- Email: [email protected]
Time dependent solutions for fractional coupled Schrödinger equations
Applied Mathematics and Computation 346 (2019) 622–632
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Time dependent solutions for fractional coupled Schrödinger equations E.K. Lenzi a,b,∗, A.S.M. de Castro a, R.S. Mendes b,c a
Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, Ponta Grossa 84030-900 PR, Brazil National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro 22290-180 RJ, Brazil c Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, Maringá 87020-900 PR, Brazil b
a r t i c l e
i n f o
Keywords: Fractional Schrödinger equation Lévy distribution Green function
a b s t r a c t We analyze dynamical properties of two fractional Schrödinger equations coupled by some classes of real time independent potentials. For this set of equations, we investigate the required conditions on the equations making it possible to retain the probabilistic interpretation of their correspondent solutions when two component wave functions are considered. We observe the presence of interference between the components during the transition processes which can be either reversible or irreversible depending on the condition imposed on the potentials. The solutions for these equations are obtained in both cases of localized and non-localized coupling potentials. © 2018 Elsevier Inc. All rights reserved.
1. Introduction The Schrödinger equation (SE) is essential for many non-relativistic problems and gives an accurate description of the true nature of the microscopic world in a probabilistic sense. It can be applied to describe the dynamical properties of the quantum system through a well defined valued-complex function which can be determined by applying suitable analytical or numerical methods [1,2]. Quantum mechanics can also be formulated as a path integral over the Brownian paths according to Feynman and Hibbs [3]. This approach has motivated Laskin to develop the so called Fractional Quantum Mechanics formulated as a path integral over the Levy flights paths [4]. As a consequence of the Laskin approach, the Fractional Schrödinger Equation (FSE) arose with fractional spatial derivatives instead of the usual ones [5,6] and, consequently, we have a path integral over non-Brownian paths. The FSE brings a wide range of problems which has been investigated from the mathematical and physical point of view. It is also considered in the context of fractional dynamics which has opened new perspectives in scientific and technological developments [7]. The problem concerning the FSE as well as approaches to the construction and analyses of its solutions have been considered in many works in different contexts [8–25]. For example, in Ref. [9], the authors use a Dirac delta function to construct a fractional operator called fractional corresponding operator, which is the general form of the momentum corresponding operator. An optical realization of the fractional Schrödinger equation, based on transverse light dynamics in a spherical optical cavities, is proposed by Longhi in Ref. [10]. In addition, the FSE seems connected to the non-Hermitian
∗ Corresponding author at: Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, 84030-900 Ponta Grossa PR, Brazil E-mail address: [email protected] (E.K. Lenzi).
https://doi.org/10.1016/j.amc.2018.10.074 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
623
Hamiltonian in Ref. [11], where solutions for FSE are considered for a periodic PT-symmetric potential in the domain of non-Hermitian operators. In Ref. [13], the authors demonstrate optical Bloch oscillation and optical Zener tunneling in the fractional Schrödinger equation with periodic and linear potentials, numerically and theoretically. There are numerous works reporting that the whole understanding of the basic approaches behind the FSE allow to go toward advanced topics on quantum physics of high level technical progress. Recently, an analytical solution for the generalized time-dependent Schrödinger equation (SE) in two dimensions under constraints was obtained on a comb with a memory kernel. The wave function was constructed by applying the Green’s function approach for several forms of the memory kernels. An exact solution of SE for the wave function and time dependent coupling of two components was considered in Ref. [12], involving two potentials coupled by a time dependent Dirac delta function potential. The study of a two-component wave function problem implies in the analysis of the system of coupled Schrödinger equations [14]. Motivated by these developments, we first present a Lagrangian density from which a fractional Schrödinger equation with multi-component wave function can be obtained. After this, we focus our attention on the problem of the fractional coupled Schrödinger equation (FCSE) with a two-component wave function and different time independent terms of coupling. The paper is organized as follows. In Section 2, we introduce the set of spatial fractional Schrödinger equations coupled by means of time independent potentials and discuss the condition to conserve the probability in time. Different potential conformations are discussed and formal solutions are formally presented in a consistent way, addressing some numerical analysis in order to illustrate the obtained results. In Section 3, we discuss the results obtained here and present our conclusions. 2. The coupled Schrödinger equations In many problems in quantum mechanics, to consider discrete degrees of freedom in addition to the continuous ones it is necessary to achieve accurate descriptions. This is the case, for instance, of the use of two-component wave functions in the quantum dynamics of an electron when its spin interacts with a magnetic field. In order to encompass the possibility of N discrete degrees of freedom and fractional quantum dynamics, we consider multi-component wave function whose dynamics is obtained from the following Lagrangian density
L(ψ1 , . . . , ∂t ψn ) = i h ¯
N
ψn∗ (x, t )
n=1
−
N n=1
∞ −∞
N,N 1 ∂ ψn (x, t ) − Vnm (x )ψn∗ (x, t )ψm (x, t ) ∂t 2 n,m=1
dx ψn∗ (x , t )Kn (x − x )ψn (x, t ) ,
(1)
with N = 2. The kernel Kn (x ) is related to the kinetic term and, in particular, we consider that F {Kn (x )} ∝ −| p|μn ∞ ∞ −i p x i px (1 < μn ≤ 2) in the Fourier space [7] (F {· · · } = −∞ dx e h¯ · · · and F −1 {· · · } = 2π1 h¯ −∞ dp e h¯ · · · ), which implies that the kinetic term is extended in order to incorporate spatial fractional derivatives [4]. The external and coupling potentials are represented by Vnn (x) and Vnm (x), n = m. It is worth mentioning that the above expressions, which are in one dimension, can directly be extended to the d dimensional case. By considering, without loss of generality, the case of two-component wave functions, i.e., N = 2, and applying the Euler– Lagrange equations, we can obtain from Eq. (1) two coupled spatial fractional Schrödinger equations (FCSE):
ih ¯
∂ ψ (x, t ) = −Dμ1 ( h¯ ∇ )μ1 ψ1 (x, t ) + V11 (x )ψ1 (x, t ) + V12 (x )ψ2 (x, t ) , ∂t 1
(2)
ih ¯
∂ ψ (x, t ) = −Dμ2 ( h¯ ∇ )μ2 ψ2 (x, t ) + V22 (x )ψ2 (x, t ) + V21 (x )ψ1 (x, t ) , ∂t 2
(3)
which are each coupled by the time independent potentials with V12 (x) and V21 (x) being the coupling potentials, as mentioned before. Note that the choices of kernels was performed in order to obtain the spatial fractional derivatives in the same way as the Riesz–Weyl ones [26], i.e., FCSE obtained from Eq. (1) could be written in terms of fractional differential operators, which for, μ1 = μ2 = 2, recover the usual ones. It is worth mentioning that the coupling terms promote a mixing between ψ 1 (x, t) and ψ 2 (x, t) with a direct influence on the probability densities (|ψ 1 (x, t)|2 and |ψ 2 (x, t)|2 ) related to these wave functions. In this regard, performing some calculations by using the Fourier transform F {ψ1(2 ) (x, t )} = ψ¯ 1(2 ) ( p, t ) and F −1 {ψ¯ ( p, t )} = ψ (x, t ), where the Fourier transform of the spatial fractional differential operator yields [7] 1 (2 )
1 (2 )
( h¯ ∇ )μ1(2) ψ1(2) (x, t ) = −
1 2π h ¯
∞ −∞
dpe−i h¯ x | p|μ1(2) ψ¯ 1(2) ( p, t ) , p
(4)
it is possible to show
ih ¯
∂ P (t ) = ∂ t 11
∞
−∞
dxV12 (x )ψ1∗ (x, t )ψ2 (x, t ) −
∞ −∞
dxV12 (x )ψ2∗ (x, t )ψ1 (x, t ) ,
(5)
624
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
ih ¯
∂ P (t ) = ∂ t 22
∞
−∞
dxV21 (x )ψ2∗ (x, t )ψ1 (x, t ) −
∞
with P11 (t) and P22 (t) defined as
P11 (t ) =
∞
−∞
dxψ1∗ (x, t )ψ1 (x, t ) ,
dxV21 (x )ψ1∗ (x, t )ψ2 (x, t ) ,
−∞
P22 (t ) =
∞
−∞
(6)
dxψ2∗ (x, t )ψ2 (x, t ) .
