of the wave function,on the surface of the region D,ψ=0,then,

According to Eq.(40),we will obtain

In other words,

Hence,the total probability is still conserved.Similarly,for Eq.(31),we can still obtain that the total probability of finding a free particle in the whole space is normalized.

4 Space Fractional Schr dinger Equation with Fractional Factor

When considering anomalous diffusion,which is generated by non-Gaussian distributions,fractional diffusion equations are produced.The space fractional diffusion equation can be constructed as[4,18]

Here,U represents the concentration of the diffusing material and c is the diffusion coefficient,α is the fractional order(0< α ≤ 2).The Schrdinger equation has the mathematical form of a diffusion equation,so when considering the Schrdinger equation that is obtainable for non-Gaussian distributions,we can revise the ordinary Schrdinger equation according to Eq.(44).As a result,the second-order differential of space coordinates in the ordinary Schrdinger equation,which is ∇2ψ(r,t),is replaced with fractional derivative of space coordinates,represented by(1<α<2).In 3D space,

This is also consistent with the local fractional Laplace operator.[19]

Nick Laskin has obtained the space FSE by using the path integral over L´evy trajectories:[2]

Here,(−~2∆)α/2is 3D quantum Riesz fractional derivative.The quantity Dαhas the physical dimension:

However,Wei pointed out that the probability continuity equation obtained from this kind of Schrdinger equation had a missing source term,which led to particle teleportation.[20]In the reply,to emphasize the features of fractional quantum mechanics,Laskin presented a comparison between standard quantum mechanics and the fractional one.[3]Here,the space FSE was presented as

Combine Eq.(45)with Eq.(48),we can obtain the space FSE that has a form of the space fractional diffusion equation(In order to avoid confusion,here,“Dα” in Eq.(48)is replaced with“Mα”below):

For one-dimensional situation,Eq.(49)can be written as

According to the two definitions of fractional derivative with FF and FT,Eq.(50)becomes,respectively,

By using the method of separation of variables in Eq.(51),the time-independent FSE can be given as

And

Similarly,the energy eigen-equation obtained from Eq.(52)is

4.1 A Particle in a One-Dimensional Infinite Square Potential Well

Considering a particle in a one-dimensional square potential well,and the depth of the potential well is infinite:

For 0

Define a parameter A as

Then we can obtain the solution of Eq.(57):

Here,J0(x)is zero-order Bessel function and Y0(x)is zeroorder Neumann function.C1,C2are coefficients that satisfy the normalizing condition

Considering the boundary conditions,we can obtain:

Define two parameters:

Then from Eq.(61),we can get

When x→∞,J0(x)and Y0(x)satisfy the following approximate formulas:[21−22]

Actually,when x reaches a critical value x0,the formulas above are appropriate enough.The degree of approximation increases with x0(In this paper,x0equals 4)(see the appendix).By applying Eqs.(64),(65),and Eq.(63)can be written as:

Simplify it,we can obtain

Then,

So,

Combine Eqs.(58)and(69),it turns out

Then we can compute the energy levels of this particle:

Considering the condition to use Eqs.(64)and(65),we will obtain the limiting condition for the width of the potential well:

Because n≥1 and 1<α<2,to ensure that the approximate formulas can be applied to all the energy levels and fractional orders,the width of the potential well must satisfy

Then,

For the potential well whose width is bigger than this value,the approximate formulas Eqs.(64)and(65)will not be accurate enough.The solution to Eq.(63)will be given by using the original forms of zero-order Bessel function and zero-order Neumann function,and the solution procedure will get more complex.However,because the zero-order Bessel function has partial similarity and approximate formula Eq.(65)is still close to the zero-order Neumann function when x

where B=nπ/[e(1−α/2)a−1],and the normalization coefficient C1can be solved by normalization condition Eq.(60).In quantum mechanics,the behavior of particles is comprehended by the concept of probability.Then we will discuss the behavior of the particle described by the wave function Eq.(75).

Firstly,define the probability density function as

Considering the approximate condition Eq.(74)and normalization condition Eq.(60)at the same time,we can obtain the normalization coefficient C1in the wave function Eq.(75)with different energy levels or different fractional orders(see Table 1).As we can see in Fig.3,when the particle is at energy level 2 and 3,the local maxima are not equal to each other.Actually,if the energy level of the particle is bigger than 1,that is to say,there are multiple peaks of the probability density distribution,the local maxima will decrease progressively with the increase of x.This is different from normal quantum mechanics,and it can be viewed as a property of attenuation.

Fig.3 The probability density distribution of a particle in a one-dimensional infinite square potential well under the description of FSE with fractional factor.The fractional order α is 1.5 and the particle is respectively at energy level 1,2,3.

Table 1 The normalization coefficients with different energy levels and different fractional orders.

In addition,when the space fractional order α changes from 2 to 1,the attenuation of the probability density will get more and more pronounced(see Fig.4).Conversely,if the fractional order α tends to 2,the distribution will get close to the one in normal quantum mechanics(see Fig.4(d)).

Fig.4 The probability density distribution of a particle in a one-dimensional infinite square potential well.The particle is at energy level 3,and the fractional order of the FSE is 1.01,1.3,1.7,1.99,which changes from 1 to 2.

