A Study of Fractional Schrödinger Equation Composed of Jumarie Fractional Derivative

Pramana – J. Phys. (2017) 88: 70 c Indian Academy of Sciences DOI 10.1007/s12043-017-1368-1

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

JOYDIP BANERJEE1, UTTAM GHOSH2,∗, SUSMITA SARKAR2 and SHANTANU DAS3

1Uttar Buincha Kajal Hari Primary School, Fulia Buincha, Nadia 741 402, India 2Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700 009, India 3Reactor Control Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India ∗Corresponding author. E-mail: [email protected]

MS received 4 April 2016; revised 4 November 2016; accepted 10 November 2016; published online 27 March 2017

Abstract. In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrödinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero – rather they are non-zero with a minimum spread. This type of non-zero spread arises in the micro- scopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (x)α (and (t)α) with 0 <α<1; called as ‘fractional differentials’. For arbitrarily small x and t (tending towards zero), these ‘fractional’ differentials are greater than x (and t), i.e. (x)α >x and (t)α >t. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.

Keywords. Jumarie fractional derivative; Mittag-Lefﬂer function; fractional Schrödinger equation; fractional wave function.

PACS Nos 02.30.Jr; 03.65.−w; 05.30.−d; 05.40.Fb; 05.45.Df; 03.65.Db

1. Introduction in the field of fractional calculus. The Mittag-Leffler function with complex argument gives the fractional Fractional-order derivatives are extensively used to study sine and cosine functions [9]. Ghosh et al [11,16,17] different natural processes and physical phenomena discussed about the solutions of various types of linear [1–14]. The mathematicians of this era are trying fractional differential equations (composed of Jumarie to construct general form of calculus by modifying fractional derivative) in terms of Mittag-Leffler func- the classical order derivative to an arbitrary order. tions. On the other hand, researchers are using different Riemann–Liouville [11] definition of fractional deriva- fractional differential equations by incorporating the tive admits non-zero value for fractional differentiation fractional order in place of classical order derivatives. of a constant. This contradicts the basic properties of classical calculus. To overcome this problem, Jumarie [9, 13, Question arises as to what will be the actual equation 15] modified Riemann–Liouville definition of frac- in fractional sense if we start with the basic fractional tional derivative to obtain zero value for the fractional equations. This is a challenging task to mathematicians. derivative of a constant. This type of derivative is also The coordinate x in x-space, originates from the dif- x = applicable for continuous but non-differentiable func- ferential dx, as its integration, i.e. 0 dy x. Now with tions. Mittag-Leffler [10] function was initially intro- a differential (dx)α, with 0 <α<1, we have (dx)α > x α ∼ α duced in classical sense but it has several applications dx,where 0 (dy) x [13,18,19]. That is, the space 1 70 Page 2 of 15 Pramana – J. Phys. (2017) 88: 70 is transformed to a fractal space where the coordi- 2.1 Riemann–Liouville (RL) definition nate x is transformed to xα. This is coarse graining of fractional derivative phenomenon in particular scale of observation. Here we come across the fractal space–time, where the nor- Riemann–Liouville (RL) fractional derivative of a function f(x)is defined as mal classical differentials dx and dt cannot be taken arbitrarily to zero. Thus, in these cases the concept of m+1 classical differentiability is lost [20]. The fractional- α = 1 d aDx f(x) − + + order α is related to roughness character of the space– ( α m 1) dx time, i.e. the fractal dimension [18,19]. In this paper, x − fractional differentiation of order α is used to study the × (x − τ)m αf(τ)dτ, dynamic systems defined by the function f(xα,tα).In a [14,18,19] the demonstration of fractional calculus on where m ≤ αquantum mechanics for the fractal region. This formulation also leads to normal quantum mechanics at limiting condi- 2.2 Jumarie-modified RL definition tion, of fractional order α tending to unity. The nature of of fractional derivative the solution at various values of fractional orders α of fractional differentiation signifies the underlying beha- To get rid of the above-mentioned problem of RL viour of quantum mechanics, in fractal space–time. We fractional derivative, Jumarie modified [9,13,15] this have modified de Broglie’s and Planck’s hypothesis definition, for a continuous (but not necessarily dif- in the fractional sense such that they remain intact if ferentiable) function f(x), with start point of function = = limiting conditions of α tending to unity are used. a 0 such that f(a) f(0) is finite at the start point This paper is divided into separate sections and sub- of the function, as follows: sections. In §2 some definitions of fractional calculus (α) = J α are described. In §3 we discuss about fractional-order f (x) 0 Dx f(x) wave equation. Section 4 is about solution of frac- ⎧ ⎪ 1 x tional wave equation. Section 5 deals with fractional ⎪ (x−ξ)−α−1f(ξ)dξ, α<0, ⎪ − Schrödinger equation. In §6 we develop ‘equation of ⎪( α) 0 ⎪ continuity’. In §6.1 we discuss about properties of ⎨⎪ 1 d x − −α − fractional wave function. Section 6.2 discusses fur- = − (x ξ) (f(ξ) f(0)) dξ, ⎪(1 α) dx 0 ther study on fractional wave function, §6.3 is about ⎪ ⎪ ‘orthogonal’ and ‘normal’ conditions of wave func- ⎪ 0 <α<1, ⎪ tions. Section 7 is about operators and expectation ⎩⎪ (α−n) (n) ≤ + ≥ values. In §8 and 8.1 we discuss simple application of f (x) ,nαprobability density. In §8.4 we discuss denotes a continuous (but not necessarily dif- → ∈ R the energy calculations. In Appendix, we have defined ferentiable) function such that x f(x)for all x . fractional quantities that are used in the paper. Let h>0 denotes a constant infinitesimal step. Define a forward operator Eh[f(x)]=f(x + h); then the right-hand fractional difference of order α (0 <α<1) is, 2. Some definitions of fractional calculus (α) α + f(x) = (Eh − 1) f(x) There are several definitions of fractional derivative. The leading definitions are Riemann–Liouville (RL) ∞ k α definition [5], Caputo definition [1] and modified RL = (−1) ( Ck)f (x + (α − k)h) , definition [9]. k=0 Pramana – J. Phys. (2017) 88: 70 Page 3 of 15 70