Note that Eqs. (5) and (6) have additional terms which can be related to processes of interference between the two components due to the coupling potential. From the previous equations, it is possible to verify that
ih ¯
∂ (P (t ) + P22 (t ) ) = 0 , ∂ t 11 P11 (t ) + P22 (t ) = constant.
(7)
when the conditions Im[V11 (x )] = Im[V22 (x )] = 0 and V12 (x ) = V21 (x ) are satisfied. This point is important and it is directly related with the feature that the Lagrangian should be real, i.e., L = L∗ . Another interesting point about this coupled set of Schrödinger equations is the possibility of obtaining situations characterized by nonlocal terms. This feature can be verified by performing some calculation, for instance, by considering that V11 (x ) = V22 (x ) = 0 with the potentials V12 (x ) = const and V21 (x ) = const. In this case, it is possible to obtain the following Schrödinger equation with a nonlocal term, i.e.,
ih ¯
∂ ψ (x, t ) = −Dμ1 ( h¯ ∇ )μ1 ψ1 (x, t ) + ∂t 1
t
dt
∞
−∞
0
dx (x − x , t − t )ψ1 (x , t ),
(8)
which is similar to the nonlocal Schrödinger equations worked out in Refs. [15,26–28]. Thus, the nonlocal term present in these equations could be considered as a consequence of the coupling between these FSE. Different scenarios may also be obtained by considering the presence of an external potential yielding, by suitable considerations, the following time independent FSE μ1
−Dμ1 ( h ¯ ∇)
ψ1 (x ) + V11 (x )ψ1 (x ) +
∞ −∞
dx ζ (x, x )ψ1 (x ) = E ψ1 (x ) ,
(9)
which is formally similar to the equation investigated in Ref. [29], assuming μ1 = 2. Below, we investigate some solutions for the previous FCSE. 2.1. Localized coupling: V12 (x ) = V21 (x ) = vδ (x − l ) with V11 = V22 = 0 Let us first investigate the solutions for the case V12 (x ) = V21 (x ) = vδ (x − l ), with v as a real parameter, and which represents a localized interaction between these two components, ψ 1 (x, t) and ψ 2 (x, t). We also consider the absence of external potential, i.e., for simplicity, V11 (x, t ) = V22 (x, t ) = 0. In order to solve Eqs. (2) and (3) under these conditions, we may use the Fourier transform and consider h ¯ = 1, without loss of generality. In this case, after applying the integral transforms, we may obtain in the Fourier space (domain) the correspondent set of coupled equations
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) + vψ2 (l, t ) , ∂t 1
(10)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) + vψ1 (l, t ) . ∂t 2
(11)
Their solutions read as
ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)e−iDμ1 | p|
μ1 t
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−iDμ1 | p|
μ1 t
− iv
t 0
− iv
t 0
μ dt ψ2 (l, t )e−iDμ1 | p| 1 (t −t ) ,
(12)
μ dt ψ1 (l, t )e−iDμ2 | p| 2 (t −t ) ,
(13)
where ψ¯ 1 ( p, 0 ) = ϕ¯ 1 ( p) and ψ¯ 2 ( p, 0 ) = ϕ¯ 2 ( p) are the Fourier transform of the initial conditions ψ1 (x, 0 ) = ϕ1 (x ) ∞ ∞ ∞ and ψ2 (x, 0 ) = ϕ2 (x ) taking into account that −∞ dx|ψ1 (x, 0 )|2 = −∞ dx|ϕ1 (x )|2 = const and −∞ dx|ψ2 (x, 0 )|2 = ∞ 2 −∞ dx|ϕ2 (x )| = const. By performing the inverse of Fourier transform on the previous equation, it is possible to obtain that
ψ1 (x, t ) = ψ2 (x, t ) =
∞
−∞
∞
−∞
dx ϕ1 (x )Gμ1 (x − x , t ) − iv dx ϕ2 (x )Gμ2 (x − x , t ) − iv
t 0
t 0
dt ψ2 (l, t )Gμ1 (x, t − t ) ,
(14)
dt ψ1 (l, t )Gμ2 (x, t − t ) ,
(15)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
625
where Gμ1(2) (x, t ) is the Green function for the free particle case, i.e., in the absence of a coupling term. In terms of the H Fox function [30], it can be written as
Gμ1(2) (x, t ) =
1
μ1 ( 2 ) | x |
H12,,12
|x| iDμ1(2) t
μ 1
1 (2 )
1, 1 ,(1, 1 ) 2 μ1(2) . (1,1),(1, 1 ) 2
(16)
For the particular case of μ1(2 ) = 2 and by taking into account Dμ1(2) = h ¯ 2 /(2m ), Eq. (16) recovers the usual form [2] of the quantum free particle propagator which is given by
G2 (x, t ) =
m 2 m e− 2i h¯ t (x−x ) . 2π i h ¯t
(17)
It is worth mentioning that the asymptotic behavior of Eq. (16) is characterized by a power law behavior, i.e., Gμ1(2) (x, t ) ∼ μ −1 1+μ1(2 ) , and consequently leads to a long tailed behavior for the wave functions. Dμ1(2) it h ¯ 1 ( 2 ) /|x| In addition, from Eqs. (14) and (15), it is possible to show that the solutions ψ k (l, t) can be written as
ψ1 (l, t ) = ψ2 (l, t ) =
∞ −∞
∞ −∞
(1 ) dx Gμ (l, x , t )ϕ1 (x ) + iv 1 ,μ2
(1 ) dx Gμ (l, x , t )ϕ2 (x ) + iv 2 ,μ1
with the functions Gα(k,)β (l, x , t ) given by
∞ −∞
∞ −∞
(2 ) dx Gμ (l, x , t )ϕ2 (x ) , 1 ,μ2
(18)
(2 ) dx Gμ (l, x , t )ϕ1 (x ) , 2 ,μ1
(19)
Gα ( l − x , s ) =L 1 + v2 Gα (l, s )Gβ (l, s ) t ∞ = Gα ( l − x , t ) + (−v2 )n dtn α,β (l , l , t − tn ) · · ·