Then we consider the situation of a symmetrical potential well,it turns out that the probability distribution loses symmetry,which is different from the conclusion in normal quantum mechanics.The potential well is given as

Then the wave function turns out

where

At the same time,we can obtain the energy levels:

Similarly,the probability density distribution of a particle in a symmetrical potential well can be computed by the wave function Eq.(78).Here,we take the situation that the fractional order is 1.5 and the particle is respectively at energy levels 1,2,3 for example.Combine the normalizing condition and the wave function,the normalization coefficient C1can be obtained(see Table 2).And then,we can get the probability density distribution(see Fig.5).

Table 2 The normalization coefficients when the fractional order is 1.5 and the particle is respectively at energy levels 1,2,3.

Fig.5 The probability density distribution of a particle in a one-dimensional in finite symmetrical square potential well.

As can be seen in Fig.5,the curve of the probability density is not symmetric with respect to the straight line “x=0” any more,and it shows the same property of attenuation with the one in the asymmetrical potential well.Therefore,under the description of the FSE with fractional factor,whether the potential well is symmetrical or not,the probability density distribution is not affected by spatial symmetry and is only related to space distance.In conclusion,the attenuation property of the probability density distribution will become more and more obvious if the space distance gets longer and the space fractional order tends to 1,or the particle is at higher energy levels.

Then,we will discuss the space FSE with fractional term FT.As we mentioned before,the result shows that this kind of equation can not describe quantum behavior successfully.Solve Eq.(55)under boundary condition Eq.(61),and we can obtain the wave function:

Then,

C1,C2satisfy normalization condition.

From the boundary condition,we can get just one limit equation

Then we can not certain the coefficient A.That is to say,the coefficient E whose physical meaning is the energy of the particle will be uncertain.Hence,although the space FSE with fractional term FT can be solved in mathematics,the solution omits some important physical information.So the conformable fractional derivative is not appropriate to construct the fractional Schrdinger equation.

4.2 A Particle in a One-Dimensional Infinite δ Potential Well

Suppose a one-dimensional infinite δ potential well:

Eq.(53)can be written as:

The wave function Φ should be continuous(condition 1).For the scattering state(E>0),the wave function is

where A=E/Mα~α(E>0).When x→−∞,

The wave function should be finite,so C2=0.When x→+∞,

are both finite.Hence,

Considering condition 1

we will get

Next we will discuss the jump condition(condition 2)of Φ′.Equation(84)can be written as

Then integrate x from −ε to ε(ε → 0)in both sides of this equation:

It turns out

In addition,

As a result,the jump condition of Φ′is obtained from Eqs.(91)and(92):

Under normal circumstances,Eq.(93)is written as

However,when Φ(0)=0,Φ′(0+)= Φ′(0−),Φ′(x)can be continuous everywhere.At this time

From the approximate formula Eq.(64),we can get

Then the energy levels are obtained:

In other words,when the particle is at these approximate energy levels,the first derivative of Φ to space coordinate x can be continuous everywhere.By contrast,in the standard quantum mechanics,Φ′(x)is discontinuous at x=0.[23]

5 Conclusion

In this paper we introduced a new definition of fractional derivative and discussed its physical meanings.By use of this definition,we constructed the time FSE with fractional factor.It was found that the total probability was still conserved.Then the space FSE with fractional factor was presented,and been applied to describing the behavior of a particle in a one-dimensional infinite square potential well.It turned out that the solution to this function is a linear combination of Bessel function and Neumann function and the particle’s energy was dispersed.The distribution of probability density changed from a normal one to the one showing properties of“fractal”when the fractional order increased from 1 to 2.When considering the situation of a one-dimensional infinite δ potential well,it was found that when the particle was at some special discrete energy levels,the first derivative of the wave function without time to space coordinate x could be continuous everywhere even at x=0 where the potential was infinite.This is a result,which will not appear in standard quantum mechanics.

During the past years,many studies have been done in different areas of physics and engineering to search for systems,which can be described by fractional order differential equations.For example,appropriate analog circuits can construct the fractional relaxation system and oscillation system.[24]In addition,based on transverse light dynamics in aspherical optical cavities,one can obtain an optical realization of the fractional Schrdinger equation.[25]So from these aspects,our next work is searching for fractional order systems to realize the phenomena appearing when we apply the fractional derivative with fractional factor to solving fractional order differential equations.

Appendix A:The Approximate Formulas of Zero-Order Bessel Function and Zero-Order Neumann Function

Bessel equation and Bessel function play important roles in many modern theoretical problems and engineering technologies.Bessel equation was produced in the process of solving the partial differential equation describing the motion of a hanging chain.It is the key to solving the Laplace equation in polar and cylindrical coordinate systems.

ν-order Bessel equation can be written as[17,23]

and its general solution is

where Jν(x)is ν-order Bessel function of the first kind,Yν(x)is ν-order Bessel function of the second kind(also called Neumann function).

ν-order Bessel function of the first kind is defined as

When ν=n,(n=0,1,2,...),integer order Bessel function is obtained:

and the approximate function of it is[17]

For zero-order Bessel function,we can get

ν-order Bessel function of the second kind is defined as

When ν=n,(n=0,1,2,...),integer order Neumann function is given as

For zero-order Neumann function,its approximate function is

The approximate functions Eqs.(A6)and(A9)will get more and more accurate with x tending to infinity.Actually,when x reaches some value x0,the approximate formulas are accurate enough(see Fig.6).In this paper,in order to get a relatively wider potential well,we take x0as 4.

Fig.6 The curves of Bessel functions and respective approximate functions.(a)The curves of zero-order Bessel function and its approximate function.The smaller picture is partial enlarged detail of the curves at x=4.(b)The curves of zero-order Neumann function and its approximate function.The smaller picture is partial enlarged detail of the curves at x=4.