α ∞ + where Ck = (α!/k!(α−k)!) are the generalized bino- α − − α (2k 1)α α Eα(it ) Eα( it ) k t mial coefficient. Then the Jumarie fractional derivative sinα(t )= = (−1) 2i (2kα+α)! is k=1 ∞ (α)[ − ] (2k+1)α (α) + f(x) f(0) k t f+ (x) = lim × (−1) . h↓0 hα (2kα+α+1) k=1 dαf(x) = . dxα One of the most important properties of Mittag-Leffler J α α = α Similarly, one can have left Jumarie derivative by function [11] is Dx Eα(ax ) aEα(ax ).This defining backward shift operator. In this Jumarie def- means that the Jumarie fractional derivative of order α inition we subtract the value of the function at the start of the Mittag-Leffler function of order α (in scaled vari- point, from the function itself and then the fractional able xα) is just returning the function itself. Thus, the α derivative is taken (in Riemann–Liouvelli sense). The function Eα(ax ) is the eigenfunction for the Jumarie offsetting of the function by subtracting the start point derivative operator. We mention here that this function α value makes the fractional derivative of constant func- Eα(ax ) is also eigenfunction of Caputo derivative α tion as zero. This also gives conjugation with classical operator. This function Eα(ax ) is also termed as α integer-order calculus, especially regarding chain rule α-exponential function e˜α(x) = Eα(x ).Thisisin for fractional derivatives, fractional derivative of the conjugation with classical calculus similar to expo- product of two functions etc. [13]. In the rest of the nential function and very useful in solving fractional paper, the derivative operator Dα will be regarded as differential equations composed of Jumarie fractional the modified Riemann–Liouville (Jumarie) derivative. derivative.

2.3 Some techniques of Jumarrie derivative 3. Fractional-order wave equation and its solution Consider a function f [u(x)] which is not differentiable in the classical sense but fractionally differentiable. In this section we consider a plane progressive wave Jumarie suggested [13] three different ways depend- propagating in the positive x direction with a constant ing upon the characteristics of the function, depicted velocity v. The general form is f(x,t) = f(x − vt) by formulas (i), (ii) and (iii), as follows: [21]. The fractional plane progressive wave propagat- α (α) α ing in fractional space–time can be considered in the (i) D (f [u(x)]) = fu (u)(u ) , x following form: α [ ] = 1−α α α (ii) D (f u(x) ) (f/u) (fu(u)) u (x), = α − α 0 ≤ 1 (1) α α−1 (α) α f(x,t) f(x vαt ), <α . (iii) D (f [u(x)]) = (1 − α)!u fu (u)u (x). Here vα is the fractional velocity (we shall elaborate 2.4 Mittag-Leffler function and fractional this in Appendix). When α tends to one, this frac- trigonometric functions tional plane progressive wave turns to one-dimensional plane progressive wave. Thus, eq. (1) represents an The one-parameter Mittag-Leffler function [10] is αth-order fractional plane progressive wave, where the defined as infinite series in the following form: space and time axes are transformed to xα and tα ∞ ≤ zk respectively (with 0 <α 1). Thus, the wave we Eα(z) = ,z∈ C, Re(α) > 0. consider here is a fractional plane wave moving in the (1 + αk) k=0 x direction. Jumarie-type fractional derivative [9,13] For α = 1, it is a simple exponential function E1(z) = is used to find various physical quantities and physical ez. The fractional sine and cosine functions [4] defined properties of the corresponding wave. Let us define the J α ≡ α α J 2α ≡ 2α 2α by Mittag-Leffler function are as follows: operators as Dx ∂ /∂x , Dx ∂ /∂x and ∞ J α ≡ α α J 2α ≡ 2α 2α α + − α 2kα Dt ∂ /∂t , Dt ∂ /∂t . These are Jumarie α Eα(it ) Eα( it ) k t cosα(t ) = = (−1) derivative operators. Now consider the following with 2 (2kα)! ≤ = α − α k=1 the condition, i.e. 0 <α 1, with u(x, t) x vαt ∞ t2kα and write function of u(x, t) as = (−1)k (2kα + 1) α α k=1 f (u(x, t)) = f(x − vαt ). 70 Page 4 of 15 Pramana – J. Phys. (2017) 88: 70