Gα(1,β) (l, x , t )
−1
×
t2 0
Gα(2,β) (l, x , t ) = L−1
dt1 α ,β (l , l , t2 − t1 )Gα (l − x , t1 ) , Gα (l, s )Gβ (l − x , s ) ∞
(−v2 )n
n=1
t2 0
(20)
1 + v2 Gα (l, s )Gβ (l, s )
= α ,β (l, l − x , t ) + ×
0
n=1
0
t
dtn α ,β (l , l , t − tn ) · · ·
dt1 α ,β (l , l , t2 − t1 ) α ,β (l , l − x , t1 ) ,
(21)
and α , β (x, x , t) determined as
α,β (x, x , t ) =
t
0
dt Gα (x, t − t )Gβ (x , t ) ,
(22)
with Gα (x, t ) defined by Eq. (16). Direct substitution of the results (18)–(22) into Eqs. (14) and (15) implies that
ψ1 (x, t ) =
∞
−∞
dx ϕ1 (x )Gμ1 (x − x , t )
−iv
−∞ ∞
−v2 and
ψ2 (x, t ) =
−∞
∞
−∞
−v2
dx ϕ2 (x )
t 0
dx ϕ1 (x )
t
0
(1 ) dt Gμ (l, x , t )Gμ1 (x, t − t ) 2 ,μ1 (2 ) dt Gμ (l, x , t )Gμ1 (x, t − t ), 1 ,μ2
(23)
dx ϕ2 (x )Gμ2 (x − x , t )
−iv
∞
∞
−∞ ∞ −∞
dx ϕ1 (x )
dx ϕ2 (x )
t 0
0
t
(1 ) dt Gμ (l, x , t )Gμ2 (x, t − t ) 1 ,μ2 (2 ) dt Gμ (l, x , t )Gμ2 (x, t − t ) . 2 ,μ1
(24)
626
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
From the previous equations, we may obtain the particular case l = 0. This case evidences the non-exponential character of the relaxation processes present in previous expressions. In fact, for this case, we may simplify the previous equations and, in particular, show that
Gα(1,β) (0, x , t ) = Gα (x , t ) t v2 dt v2 )α¯ G (x , t ) , − E − ( t − t α α¯ ,α¯ D¯ α D¯ β 0 (t − t )1−α¯ D¯ α D¯ β 1 D¯ α
Gα(2,β) (0, x , t ) =
dt
t
Eα¯ ,1− 1
(t − t ) α 1
0
β
−
v2 D¯ α D¯ β
(t − t )α¯ Gβ (x , t ) ,
(25)
(26)
with α¯ = 2 − 1/α − 1/β ,
1 1 π iDβ β π iDα ) α ( , and D¯ β = . D¯ α = (1 + α ) (1 − α ) (1 + β ) (1 − β )
(27)
Note, in previous equations, the presence of the generalized Mittag–Leffler function [30,31] which is related to nonexponential relaxation. This feature can be verified by analyzing the asymptotic behavior of this function which, for long times, i.e., t → ∞, leads to Eα ,β (−t γ ) ∼ 1/t γ for α = β and Eα ,β (−t γ ) ∼ 1/t 2γ for α = β = γ , different from the exponential behavior. The first term of Eqs. (23) and (24) corresponds to the evolution of the initial condition and the other terms represent the effect of the coupling potential on the time evolution of the initial condition. Thus, these terms influence the evolution on account of the coupling promoted by the potential which results in terms with power - law relaxations. 2.2. Coupling potential: V12 (x, t ) = V21 (x, t ) = v with V11 = V22 = 0 Now, we analyze a different situation from the previous case. We consider that the coupling potentials are time independent function, i.e., V12 (x, t ) = V21 (x, t ) = v with V11 = V22 = 0, where v is a real constant. This choice implies that the coupling potential is not localized as the one worked out in the previous case, which was defined in terms of a Dirac delta. In this case, the coupled equations read as
i
∂ ψ (x, t ) = −Dμ1 ∇ μ1 ψ1 (x, t ) + vψ2 (x, t ) , ∂t 1
(28)
i
∂ ψ (x, t ) = −Dμ2 ∇ μ2 ψ2 (x, t ) + vψ1 (x, t ) . ∂t 2
(29)
Following the procedure applied above, after applying the Fourier transform, we obtain in the Fourier space (domain) the equations
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) + vψ¯ 2 ( p, t ) , ∂t 1
(30)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) + vψ¯ 1 ( p, t ) , ∂t 2
(31)
where arbitrary initial conditions are considered as in the previous case. By performing some calculations, from the previous set of equations, it is possible to obtain the following solutions
¯ p, t ) − iv ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)(
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−iDμ2 | p|
μ2 t
t
dt e−iDμ2 | p|
0
− iv
with
¯ p, t ) = e−iDμ1 | p| (
μ1 t
+
∞
(−v2 )n
n=1
t
¯ p, t − t ) , ϕ¯ 2 ( p)(
dt e−iDμ2 | p|
0
μ2 t
t
0
μ2 t
(32)
ψ¯ 1 ( p, t − t ) ,
dt t n (t − t )n−1 e−iDμ1 | p|
(33)
μ1 t −iD μ
e
2
| p|μ2 (t −t ) .
(34)
Performing the inverse of Fourier transform on Eqs. (32) and (33), we obtain that
ψ1 (x, t ) =
∞ −∞
dx ϕ1 (x )(x − x , t )
−iv
t 0
dt
∞
−∞
d x
∞ −∞
dx ϕ2 (x )Gμ2 (x − x , t )(x − x , t − t ) ,
(35)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
627
1.2
0.9
0.6
0.3
0.0 -3
0
x
3
6
Fig. 1. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ1 = 6/5 and μ2 = 5/3. We consider, for simplicity, the initial condition ϕ1(2) (x ) =
2/ πσ12(2)
1/4
2 2 e−(x−x1(2) ) /σ1(2) , Dμ1 = 1, Dμ2 = 1, v = 1, x1 = 2, x2 = 1, σ12 = 0.1, σ22 = 2, and t = 0.2.
and
ψ2 (x, t ) =
∞
−∞
dx ϕ2 (x )Gμ2 (x − x , t )
−iv
t
dt
∞ −∞
0
dx Gμ2 (x − x , t − t )ψ¯ 1 (x , t ) ,
(36)
with
(x, t ) = Gμ1 (x, t ) + ×
t 0
dt
∞
(−v2 )n (n ) (1 + n ) n=1
∞ −∞
dx t n (t − t )n−1 Gμ1 (x − x , t )Gμ2 (x , t − t ) .