We now choose the differential trick that is Dα(f [u(x)])= where left-hand side is space-dependent and right-hand α−1 (α) α (1 − α)!u fu (u)u (x); the formula (iii), of Jumarie side is time-dependent. This is possible if and only if [22]. We know the following expression: both sides of this equation is constant. Let the constant be k2. (α) = α = α[ α − α]= α[ α] α ux Dx u(x, t) Dx x vαt Dx x The space part of the equation is now = != + α (α 1). 1 D2αg(xα) = k2 or D2αg(xα) = k2g(xα) Therefore, applying Jumarie fractional derivative with g(xα) x α x α respect to x; with formula (iii) we write (2) (3a) α[ ]= ! − ! α−1 (α) Dx f (u(x, t)) α (1 α) u fu (u). (2) Solution of this equation [11,23] is α α Doing similar operation with respect to t, we write the g(x ) = bEα(±ikαx ), following equation: where b is a constant. Similarly, the time part of the α[ ]= ! − ! α−1 (α) 2α α = 2 2 α Dt f (u(x, t)) vαα (1 α) u fu (u). (2a) equation is Dt h(t ) kαvαh(t ). The solution to From eqs (2) and (2a) we get the following equations: this is α α α[ ]= α [ ] h(t ) = BEα(±iωαt ). Dt f (u(x)) vα(Dx f u(x) ). α Here we put ω = kαvα and assume B as a constant. On operating Dx on both sides of the above expression α we get Thus, the general solution of (3a) is of the following form: DαDαf[u(x)]=v (DαDαf[u(x)]) x t α x x = ± α ± α = 2α [ ] f(x,t) AEα( ikαx )Eα( iωαt ), (4) vα(Dx f u(x) ). (2b) = α where A bB is a constant. Now on operating Dt on both sides of (2b) we get α α [ ]= 2α [ ] Dt Dt f u(x) Dt f u(x) 5. Fractional Schrödinger equation: Derivation = α α [ ] vα(Dt Dx f u(x) ). (2c) and solution Using Theorem A.8 and combining eqs (2b) and (2c) Consider a particle of mass moving with veloc- we get the following expression: m ity v. According to de Broglie hypothesis [24], there 1 D2αf [u(x)]= (D2αf [u(x)]) is a wave associated with every moving particle. The x 2 t = vα mathematical form of de Broglie hypothesis is p J 2α α α 2 J 2α α α D [f(x − vαt )]=v ( D [f(x − vαt )]). k.Herep is the momentum of the particle and k is t α x (3) the wave vector in one dimension; is the reduced Planck’s constant. Now Planck’s hypothesis [24] shows Equation (3) represents the fractional wave equation that the energy ε of a particle in a particular quantum of α order. If α = 1 the equation changes to one- level is proportional to the angular frequency, that is, dimensional classical wave equation for the plane ε = ω. In this context, it is assumed that de Broglie progressive wave. hypothesis and Planck’s hypothesis are also valid in fractional αth order with a modiﬁed form 4. Solution of the fractional wave equation pα = αkα (5)

To find solution of the fractional wave equation (3) by εα = αωα. (6) separation of the variable method, we consider its solu- It is clear that if α = 1, eqs (5) and (6) reduce to tion of the form f(x,t) = g(xα)h(tα). This reduces the original form of de Broglie and Planck’s hypothe- eq. (3) to ses. Here α is the reduced Planck’s constant of α 1 2α α = 1 1 2α α order; = h/2π,andh is the Planck’s constant. Here, α Dx g(x ) 2 α Dt h(t ) g(x ) vα h(t ) ωα is the fractional-order angular frequency and kα is which implies the following expression: the fractional-order wave vector (which is described in Appendix). 1 α h(tα)D2αg(xα) = g(xα)D2αh(tα), The general solution for eq. (3) is u = Af (x − x 2 t α vα vαt ),0<α≤ 1whereA is a constant. To find Pramana – J. Phys. (2017) 88: 70 Page 5 of 15 70 the explicit form of this solution, Mittag-Leffler [10] This gives the following expression: function is taken as a trial solution [11] similar to 2 the classical differential equation where we consider − α J 2α α α = α α (11) α Dx f(x ,t ) εαKf(x ,t ). exp(x) as the trial solution [25]. Thus, we can take 2 mα α α α α The variation of the function (8) with time is studied. f(x ,t ) = AEα(ikαx − iωαt ) Repeating the above steps now by taking Jumarie frac- = AE (ik xα)E (−iω tα), (7) α α α α tional derivative of order α with respect to time (t), we α α where Eα(ikαx ) and Eα(−iωαt ) are one-parameter have the following equation: Mittag-Leffler functions in complex variable. Thus, (7) J α α α =− i α α is a trial solution of eq. (3). Now fractional velocity Dt f(x ,t ) εαf(x ,t ). (12) α vα can be defined as vα = ωα/kα, which is the fractional velocity of the particle as well as the fractional Here εα is the total energy of the system. From the phase velocity of the wave. This fractional velocity is conservation of energy in fractal space we write assumed to be constant. = Consider a particle that possesses constant momen- Total energy (εα) (kinetic energy εαk) + (potential energy V(xα,tα)). tum pα and constant energy εα, i.e. its energy and momentum do not vary with the propagation of the That is, wave in space and time. Using conditions (5) and (6), = + α α solution (7) can be written as εα εαK V(x ,t ). (13) α α = i α − i α Substituting (13) in eq. (12) and combining them with f(x ,t ) AEα pαx Eα εαt . (8) α α eq. (11) we get the following expression: This particle has some hidden physical properties. To 2 investigate these properties we derive the space and − α J 2α[ α α ] = − α α α α α α α ( Dx f(x ,t ) ) ((εα V(x ,t ))f (x ,t ). time evolution of the function f(x ,t ). 2 mα

Now by rearranging the above expression we obtain the 5.1 Derivation of fractional Schrödinger equation following form: To derive α-order fractional Schrödinger equation we 2 first find the α-order partial derivative of (8) with − α J 2α α α + α α α α α ( Dx f(x ,t )) V(x ,t )f (x ,t ) respect to space coordinate in the form 2 mα = i (J Dαf(xα,tα)), (14) J α α α = ipα i α α t Dx f(x ,t ) A Eα pαx α α where i ×E − ε tα α α α α i α i α α f(x ,t ) = AEα pαx Eα − εαt . α α J α α α = i α α Dx f(x ,t ) (pαf(x ,t )). (9) This is the α-order fractional Schrödinger equation. In α the limit α = 1 this equation reduces to the ‘clas- Repeating the previous operation once again to eq. (9) sical’ Schrödinger equation in one-dimensional space we get the following expression: and time. Equation (14) has the solution which will 1 J D2αf(xα,tα) =− p2 f(xα,tα), (10) lead to certain interesting physical properties. x 2 α α where 5.2 Solution of fractional Schrödinger equation J α α = α [ ] 0 Dx Eα(ax ) aEα(ax ) 11 . The basic method of the solution of eq. (14) is the 2 = α Let us define pα 2 mαεαK,wheremα is the mass method of separation of variables. In this method the (in fractional sense) and εαK is the kinetic energy of solution is identified as the product of two different fractional order α. Then expression (10) can be written functions (xα) and T(tα). Here, (xα) depends on as the transformed space variable xα and T(tα) depends 1 J D2αf(xα,tα) =− (2αm ε )f (xα,tα). on transformed time variable tα. Thus, the function x 2 α αK α f(xα,tα) = (xα)T (tα). Substituting f(xα,tα) = 70 Page 6 of 15 Pramana – J. Phys. (2017) 88: 70

(xα)T (tα) in eq. (14), we obtain the following This equation is called the time-independent fractional expression: Schrödinger equation. It is potential-dependent. So it 2 1 d2α [ (xα)] is not possible to solve eq. (19) without knowing the − α + V(xα) (xα) character of the potential function. However, it can (2)αm (xα) dx2α α be confirmed that solution of (19) has only space i dα [T(tα)] = α (15) dependency. A Hamiltonian can be constructed with α α . T(t ) dt the analogy of classical Schrödinger’s one-dimensional Left-hand side of eq. (15) is only space-dependent and quantum wave equation. The Hamiltonian in terms of right-hand side is only time-dependent. Thus, to satisfy non-integer order derivative is thus defined as follows: eq. (15) both sides must be equal to some constant. On 2 d2α the left side of eq. (15) we have a fractional potential ˆ =− α − α (20) Hα α 2α V(x ). term. This has the dimension of fractional energy, that (2) mα dx is Therefore, eq. (19) can be written in terms of Hamilto- − − [ML2T 2]α =[MαL2αT 2α]. nian as depicted below: ˆ = Clearly the constant must have the dimension of Hα εα . (21) fractional energy due to homogeneity of dimension. Equation (21) is an eigenequation with the eigenvalue Observing the right-hand side of the equation, the ε . The eigenfunction of the equation is .Thiseigen- dimension analysis allows us to choose the unit of α function is the ‘information centre’ of a particle. One the constant as (Joule)α for fractional values of εα α can operate it in various ways to find the correspond- which is also supported by eq. (12). Equation (15) can ing physical property. The Hamiltonian (21) gives the be written as the following two different equations, correct information about the energy of the particle. of which one is solely time-dependent and other is solution space-dependent, as described below: α α 6. Continuity equation: Conservation iα d [T(t )] = εα, (16) of probability current density T(tα) dtα 2 1 d2α [ (xα)] Consider the Schrödinger equation previously derived − α + V(xα) = ε . (17) α α 2α α in eq. (14) (2) mα (x ) dx Solution of equation of type (16) was found by Ghosh 2 − α J 2α[ α α ] + α α α α et al [11] using the Mittag-Leffler functions in the form α ( Dx f(x ,t ) ) V(x ,t )f (x ,t ) 2 mα α i α = i (J Dα[f(xα,tα)]). T(t ) ≈ Eα − εαt . α t α Multiply the above equation with the complex con- Thus, the solution of eq. (15) is (by omitting the jugate of the solution, say f ∗(xα,tα), to write the integral constant) the following: following expression: α α α i α f(x ,t ) = α = (x )Eα − εαt . (18) 2 α − α ∗ α α J 2α[ α α ] α f (x ,t )( Dx f(x ,t ) ) For α = 1, i.e. in limiting case, the solution (18) 2 mα ∗ turns to the solution of one-dimensional classical + V(xα,tα)f (xα,tα)f (xα,tα) Schrödinger wave equation. = ∗ α α J α[ α α ] i αf (x ,t )( Dt f(x ,t ) ). (22)

5.3 Time-independent fractional Schrödinger Let us take complex conjugate of eq. (14) and multi- equation and fractional Hamiltonian ply with the function f(xα,tα) and write the following equation: Equation (17) has no time-dependent part. This can be 2 rearranged as follows: ∗ − α f(xα,tα)(J D2α[f (xα,tα)]) 2αm x 2 d2α [ (xα)] α − α − − α α = 0 + α α α α ∗ α α α 2α εα V(x ) (x ) . V(x ,t )f (x ,t )f (x ,t ) (2) mα dx =− α α J α[ ∗ α α ] (19) i αf(x ,t )( Dt f (x ,t ) ). (23) Pramana – J. Phys. (2017) 88: 70 Page 7 of 15 70