(37)
Fig. 1 illustrates the behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| on the x coordinate at the instant of time t = 0.2 assuming μ1 = 6/5 and μ2 = 5/3. 2.3. Coupling and external potential Now, we consider the presence of an external potential in each Schrödinger equation, i.e., V11 (x, t ) = −u and V22 (x, t ) = −u with u = const. Then, we analyze, from the mathematical point of view, the consequences which arise for the solutions when the condition for the probability conservation is not verified. For the first case, the set of Schrödinger equations in the Fourier space can be written as
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) − uψ¯ 1 ( p, t ) + vψ¯ 2 ( p, t ) , ∂t 1
(38)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) − uψ¯ 2 ( p, t ) + vψ¯ 1 ( p, t ) . ∂t 2
(39)
628
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
1.2
1.0
0.8
0.6
0.4
0.2
0.0 -2
0
2
4
x Fig. 2. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ1 = 6/5 and μ2 = 5/3. We consider, for simplicity, the initial condition ϕ1(2) (x ) =
2/ πσ12(2)
1/4
2 2 e−(x−x1(2) ) /σ1(2) , Dμ1 = 1, Dμ2 = 1, u = 2, v = 1, x1 = 2, x2 = 1, σ12 = 0.1, σ22 = 2, and t = 0.2.
The solution for this set of equations is given by
¯ v,u ( p, t ) − iv ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−i(Dμ2 | p|
t
dt e−i(Dμ2 | p|
) − iv
μ2 +u t
t
) ¯ v,u ( p, t − t ) ,
μ2 +u t
0
dt e−i(Dμ2 | p|
) ψ¯ 1 ( p, t − t ) ,
(41)
| p|μ2 +u )(t −t )
(42)
μ2 −v t
0
(40)
where
¯ v,u ( p, t ) = e−i(Dμ1 | p|
∞
(−v2 )n
n=1
×
) +
μ1 +u t
t
dt t n (t − t )n−1 e−i(Dμ1 | p|
0
) e (
μ1 +u t −i D μ
2
.
Now, performing the inverse of Fourier transform, we obtain that
ψ1 (x, t ) =
∞ −∞
×
ψ2 (x, t ) =
dx ϕ1 (x )v,u (x − x , t ) − iv ∞
d x
−∞ ∞
−∞
∞
−∞
t
dt eivt
0
dx ϕ˜ 2 (x )Gμ2 (x − x , t )v,u (x − x , t − t ) ,
(43)
dx ϕ2 (x )Gμ2 (x − x , t )eivt
−iv
t 0
dt eiv(t −t )
∞ −∞
d x G μ2 ( x − x , t − t ) ψ 1 ( x , t ) .
(44)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
629
Fig. 3. Behavior of 1/|ψ 1 (0, t)|2 and 1/|ψ 2 (0, t)|2 versus x for μ1 = μ2 = 3/2. We consider, for simplicity, the initial condition ϕ1 (x ) = 2/ πσ12
1/4
2 2 e−x /σ1 ,
ϕ2 (x ) = 0, Dμ = 1, u = v = 1, and σ12 = 0.1.
with
v,u (x, t ) = Gμ1 (x, t )e−iut + ×e−iut
t
dt
0
∞
(−v2 )n (n ) (1 + n ) n=1
∞
−∞
dx t n (t − t )n−1 Gμ1 (x − x , t )Gμ2 (x , t − t ) .
(45)
Fig. 2 shows the behavior of the solutions obtained for this case of a constant external potential. Fig. 3 shows the behavior of 1/|ψ 1 (0, t)|2 and |ψ 2 (0, t)|2 as a measure of the spreading of each distribution. In this figure, it is also interesting to note, after the initial transient, the oscillations between the ψ 1 (x, t) and ψ 2 (x, t) and, in addition, the feature of the spreading of these distributions growing with tμ/2 , which reveal a non-usual spreading. Let us discuss some mathematical results concerning the cases in which the conserving probability is not a necessary condition. This means a relaxation on the conditions of the hermiticity of the Lagrangean defined by Eq. (1). In this sense, we consider the situation characterized by V11 (x, t ) = −u, V22 (x, t ) = −v, V21 (x, t ) = u, and V12 = v, where u and v are arbitrary constants. This scenario in a diffusive context may characterize a reaction diffusion process, where 1 2, i.e., a reversible reaction. The set of Schrödringer equations, for this case, in the Fourier space are
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) − uψ¯ 1 ( p, t ) + vψ¯ 2 ( p, t ) , ∂t 1
(46)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) − vψ¯ 2 ( p, t ) + uψ¯ 1 ( p, t ) . ∂t 2
(47)
Their correspondent solutions read as
¯ v,u ( p, t ) − iv ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−i(Dμ2 | p|
t
dt e−i(Dμ2 | p|
0
) − iv
μ2 −v t
t 0
) ¯ v,u ( p, t − t ) ,
μ2 −v t
dt e−i(Dμ2 | p|
) ψ¯ 1 ( p, t − t ) ,
μ2 −v t
(48)
(49)
630
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
Fig. 4. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ = 3/2. We consider, for simplicity, the initial condition ϕ1(2) (x ) = 2/ πσ12(2) Dμ = 1, v = 2u = 2, x1 = −1, x2 = 3, σ12 = 0.1, σ22 = 1.5, and t = 0.2.
1/4
2 2 e−(x−x1(2) ) /σ1(2) ,
where
¯ v,u ( p, t ) = e−i(Dμ1 | p|
∞
(−uv )n
n=1
×
) +
μ1 −u t
t 0
dt t n (t − t )n−1 e−i(Dμ1 | p|
) e (
μ1 −u t −i D μ
2
| p|μ2 −v )(t −t )
.
(50)
By performing the inverse of Fourier transform, we obtain that
ψ1 (x, t ) =
t dx ϕ1 (x )v,u (x − x , t ) − iv dt eivt −∞ 0 ∞ ∞ × d x dx ϕ2 (x )Gμ2 (x − x , t )v,u (x − x , t − t ) , ∞
−∞
ψ2 (x, t ) =
∞ −∞
dxϕ2 (x )Gμ2 (x − x , t )eivt − iv
with
v,u (x, t ) = Gμ1 (x, t )eiut + ×
(51)
−∞
∞
t
dt eiv(t −t )
∞ −∞
0
(−uv )n eivt ( n ) (1 + n ) n=1
t
dt ei(u−v )t
d x G μ2 ( x − x , t − t ) ψ 1 ( x , t ) ,
(52)
0
∞
dx t n (t − t )n−1 Gμ1 (x − x , t )Gμ2 (x , t − t ) .
(53)
−∞
For the case μ1 = μ2 = μ the kinetic term of both Schrödinger equations are equal; consequently, the coupling terms have a pronounced effect on the spreading of the system. For this case, after performing some calculations, it is possible to write the previous solutions for ψ 1 (x, t) and ψ 2 (x, t) as follows
ψ1 (x, t ) =
v
∞
dx
ϕ1 ( x ) + ϕ2 ( x ) Gμ ( x − x , t )
u + v −∞ 1 i(u+v )t ∞ + e dx uϕ1 (x ) − vϕ2 (x ) Gμ (x − x , t ) , u+v −∞
(54)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
631
a 1.2 μ1 = 3/2
1.0 0.8 0.6 0.4 0.2 0.0 -4
-2
0
2
4
6
2
4
6
x
b 0.8
0.6
0.4
μ2 = 3/2
0.2
0.0 -4
-2
0
x Fig. 5. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ1 = 3/2 and μ2 = 2. We consider, for simplicity, the initial condition ϕ1(2) (x ) =
2/ πσ12(2)
1/4
2 2 e−(x−x1(2) ) /σ1(2) , Dμ = 1, v = 2u = 2, x1 = −1, x2 = 3, σ12 = 0.1, σ22 = 1.5, and t = 0.2.