Subtracting eq. (22) from eq. (23) we have following will vanish in the middle of its trajectory which is not (by dropping xα,tα) equation: possible at all. Fractional wave function must be single valued, i.e. for every position of space–time the 2 − α ∗ J 2α[ ] − J 2α[ ∗] property of the particle is unique. α (f ( Dx f ) f( Dx f )) 2 mα (b) The fractional wave function must be square inte- = ∗ J α[ ] + J α ∗ grable in fractional sense in the region a ≤ x ≤ b, i α(f ( Dt f ) f( Dt f )). (24) i.e. Equation (24) can be rewritten in the following form: b ∗ 2 α ∞ ∗ ∗ α αdx < . − α J α[ J α[ ] − J α[ ] ] a α Dx f ( Dx f ) f( Dx f ) 2 mα α = ∗ J α[ ] + J α[ ∗] The notation f(x)dx implies fractional integration i α(f ( Dt f ) f( Dt f )). (25) that is given as follows: Let us define x x 1 − f(x)dxα = (x − ξ)α 1f(ξ)dξ i α ∗J α − J α ∗ = −∞ (α) −∞ α (f Dx f f Dx f ) jα 2 mα with α>0. as the probability current density of α order. Define (c) Linear combination of solutions of the fractional ∗ = f f ρα as the probability density of α order. For Schrödinger wave equation itself is a solution. Thus, = ∗ = α 1 the fractal probability density f f ρα turns to linear combination of the wave function is another the one-dimensional probability density. Thus, eq. (25) wave function. reduces to (d) The fractional wave function must vanish at the J α[ ]=J α[ ] boundary. If it is does not vanish at the boundary, then Dx jα Dt ρα . (26) the boundary itself loses its significance. The bound- This is the equation of continuity of α order in one aries have property to seize the motion of particle to ∗ = dimension. If probability mass density f f ρα is go further away from it. As a result, the particle has to independent of time, right-hand side of (26) is zero. stop at the boundary and consequently the fractional Thus, the left-hand side is also equal to zero. This wave function vanishes. Here boundary means per- implies that the one-dimensional variation of current fectly rigid boundary. If we have an analogy with a density with space is zero. Physical significance of the vibrating string bounded by two points, then we cannot fact is that there is no source or sink of probability cur- get amplitude of vibrating string at the two end points. rent density. This is the condition of stationary state. Mathematically, the condition for the wave function in To satisfy the above condition of ρα the solution must the region a ≤ x ≤ b is thus described as follows: be of type α α α α α(a) = α(b) = 0. f(x ,t ) = α = (x )Eα(−i(εαt /α)). This is the stationary state solution of α order. For α = (e) The fractional Schrödinger equation suggests that J α ≡ α α 1 the state is the same as that of the one-dimensional the α-order fractional derivative Dx ∂ /∂x of the stationary state solution. wave function α is continuous and single valued. (f) The α-order fractional derivative of the wave function must vanish at the boundary. If not, condi- 6.1 Some properties of fractional wave function tion of stationary state will violate as suggested in the For further investigation it is needed to characterize equation of continuity. the basic properties of the solution of fractional (g) The wave function must be normalized. This sig- Schrödinger equation. We list them as follows: nifies the existence of the particle with certainty within the considered boundary. (a) The fractional wave function must be continuous and should be single valued. As the particle has 6.2 Further study on fractional wa e function physical existence, the fractional wave function of the v particle must be continuous at every position of space The general solution of fractional wave equation is and time. If the fractional wave function is not con- α α α α tinuous for some position or time, then the particle f(x ,t ) = α = (x )Eα(−iεαt /α). 70 Page 8 of 15 Pramana – J. Phys. (2017) 88: 70

Its complex conjugate is ∗ α is defined as the inner product of order α where F (x ) ∗ = ∗ α α is the complex conjugate of F(xα). In the same way, α (x )Eα iεαt / α . ∗ the orthonormal condition is defined by the following Multiplying α with α we get fractional integration: ∗ = α ∗ α α α (x ) (x ). b ∗ α α α This quantity is independent of time. We define this F |G = F (x )G(x )dx = 1. (28) quantity as ‘existence intensity’ and as existence a amplitude. In a certain considered boundary the par- The general solution of wave function is ticle exists with certainty. Let be defined in the ∞ = boundary −∞ ≤ x ≤+∞.Thenwehave α cnψn, +∞ n ∗ α = α αdx constant. where cn is some constant. Here n is a dummy index. −∞ The complex conjugate of the solution is α Note that the notation f(x)dx implies fractional ∞ integration, that is, ∗ = ∗ ∗ α cmψm x x 1 − m f(x)dxα = (x − ξ)α 1f(ξ)dξ −∞ (α) −∞ and the inner product using Dirac bracket notation is with α>0. ∞ ∞ ∗ ∗ | = c c ψ ψ . (29) If the particle does not exist at the boundary, the α α m n mα nα +∞ ∗ m n integration vanishes. We can now define −∞ α α α +∞ From orthogonal and orthonormal conditions we have dx = 1 if the particle exists with certainty and −∞ ∗ α = the following equation: α αdx 0 if the particle does not exist any- +∞ ∗ ∞ ∞ where. Clearly, the existence parameter dxα ∗ ∗ −∞ α α | = c c ψ ψ =0, if n = m ≤ +∞ ∗ α ≤ α α m n mα nα is such that the condition 0 −∞ α αdx 1, m n gets satisfied. and Suppose we have to find the information about the ∞ ∞ ∗ ∗ existence over a certain region considered inside the | = c c ψ ψ =1, if n = m. +b ∗ α = α α m n mα nα boundary, and then the quantity −a α αdx m n l<1 should be less than 1. This defines that parti- In a similar way we write cle is not localized and behaves as waves that are not | = = localized. If all these existence parameters (or proba- ψmα ψnα 1, if n m. bility) are added up, the total probability is unity. From Clearly eqs (9), (10), (12), (17) we find that wave function is ∞ an eigenfunction of various operators. | = ∗ = α α cncn 1. n 6.3 Orthogonal and orthonormal conditions More precisely, it can be written as of wave functions ∞ | = | |2 = Two functions F(x) and G(x) defined in the region α α cn 1. a ≤ x ≤ b are orthogonal if their inner product is zero n [25]. From the analogy of this orthogonal condition in We can therefore define cn as the existence coefficient {x} space we can define the orthogonal condition for or probability coefficient. {xα} space with the following fractional integration operation such that 7. Operators and expectation values b ∗ α α α F |G = F (x )G(x )dx = 0. (27) In quantum mechanics all the measureable quantities a which cannot be measured directly are measured by Here b their expectation values [25]. So in the case of α-order ∗ F |G = F (xα)G(xα)dxα quantum mechanics we need to define operators for a every measurable quantity. For this purpose, there must Pramana – J. Phys. (2017) 88: 70 Page 9 of 15 70 be some rules of choosing operators which we list as Let us take the following constant: follows: (2)αm ε α α = k2. (31a) 2 α (i) Every operator must be an eigenoperator of the α wave function. With this, the equation now is as follows: (ii) Eigenvalue of the operator defines measurable d2α (xα) + 2 α = 0 (32) quantity. 2α kα (x ) . (iii) Expectation value of an operator is the measure dx of the corresponding operator. This equation has solution as suggested by Ghosh et al [11] ˆ Consider an operator Aα that operates on a certain α α α (x ) = AEα(−ikαx ) + BEα(ikαx ). (33) function nα such that Using boundary condition (0) = (aα) = 0, we get ˆ = Aα nα λn nα. (30) A + B = 0. Thus, the solution of (32) is From the general form of , eq. (30) turns to be α (xα) = B(E (−ik xα) − E (ik xα)). ∞ ∞ α α α α ˆ ˆ Aα[ψα]= cnAαψnα = λncnψnα. Using the definition of fractional sine function [9] we n n write the following equation: Thus, λn cannot be determined directly. For the correct α α (x ) = C sinα(kαx ). (34) information of the system we have to find the mean value (or expectation) of the system. Expectation value Using boundary condition on eq. (34) we get of an operator is defined as follows: α α (a ) = C sinα(kαa ) = (0) = 0. (34a) +∞ ˆ ∗ d α −∞ ψnαAαψnα x As defined by Jumarie [9] A= +∞ . (31) ∗ α −∞ ψnαψnαdx α α sinα(x ) = sinα((x + Mα) ). For every physical measurable quantity there is a corresponding expectation value. Here we defined Mα as first-order zero or first zero crossing for the fractional sine function [26]. As sinα(0) = 0, we have 8. Simple application: Particles α = in one-dimensional infinite potential well sinα((Mα) ) 0. (34b) α Comparing eqs (34a) and (34b), sinα(kαa ) = Consider a particle bounded by a one-dimensional infi- sin ((M )α),wegetkaα = (M )α which gives the = = α α α nite potential well with length from x 0tox a following expression: ≤ ≤ for 0 x a. The potential defined here is of the α type of V = 0if0≤ x ≤ a and V =∞otherwise. Mα kα = . (34c) Thus, the particle is strictly bounded by the poten- a α α tial well in the transformed scale, i.e. x too. The Using kα = (Mα/a) in eq. (34) the solution is wave function is zero outside the well. For continuity, α α Mα α the wave function must vanish at the boundaries also, (x ) = C sinα x . i.e. (0) = (aα) = 0. The fractional Schrödinger a equation as suggested in eq. (19) is as follows: 2 d2α (xα) 8.1 Normalization of wave function − α − − α α = 0 α 2α (εα V(x )) (x ) . (2) mα dx The normalization condition for wave function of α With α = 0, this equation takes the form order is as follows: V(x ) a 2 2α α ∗ α d (x ) α α = − − = 0 α dx 1. (35) α 2α εα (x ) . α (2) mα dx 0 α α α Rearranging the above expression we write the follow- As α(x ) = (x ) = C sinα(kαx ) is a real function, ∗ 2 ing equation: α = | α| . Then, α d2α (xα) (2)αm ε a + α α (xα) = 0. | |2 α = 2α 2 α dx 1. dx α 0 70 Page 10 of 15 Pramana – J. Phys. (2017) 88: 70 Here ∗ = | |2 = | (xα)|2 = C2 sin2 (k xα). 1 a 1 a α α α α α C2 dxα − C2 cos (2k xα)dxα = 1 Thus, the integration is 2 2 α α 0 0 a = 2 2 α α = 2 α sin 2 α x a C sinα(kαx )dx 1. C x − α( kαx ) = 0 + 1 2 (1 α) 2kα x=0 To integrate the above equation an identity must be developed, which is described next. By definition we C2 aα sin (2k aα) − α α = 1. have following expression: 2 (1 + α) 2kα E (2ixα) + E (−2ixα) α α = α α The term sinα(2kαa )/2kα is zero as suggested by cosα(2x ) . 2 2 boundary√ condition. Thus, (C /2)(aα/(1 + α)) = 1 We also have the following identities [11]: or C = 2(1 + α)/aα. With all these derivations we get the solution as follows: E (2ixα) + E (−2ixα) cos (2xα) − 1 = α α − 1 α 2 + α α = 2(1 α) Mα α α + − α − (x ) sinα x . Eα(2ix ) Eα( 2ix ) 2 aα a cos (2xα) − 1 = . α 2 For α = 1 the solution is converted to one-dimensional The Mittag-Leffler function E (axα) satisfies the rela- α solution for one-dimensional Schrödinger equation of tion infinite potential well. α α α Eα(ax )Eα(bx ) = Eα((a + b)x ). = α = α 2 Putting a b we get Eα(2x ) (Eα(x )) (Theorem 1 8.2 Graphical representation of wave function of [11]). With this, we manipulate the above expression to get the following equation: Graphical presentation of cos (2xα)−1 + α α 2(1 α) α α 2 + − α 2 − α − α (x ) = sinα(kαx ) = (Eα(ix )) (Eα( ix )) 2Eα(ix )Eα( ix ) aα 2 = α − − α 2 for different values of fractional order for a 10 unit α (Eα(ix ) Eα( ix )) cosα(2x ) − 1 = . is shown in figure 1. We need to know the values of Mα 2 for various α before plotting the graph. We found using Again, by definition [11] we have the following Wolfram Mathematica-9 the various approximate val- expression: ues of (Mα) for 10,000 terms of Mittag-Leffler function α α α Eα(ix ) − Eα(−ix ) and sinα(x ) function. Numerically, it is observed that sin (xα) = . (36) α α 2i sinα(x ) losses periodicity for α<1. They are listed in table 1. So, we get the following identity: √ The plot is drawn for (xα) = 2(1 + α)/aα − α = 2 α α 1 cosα(2x ) 2sinα(x ). (37) sinα(kαx ) against x. Here the box width is taken as = Using the identity of eq. (37), and by using the formula 10 units, that is, a 10. From numerical analysis we of fractional integration, i.e. found that the quantum boundary conditions are satis- fieduptoα ≈ 0.736 from α = 1. The plot suggests that x α α −1 α the maxima of the wave function shift to the right with cosα(λy ) (dy) = λ sinα(λx ) 0 the increase in value of order α. The nature of wave function also changes with value of α.Whenα = 1, and x the plot is the same as suggested by one-dimensional − (dy)α = (α!) 1 xα Schrödinger potential box problem. When α ≈ 0.736, 0 the curve is not symmetrical and more area gets cov- we write following steps: ered at the left side of the graph. Less the value of α, a more asymmetrical is the plot. These plots have one 2 2 α α = C sinα(kαx )dx 1 zero crossing [26]. This means that the quantum num- 0 a ber of the system is 1. The system is in the ground state. 2 1 α α To compare the wave functions for various values we C (1 − cosα 2kαx )dx = 1 α 2 0 have another plot (see figure 2). Pramana – J. Phys. (2017) 88: 70 Page 11 of 15 70