ψ2 (x, t ) =
∞ u dx ϕ1 (x ) + ϕ2 (x ) Gμ (x − x , t ) u + v −∞ 1 i(u+v )t ∞ − e dx uϕ1 (x ) − vϕ2 (x ) Gμ (x − x , t ) . u+v −∞
(55)
Fig. 4 illustrates the behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| on the x coordinate at the instant of time t = 0.2 assuming μ = 3/2. In Fig. 5, we perform a comparison between the fractional case (μ1 = μ2 = 3/2 ) and usual (μ1 = μ2 = 2) cases. From this case, it is also possible to consider the situation for which v = 0 with u = 0 which yields
ψ1 (x, t ) = eiut
ψ2 (x, t ) =
∞
−∞
∞ −∞
dx ϕ1 (x )Gμ (x − x , t ) ,
dx ϕ2 (x )G (x − x , t ) + 1 − eiut
(56)
∞ −∞
dx ϕ1 (x )Gμ (x − x , t ) .
(57)
These equations show an oscillation between the components 1 and 2 when Im[u] = 0. It is interesting to note that by assuming, for this case, Im[u] > 0, we may observe an irreversible transition from the component 1 to component 2, i.e., P11 → P22 , with the presence of interference, i.e., P12 and P21 , during the transition process. Thus, this case directly implies that, there is a transfer of the density probability from one component to another, similar to an irreversible reaction process, i.e., 1 → 2.
632
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
3. Discussion and conclusions We analyzed a set of coupled Schrödinger equations in the context of fractional quantum mechanics, i.e., the spatial differential operators were extended to non-integer order by incorporating spatial fractional derivatives in the same way as the Riez–Wely ones [26]. The solutions for these equations were obtained by taking different scenarios into account. The first one considers a localized coupling between the Schrödinger equations. Next, in the second scenario, this coupling was extended by considering the coupling potential between these equations as constant; thus, the interplay between the wave functions is not localized as in the previous case. In the last scenario, the third case, we considered the presence of external potentials in each fractional Schrödinger equation. For these cases, we obtained solutions in terms of the H - Fox functions and generalized Mittag–Leffler functions. They have a long tailed behavior which is asymptotic characterized by power laws, in contrast to the usual case usually characterized by a Gaussian behavior. Furthermore, it was possible to observe non-exponential relaxations connected to the Mittag–Leffler functions present in these solutions. Acknowledgments E.K.L. thanks the CNPq for the partial financial support under the Grant no. 303642/2014-9. A.S.M. de Castro acknowledges the financial support Grant no. 453835/2014-7 from the Brazilian agency CNPq. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, second ed., World Scientific, Sigapore, 2013. J.J. Sakurai, Modern Quantum Mechanic, Addison-Wesley Publishing Company, 1994. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62 (20 0 0) 3135–3145. N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002) 056108. N.N. Achar, B.T. Yale, J.W. Hanneken, Time fractional Schrödinger equation revisited, Adv. Math. Phys. 2013 (2013) 290216. J. Klafter, S.C. Lim, R. Metzer, Fractional Dynamics, Higher Education Press, Beijing, 2010. V.E. Tarasov, Review of some promising fractional physical models, Int. J. Mod. Phys. 27 (2013) 1330 0 05. Z. Xiao, W. Chaozhen, L. Yingming, L. Maokang, Fractional corresponding operator in quantum mechanics and applications: a uniform fractional Schrödinger equation in form and fractional quantization methods, Ann. Phys. 350 (2014) 124–136. S. Longhi, Fractional Schrödinger equation in optics, Opt. Let. 40 (2015) 1117–1120. Y. Zhang, H. Zhong, M.R. Belic´ , Y. Zhu, W. Zhong, Y. Zhang, D.N. Christodoulides, M. Xiao, PT symmetry in a fractional Schrödinger equation, Las. Phot. Rev. 10 (2016) 526–531. Diwaker, B. Panda, A. Chakraborty, Exact solution of Schrödinger equation for two state problem with time dependent coupling, Physica A 442 (2016) 380–387. Y. Zhang, R. Wang, H. Zhong, J. Zhang, Milivoj, R. Belic´ , Y. Zhang, Optical Bloch oscillation and zener tunneling in the fractional Schrödinger equation, Sci. Rep. 7 (2017) 17872. T. Sandev, I. Petreska, E.K. Lenzi, Generalized time-dependent Schrödinger equation in two dimensions under constraints, J. Math. Phys. 59 (2018) 012104. E.K. Lenzi, B.F. de Oliveira, L.R.d. Silva, L.R. Evangelista, Solutions for a Schrödinger equation with a nonlocal term, J. Math. Phys. 49 (2008) 032108. S.I. Muslih, Solutions of a particle with fractional δ -potential in a fractional dimensional space, Int. J. Theor. Phys. 49 (2010) 2095–2104. E.K. Lenzi, H.V. Ribeiro, H. Mukai, R.S. Mendes, Continuous-time random walk as a guide to fractional Schröinger equation, J. Math. Phys. 51 (2010) 092102. M. Jeng, S.L.Y. Xu, E. Hawkins, J.M. Schwarz, On the non-locality of the fractional Schrödinger equation, J. Math. Phys. 51 (2010) 062102. H. Ertik, D. Demirhan, H. Sirin, F. Büyukkilic̎, Time fractional development of quantum systems, J. Math. Phys. 51 (2010) 082102. E.C. de Oliveira, J. Vaz Jr., Tunneling in fractional quantum mechanics, J. Phys. A Math. Theor. 44 (2011) 185303. X. Jiang, H. Qi, M. Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J. Math. Phys. 52 (2011) 042105. J. Dong, Green’s function for the time-dependent scattering problem in the fractional quantum mechanics, J. Math. Phys. 52 (2011) 042103. S.S¸ . Bayn, On the consistency of the solutions of the space fractional Schrödinger equation, J. Math. Phys. 53 (2012) 042105. S.S¸ . Bayn, Time fractional Schrödinger equation: Fox’s h-functions and the effective potential, J. Math. Phys. 54 (2013) 012103. Y. Luchko, Fractional Schrödinger equation for a particle moving in a potential well, J. Math. Phys. 54 (2013) 012111. L.R. Evangelista, E.K. Lenzi, Fractional Diffusion Equations and Anomalous Diffusion, Cambridge University Press, Cambridge, 2018. T. Sandev, I. Petreska, E.K. Lenzi, Time-dependent Schrödinger-like equation with nonlocal term, J. Math. Phys. 55 (2014) 092105. X. Jiang, H. Qi, M. Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J. Math. Phys. 52 (2011) 042105. R.B. Gerber, Semiclassical approximations for scattering by nonlocal potentials. I the effective-mass method, J. Chem. Phys. 58 (1973) 4936–4948. A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2009. F. Mainardi, G. Pagnini, R.K. Saxena, Fox h functions in fractional diffusion, J. Comput. Appl. Math. 178 (2005) 321–331.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Time dependent solutions for fractional coupled Schrödinger equations E.K. Lenzi a,b,∗, A.S.M. de Castro a, R.S. Mendes b,c a
Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, Ponta Grossa 84030-900 PR, Brazil National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro 22290-180 RJ, Brazil c Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, Maringá 87020-900 PR, Brazil b
a r t i c l e
i n f o
Keywords: Fractional Schrödinger equation Lévy distribution Green function
a b s t r a c t We analyze dynamical properties of two fractional Schrödinger equations coupled by some classes of real time independent potentials. For this set of equations, we investigate the required conditions on the equations making it possible to retain the probabilistic interpretation of their correspondent solutions when two component wave functions are considered. We observe the presence of interference between the components during the transition processes which can be either reversible or irreversible depending on the condition imposed on the potentials. The solutions for these equations are obtained in both cases of localized and non-localized coupling potentials. © 2018 Elsevier Inc. All rights reserved.