√ Figure 1. Graphical presentation of one-dimensional Schrödinger potential box problem. (xα) = 2(1 + α)/aα α sinα(kαx ) for α = 0.736, 0.75, 0.8, 0.85, 0.9, 0.95 and 1.0. 70 Page 12 of 15 Pramana – J. Phys. (2017) 88: 70

α Table 1. First zeros of function sinα(x ) after x = 0 for different α.

α (Mα)

0.736 3.19590 0.75 2.96354 0.80 2.80104 0.85 2.80556 0.90 2.87596 0.95 2.99051 1.0 3.14159

Figure 4. Graphical presentation of (xα) for different values of α for a = 5.7.

8.4 Energy calculation

Now we shall calculate the fractional energy εα of the particle. Use of (31a) and (34c) gives α 2 = 2α (2 mαεα)/ α (Mα/a) implying = 2 α 2α εα ( α/2 mα) (Mα/a) . 2 2 For α = 1 the energy is ε1 = ( /2m) (π/a) .This is the energy for the ﬁrst quantum state as described in Figure 2. Graphical presentation of (xα) for different quantum mechanics. values of α for a = 10. If we choose the box length as 5.7 units, the plot is as in ﬁgure 4.

9. Conclusions

Using fractional derivative of Jumarie-type in the fractional Schrödinger equation, we found that in the range 0.736 <α≤ 1 of order of fractional derivative, the quantum behaviour of the particle in a one-dimensional box changes dramatically. In this range of fractional derivative, the equation of continuity is successfully maintained and the stationary condition also holds. For α = 1 all the equations are reduced to the classical Schrödinger equation. Here, we studied particle ∗ in a box problem and found that the wave equation Figure 3. Graphical presentation of ψα ψα for different = in fractional sense also meets the condition described values of α for a 10. in the classical sense. Further, the wave function is not symmetric till α = 1. When 0.736 <α<1 the peak of the wave function is left sided, i.e. the 8.3 Probability density peak is on the left side of the middle point of the box. The existence amplitude, i.e. wave function has As we got wave functions for various values of α,we higher amplitude for higher α and it is maximum when = ∗ can also get probability density, i.e. ρα ψαψα. Prob- α = 1. Thus, the wave function or existence amplitude ability density plot for various α is given in figure 3. is α-dependent. We need to further study the fractional Pramana – J. Phys. (2017) 88: 70 Page 13 of 15 70 quantum mechanics of α<0.736 to understand the A.3 Fractional wavelength internal behaviour of the quantum states. Fractional wavelength is demonstrated in figure A1 of a fractional wave of the order α = 0.8. Acknowledgement The wavelength is the distance AB. That is the dis- The authors are grateful to the anonymous referee tance covered by a fractional wave in a full fractional for a careful checking of the details and for helpful cycle. Fractional wavelength is not a fixed quantity. comments that improved this paper. It changes with the evolution of fractional time like a damped oscillating wave. Appendix A.4 Fractional time period A.1 Fractional mass The time taken N for a wave to cover the distance = α α Fractional mass mα is defined as mα ρdx ,where AB (figure A1) is the fractional time period. We should ρ is the fractional linear mass density in one dimen- note that this is the first-order time period. As wave- = α sion; when ρ is constant, mα ρ dx .Wehave length changes, the time period also changes with the considered that the density is the same as the case wave propagation. But we assume that λα = vαNα. α = 1.

A.5 Fractional angular frequency A.2 Fractional velocity Figure A2 is the polar plot of the fractional wave of the The fractional change of displacement, i.e. dαx = order α = 0.8. In this polar plot we can easily see that (1 + α)dx per unit change in fractional time differ- the wave is returned to the same point after completing ential (dt)α is the fractional velocity, i.e. with 0 <α<1 a fractional cycle, i.e. to its origin. From the polar plot dαx dx we can say that the angle traversed in a full fractional vα = = α! . dtα (dt)α cycle is 2π. Thus, the fractional angular frequency can be assigned as Then we can write the following equation: 1−α 2π d = dx = = − (vα) v. ωα . dt1 α dt Nα

Figure A1. Fractional wavelength of a fractional wave of the order α = 0.8. 70 Page 14 of 15 Pramana – J. Phys. (2017) 88: 70

Here, υ is the integer-order frequency and υα is the fractional-order frequency. We write the following expression with the above description: υα ωα h = lim hα, = lim α. α→1 υ α→1 ω

Using Appendix A.5, we have, = limα→1 α.We assume that the energy is proportional to angular frequency and this assumption does not depend upon the value of α. Therefore, we have, E ∝ ω and εα ∝ ωα.If we remove the proportionality, we can have a constant, such that the following condition is satisﬁed:

ε0 73 ε0 79 ε0 8 E . = . = . =···= = . ω0.73 ω0.79 ω0.8 ω

Figure A2. The plot showing the concept of fractional Therefore, we have α = and hence hα = h.We angular frequency. can conclude that the fractional Planck’s constant is nothing but Planck’s constant. The reduced fractional Planck’s constant is also the reduced Planck’s constant. In ﬁgure A2 the initial line is along the horizontal line. From this, we can see that the product of frac- A.8 Theorem tional angular momentum and fractional time period If a function, i.e. f(x,y) is fractionally differentiable, Nα is always 2π though both are varying. In limiting condition we have with order α with respect to both the variables x and y, α α = α α 2α = then Dy Dx f(x,y) Dx Dy f(x,y) or fxy (x, y) ωα f 2α(x, y) are equivalent. where 0 ≤ α ≤ 1 and Dα is lim ωα = ω or lim = 1. yx α→1 α→1 ω the Jumarie derivative operator. Proof. Consider a function φ(x) = f(x,y + k) − A.6 Fractional wave constant (or vector in three f(x,y),k > 0. The fractional mean value theorem dimensions) states, for 0 <θ<1and0≤ α ≤ 1, the following From the analysis of fractional wave of eq. (8) if [14] equation: α − α = kαx ωαt 0, that is the phase part of the wave is hα zero, we get k = ω (tα/xα) = ω /v as xα/tα = v φ(x + h) − φ(x) = φα(x + θh), α α α α α (1 + α) x is the fractional velocity. We have kα = (ωα/vα) = hα 2π/(vαNα), using Appendix (A.5). Now by using + − = [ α + + φ(x h) φ(x) + fx (x θh,y k) Appendix (A.4) we have kα = 2π/λα. (1 α) − α + ] fx (x θh,y) . A.7 Fractional reduced Planck’s constant = α + Let F(y) fx (x θh,y). Then using fractional mean In this paper we have introduced fractional Planck’s value theorem, we have the following expression: constant α as a basic constant. For the limiting condition of α this constant is of the form of reduced hα φ(x + h) − φ(x) = [F(y + k) − F(y)] Planck’s constant . Consider integer (i.e. classical) (1 + α) and fractional energies, which are E = ω = hυ and hα kα ε = ω = h υ respectively. Now we can write = [ α + − α ] α α α α α + + Fy (y θ1k) Fy (y) for the limiting condition, i.e. with α → 1, (1 α) (1 α) α α h k α = [f (x + θh,y + θ1k)] (1 + α) (1 + α) yx E = hυ = lim (hαυα). α→1 where 0 <θ,θ1 < 1, 0 ≤ α ≤ 1. Pramana – J. Phys. (2017) 88: 70 Page 15 of 15 70

On the other hand, we have

φ(x + h) = f(x+ h, y + k) − f(x+ h, y).

Therefore,

φ(x + h) − φ(x) = f(x+ h, y + k) − f(x+ h, y) − f(x,y+ k) + f(x,y) + − α = + f(x,y k) f(x,y) fy (x, y) (1 α) lim k→0 kα or f α(x, y + h) − f α(x, y) 2α = + y y fxy (x, y) (1 α) lim h→0 hα f(x+ h, y + k) − f(x+ h, y) − f(x,y+ k) + f(x,y) = ((1 + α))2 lim lim h→0 k→0 kαhα φ(x + h) − φ(x) = ((1 + α))2 lim lim h→0 k→0 kαhα = + 2 2α + + ((1 α)) lim lim fyx (x θh,y θ1k) h→0 k→0 or 2α = 2α fxy (x, y) fyx (x, y).

Hence the theorem is proved. [12] S Das, Int. J. Math. Comput. 19(2), 732 (2013) [13] G Jumarie, Cent. Eur. J. Phys. 11(6), 617 (2013) We have used this theorem in our derivation. [14] G Jumarie, Comput. Math. Appl. 51, 1367 (2006) [15] U Ghosh, S Sarkar and S Das, Adv. Pure Math. 5, 717 References (2015) [16] U Ghosh, S Sarkar and S Das, Am. J. Math. Anal. 3(3), 54 [1] S Das, Functional fractional calculus, 2nd edn (Springer- (2015) Verlag, 2011) [17] U Ghosh, S Sarkar and S Das, Am. J. Math. Anal. 3(3), 72 [2] S Zhang and H Q Zhang, Phys. Lett. A 375, 1069 (2011) (2015) [3] J F Alzaidy, Am. J. Math. Anal. 1(1), 14 (2013) [18] Abhay Parvate and A D Gangal, Calculus on fractal subset [4] H Jafari and S Momani, Phys. Lett. A 370, 388 (2007) of real-line-I: Formulation, fractals, Vol 17, No. 1 (2009), [5] K S Miller and B Ross, An introduction to the fractional 53–81 calculus and fractional differential equations (John Wiley & [19] Abhay Parvate and A D Gangal, Pramana – J. Phys. 64(3), Sons, New York, USA, 1993) 389 (2005) [6] I Podlubny, Fractional differential equations, mathematics [20] L Nottale, Fractal space time in microphysics (World Scien- in science and engineering (Academic Press, San Diego, tific, Singapore, 1993) California, USA, 1999) p. 198 [21] D P Ray-Chaudhuri, Adv. Acoustics (The New Book Stall, [7] K Diethelm, The analysis of fractional differential equations 2001) (Springer-Verlag, 2010) [22] G Jumarie, Acta Math. Sinica 28(9), 1741 (2012) [8] A Kilbas, H M Srivastava and J J Trujillo, Theory and appli- [23] S Das, Kindergarten of fractional calculus, in: A book of cations of fractional differential equations (North-Holland lecture notes in limited prints (Dept. of Physics, Jadavpur Mathematics Studies, Elsevier Science, Amsterdam, The University, Kolkata) Netherlands, 2006) p. 2014 [24] J L Powell and B Crasemann, Quantum mechanics (Addison- [9] G Jumarie, Comput. Math. Appl. 51(9–10), 1367 (2006) Wesley, 1965) [10] G M Mittag-Leffler, C. R. Acad. Sci. Paris (Ser. II) 137, 554 [25] G B Arfken, H J Weber and F E Harris, Mathematical (1903) methods for physicist, 7th edn (Academic Press, 2012) [11] U Ghosh, S Sengupta, S Sarkar and S Das, Am. J. Math. Anal. [26] D J Griffiths, Introduction to quantum mechanics, 2nd edn, 3(2), 32 (2015) 9th impression (Pearson Education, Inc, 2011)