1. Introduction The Schrödinger equation (SE) is essential for many non-relativistic problems and gives an accurate description of the true nature of the microscopic world in a probabilistic sense. It can be applied to describe the dynamical properties of the quantum system through a well defined valued-complex function which can be determined by applying suitable analytical or numerical methods [1,2]. Quantum mechanics can also be formulated as a path integral over the Brownian paths according to Feynman and Hibbs [3]. This approach has motivated Laskin to develop the so called Fractional Quantum Mechanics formulated as a path integral over the Levy flights paths [4]. As a consequence of the Laskin approach, the Fractional Schrödinger Equation (FSE) arose with fractional spatial derivatives instead of the usual ones [5,6] and, consequently, we have a path integral over non-Brownian paths. The FSE brings a wide range of problems which has been investigated from the mathematical and physical point of view. It is also considered in the context of fractional dynamics which has opened new perspectives in scientific and technological developments [7]. The problem concerning the FSE as well as approaches to the construction and analyses of its solutions have been considered in many works in different contexts [8–25]. For example, in Ref. [9], the authors use a Dirac delta function to construct a fractional operator called fractional corresponding operator, which is the general form of the momentum corresponding operator. An optical realization of the fractional Schrödinger equation, based on transverse light dynamics in a spherical optical cavities, is proposed by Longhi in Ref. [10]. In addition, the FSE seems connected to the non-Hermitian
∗ Corresponding author at: Departamento de Física, Universidade Estadual de Ponta Grossa, Av. Gen. Carlos Cavalcanti 4748, 84030-900 Ponta Grossa PR, Brazil E-mail address: [email protected] (E.K. Lenzi).
https://doi.org/10.1016/j.amc.2018.10.074 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
623
Hamiltonian in Ref. [11], where solutions for FSE are considered for a periodic PT-symmetric potential in the domain of non-Hermitian operators. In Ref. [13], the authors demonstrate optical Bloch oscillation and optical Zener tunneling in the fractional Schrödinger equation with periodic and linear potentials, numerically and theoretically. There are numerous works reporting that the whole understanding of the basic approaches behind the FSE allow to go toward advanced topics on quantum physics of high level technical progress. Recently, an analytical solution for the generalized time-dependent Schrödinger equation (SE) in two dimensions under constraints was obtained on a comb with a memory kernel. The wave function was constructed by applying the Green’s function approach for several forms of the memory kernels. An exact solution of SE for the wave function and time dependent coupling of two components was considered in Ref. [12], involving two potentials coupled by a time dependent Dirac delta function potential. The study of a two-component wave function problem implies in the analysis of the system of coupled Schrödinger equations [14]. Motivated by these developments, we first present a Lagrangian density from which a fractional Schrödinger equation with multi-component wave function can be obtained. After this, we focus our attention on the problem of the fractional coupled Schrödinger equation (FCSE) with a two-component wave function and different time independent terms of coupling. The paper is organized as follows. In Section 2, we introduce the set of spatial fractional Schrödinger equations coupled by means of time independent potentials and discuss the condition to conserve the probability in time. Different potential conformations are discussed and formal solutions are formally presented in a consistent way, addressing some numerical analysis in order to illustrate the obtained results. In Section 3, we discuss the results obtained here and present our conclusions. 2. The coupled Schrödinger equations In many problems in quantum mechanics, to consider discrete degrees of freedom in addition to the continuous ones it is necessary to achieve accurate descriptions. This is the case, for instance, of the use of two-component wave functions in the quantum dynamics of an electron when its spin interacts with a magnetic field. In order to encompass the possibility of N discrete degrees of freedom and fractional quantum dynamics, we consider multi-component wave function whose dynamics is obtained from the following Lagrangian density
L(ψ1 , . . . , ∂t ψn ) = i h ¯
N
ψn∗ (x, t )
n=1
−
N n=1
∞ −∞
N,N 1 ∂ ψn (x, t ) − Vnm (x )ψn∗ (x, t )ψm (x, t ) ∂t 2 n,m=1
dx ψn∗ (x , t )Kn (x − x )ψn (x, t ) ,
(1)
with N = 2. The kernel Kn (x ) is related to the kinetic term and, in particular, we consider that F {Kn (x )} ∝ −| p|μn ∞ ∞ −i p x i px (1 < μn ≤ 2) in the Fourier space [7] (F {· · · } = −∞ dx e h¯ · · · and F −1 {· · · } = 2π1 h¯ −∞ dp e h¯ · · · ), which implies that the kinetic term is extended in order to incorporate spatial fractional derivatives [4]. The external and coupling potentials are represented by Vnn (x) and Vnm (x), n = m. It is worth mentioning that the above expressions, which are in one dimension, can directly be extended to the d dimensional case. By considering, without loss of generality, the case of two-component wave functions, i.e., N = 2, and applying the Euler– Lagrange equations, we can obtain from Eq. (1) two coupled spatial fractional Schrödinger equations (FCSE):
ih ¯
∂ ψ (x, t ) = −Dμ1 ( h¯ ∇ )μ1 ψ1 (x, t ) + V11 (x )ψ1 (x, t ) + V12 (x )ψ2 (x, t ) , ∂t 1
(2)
ih ¯
∂ ψ (x, t ) = −Dμ2 ( h¯ ∇ )μ2 ψ2 (x, t ) + V22 (x )ψ2 (x, t ) + V21 (x )ψ1 (x, t ) , ∂t 2
(3)
which are each coupled by the time independent potentials with V12 (x) and V21 (x) being the coupling potentials, as mentioned before. Note that the choices of kernels was performed in order to obtain the spatial fractional derivatives in the same way as the Riesz–Weyl ones [26], i.e., FCSE obtained from Eq. (1) could be written in terms of fractional differential operators, which for, μ1 = μ2 = 2, recover the usual ones. It is worth mentioning that the coupling terms promote a mixing between ψ 1 (x, t) and ψ 2 (x, t) with a direct influence on the probability densities (|ψ 1 (x, t)|2 and |ψ 2 (x, t)|2 ) related to these wave functions. In this regard, performing some calculations by using the Fourier transform F {ψ1(2 ) (x, t )} = ψ¯ 1(2 ) ( p, t ) and F −1 {ψ¯ ( p, t )} = ψ (x, t ), where the Fourier transform of the spatial fractional differential operator yields [7] 1 (2 )
1 (2 )
( h¯ ∇ )μ1(2) ψ1(2) (x, t ) = −
1 2π h ¯
∞ −∞
dpe−i h¯ x | p|μ1(2) ψ¯ 1(2) ( p, t ) , p
(4)
it is possible to show
ih ¯
∂ P (t ) = ∂ t 11
∞
−∞
dxV12 (x )ψ1∗ (x, t )ψ2 (x, t ) −
∞ −∞
dxV12 (x )ψ2∗ (x, t )ψ1 (x, t ) ,
(5)
624
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
ih ¯
∂ P (t ) = ∂ t 22
∞
−∞
dxV21 (x )ψ2∗ (x, t )ψ1 (x, t ) −
∞
with P11 (t) and P22 (t) defined as
P11 (t ) =
∞
−∞
dxψ1∗ (x, t )ψ1 (x, t ) ,
dxV21 (x )ψ1∗ (x, t )ψ2 (x, t ) ,
−∞
P22 (t ) =
∞
−∞
(6)
dxψ2∗ (x, t )ψ2 (x, t ) .
Note that Eqs. (5) and (6) have additional terms which can be related to processes of interference between the two components due to the coupling potential. From the previous equations, it is possible to verify that
ih ¯
∂ (P (t ) + P22 (t ) ) = 0 , ∂ t 11 P11 (t ) + P22 (t ) = constant.
(7)
when the conditions Im[V11 (x )] = Im[V22 (x )] = 0 and V12 (x ) = V21 (x ) are satisfied. This point is important and it is directly related with the feature that the Lagrangian should be real, i.e., L = L∗ . Another interesting point about this coupled set of Schrödinger equations is the possibility of obtaining situations characterized by nonlocal terms. This feature can be verified by performing some calculation, for instance, by considering that V11 (x ) = V22 (x ) = 0 with the potentials V12 (x ) = const and V21 (x ) = const. In this case, it is possible to obtain the following Schrödinger equation with a nonlocal term, i.e.,
ih ¯
∂ ψ (x, t ) = −Dμ1 ( h¯ ∇ )μ1 ψ1 (x, t ) + ∂t 1
t
dt
∞
−∞
0
dx (x − x , t − t )ψ1 (x , t ),
(8)
which is similar to the nonlocal Schrödinger equations worked out in Refs. [15,26–28]. Thus, the nonlocal term present in these equations could be considered as a consequence of the coupling between these FSE. Different scenarios may also be obtained by considering the presence of an external potential yielding, by suitable considerations, the following time independent FSE μ1
−Dμ1 ( h ¯ ∇)
ψ1 (x ) + V11 (x )ψ1 (x ) +
∞ −∞
dx ζ (x, x )ψ1 (x ) = E ψ1 (x ) ,
(9)
which is formally similar to the equation investigated in Ref. [29], assuming μ1 = 2. Below, we investigate some solutions for the previous FCSE. 2.1. Localized coupling: V12 (x ) = V21 (x ) = vδ (x − l ) with V11 = V22 = 0 Let us first investigate the solutions for the case V12 (x ) = V21 (x ) = vδ (x − l ), with v as a real parameter, and which represents a localized interaction between these two components, ψ 1 (x, t) and ψ 2 (x, t). We also consider the absence of external potential, i.e., for simplicity, V11 (x, t ) = V22 (x, t ) = 0. In order to solve Eqs. (2) and (3) under these conditions, we may use the Fourier transform and consider h ¯ = 1, without loss of generality. In this case, after applying the integral transforms, we may obtain in the Fourier space (domain) the correspondent set of coupled equations
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) + vψ2 (l, t ) , ∂t 1
(10)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) + vψ1 (l, t ) . ∂t 2
(11)
Their solutions read as
ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)e−iDμ1 | p|
μ1 t
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−iDμ1 | p|
μ1 t
− iv
t 0
− iv
t 0
μ dt ψ2 (l, t )e−iDμ1 | p| 1 (t −t ) ,
(12)
μ dt ψ1 (l, t )e−iDμ2 | p| 2 (t −t ) ,
(13)
where ψ¯ 1 ( p, 0 ) = ϕ¯ 1 ( p) and ψ¯ 2 ( p, 0 ) = ϕ¯ 2 ( p) are the Fourier transform of the initial conditions ψ1 (x, 0 ) = ϕ1 (x ) ∞ ∞ ∞ and ψ2 (x, 0 ) = ϕ2 (x ) taking into account that −∞ dx|ψ1 (x, 0 )|2 = −∞ dx|ϕ1 (x )|2 = const and −∞ dx|ψ2 (x, 0 )|2 = ∞ 2 −∞ dx|ϕ2 (x )| = const. By performing the inverse of Fourier transform on the previous equation, it is possible to obtain that
ψ1 (x, t ) = ψ2 (x, t ) =
∞
−∞
∞
−∞
dx ϕ1 (x )Gμ1 (x − x , t ) − iv dx ϕ2 (x )Gμ2 (x − x , t ) − iv
t 0
t 0
dt ψ2 (l, t )Gμ1 (x, t − t ) ,
(14)
dt ψ1 (l, t )Gμ2 (x, t − t ) ,
(15)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
625
where Gμ1(2) (x, t ) is the Green function for the free particle case, i.e., in the absence of a coupling term. In terms of the H Fox function [30], it can be written as
Gμ1(2) (x, t ) =
1
μ1 ( 2 ) | x |
H12,,12
|x| iDμ1(2) t
μ 1
1 (2 )
1, 1 ,(1, 1 ) 2 μ1(2) . (1,1),(1, 1 ) 2
(16)
For the particular case of μ1(2 ) = 2 and by taking into account Dμ1(2) = h ¯ 2 /(2m ), Eq. (16) recovers the usual form [2] of the quantum free particle propagator which is given by
G2 (x, t ) =
m 2 m e− 2i h¯ t (x−x ) . 2π i h ¯t
(17)
It is worth mentioning that the asymptotic behavior of Eq. (16) is characterized by a power law behavior, i.e., Gμ1(2) (x, t ) ∼ μ −1 1+μ1(2 ) , and consequently leads to a long tailed behavior for the wave functions. Dμ1(2) it h ¯ 1 ( 2 ) /|x| In addition, from Eqs. (14) and (15), it is possible to show that the solutions ψ k (l, t) can be written as
ψ1 (l, t ) = ψ2 (l, t ) =
∞ −∞
∞ −∞
(1 ) dx Gμ (l, x , t )ϕ1 (x ) + iv 1 ,μ2
(1 ) dx Gμ (l, x , t )ϕ2 (x ) + iv 2 ,μ1
with the functions Gα(k,)β (l, x , t ) given by
∞ −∞
∞ −∞
(2 ) dx Gμ (l, x , t )ϕ2 (x ) , 1 ,μ2
(18)
(2 ) dx Gμ (l, x , t )ϕ1 (x ) , 2 ,μ1
(19)
Gα ( l − x , s ) =L 1 + v2 Gα (l, s )Gβ (l, s ) t ∞ = Gα ( l − x , t ) + (−v2 )n dtn α,β (l , l , t − tn ) · · ·
Gα(1,β) (l, x , t )
−1
×
t2 0
Gα(2,β) (l, x , t ) = L−1
dt1 α ,β (l , l , t2 − t1 )Gα (l − x , t1 ) , Gα (l, s )Gβ (l − x , s ) ∞
(−v2 )n
n=1
t2 0
(20)
1 + v2 Gα (l, s )Gβ (l, s )
= α ,β (l, l − x , t ) + ×
0
n=1
0
t
dtn α ,β (l , l , t − tn ) · · ·
dt1 α ,β (l , l , t2 − t1 ) α ,β (l , l − x , t1 ) ,
(21)
and α , β (x, x , t) determined as
α,β (x, x , t ) =
t
0
dt Gα (x, t − t )Gβ (x , t ) ,
(22)
with Gα (x, t ) defined by Eq. (16). Direct substitution of the results (18)–(22) into Eqs. (14) and (15) implies that
ψ1 (x, t ) =
∞
−∞
dx ϕ1 (x )Gμ1 (x − x , t )
−iv
−∞ ∞
−v2 and
ψ2 (x, t ) =
−∞
∞
−∞
−v2
dx ϕ2 (x )
t 0
dx ϕ1 (x )
t
0
(1 ) dt Gμ (l, x , t )Gμ1 (x, t − t ) 2 ,μ1 (2 ) dt Gμ (l, x , t )Gμ1 (x, t − t ), 1 ,μ2
(23)
dx ϕ2 (x )Gμ2 (x − x , t )
−iv
∞
∞
−∞ ∞ −∞
dx ϕ1 (x )
dx ϕ2 (x )
t 0
0
t
(1 ) dt Gμ (l, x , t )Gμ2 (x, t − t ) 1 ,μ2 (2 ) dt Gμ (l, x , t )Gμ2 (x, t − t ) . 2 ,μ1
(24)
626
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
From the previous equations, we may obtain the particular case l = 0. This case evidences the non-exponential character of the relaxation processes present in previous expressions. In fact, for this case, we may simplify the previous equations and, in particular, show that
Gα(1,β) (0, x , t ) = Gα (x , t ) t v2 dt v2 )α¯ G (x , t ) , − E − ( t − t α α¯ ,α¯ D¯ α D¯ β 0 (t − t )1−α¯ D¯ α D¯ β 1 D¯ α
Gα(2,β) (0, x , t ) =
dt
t
Eα¯ ,1− 1
(t − t ) α 1
0
β
−
v2 D¯ α D¯ β
(t − t )α¯ Gβ (x , t ) ,
(25)
(26)
with α¯ = 2 − 1/α − 1/β ,
1 1 π iDβ β π iDα ) α ( , and D¯ β = . D¯ α = (1 + α ) (1 − α ) (1 + β ) (1 − β )
(27)
Note, in previous equations, the presence of the generalized Mittag–Leffler function [30,31] which is related to nonexponential relaxation. This feature can be verified by analyzing the asymptotic behavior of this function which, for long times, i.e., t → ∞, leads to Eα ,β (−t γ ) ∼ 1/t γ for α = β and Eα ,β (−t γ ) ∼ 1/t 2γ for α = β = γ , different from the exponential behavior. The first term of Eqs. (23) and (24) corresponds to the evolution of the initial condition and the other terms represent the effect of the coupling potential on the time evolution of the initial condition. Thus, these terms influence the evolution on account of the coupling promoted by the potential which results in terms with power - law relaxations. 2.2. Coupling potential: V12 (x, t ) = V21 (x, t ) = v with V11 = V22 = 0 Now, we analyze a different situation from the previous case. We consider that the coupling potentials are time independent function, i.e., V12 (x, t ) = V21 (x, t ) = v with V11 = V22 = 0, where v is a real constant. This choice implies that the coupling potential is not localized as the one worked out in the previous case, which was defined in terms of a Dirac delta. In this case, the coupled equations read as
i
∂ ψ (x, t ) = −Dμ1 ∇ μ1 ψ1 (x, t ) + vψ2 (x, t ) , ∂t 1
(28)
i
∂ ψ (x, t ) = −Dμ2 ∇ μ2 ψ2 (x, t ) + vψ1 (x, t ) . ∂t 2
(29)
Following the procedure applied above, after applying the Fourier transform, we obtain in the Fourier space (domain) the equations
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) + vψ¯ 2 ( p, t ) , ∂t 1
(30)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) + vψ¯ 1 ( p, t ) , ∂t 2
(31)
where arbitrary initial conditions are considered as in the previous case. By performing some calculations, from the previous set of equations, it is possible to obtain the following solutions
¯ p, t ) − iv ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)(
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−iDμ2 | p|
μ2 t
t
dt e−iDμ2 | p|
0
− iv
with
¯ p, t ) = e−iDμ1 | p| (
μ1 t
+
∞
(−v2 )n
n=1
t
¯ p, t − t ) , ϕ¯ 2 ( p)(
dt e−iDμ2 | p|
0
μ2 t
t
0
μ2 t
(32)
ψ¯ 1 ( p, t − t ) ,
dt t n (t − t )n−1 e−iDμ1 | p|
(33)
μ1 t −iD μ
e
2
| p|μ2 (t −t ) .
(34)
Performing the inverse of Fourier transform on Eqs. (32) and (33), we obtain that
ψ1 (x, t ) =
∞ −∞
dx ϕ1 (x )(x − x , t )
−iv
t 0
dt
∞
−∞
d x
∞ −∞
dx ϕ2 (x )Gμ2 (x − x , t )(x − x , t − t ) ,
(35)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
627
1.2
0.9
0.6
0.3
0.0 -3
0
x
3
6
Fig. 1. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ1 = 6/5 and μ2 = 5/3. We consider, for simplicity, the initial condition ϕ1(2) (x ) =
2/ πσ12(2)
1/4
2 2 e−(x−x1(2) ) /σ1(2) , Dμ1 = 1, Dμ2 = 1, v = 1, x1 = 2, x2 = 1, σ12 = 0.1, σ22 = 2, and t = 0.2.
and
ψ2 (x, t ) =
∞
−∞
dx ϕ2 (x )Gμ2 (x − x , t )
−iv
t
dt
∞ −∞
0
dx Gμ2 (x − x , t − t )ψ¯ 1 (x , t ) ,
(36)
with
(x, t ) = Gμ1 (x, t ) + ×
t 0
dt
∞
(−v2 )n (n ) (1 + n ) n=1
∞ −∞
dx t n (t − t )n−1 Gμ1 (x − x , t )Gμ2 (x , t − t ) .
(37)
Fig. 1 illustrates the behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| on the x coordinate at the instant of time t = 0.2 assuming μ1 = 6/5 and μ2 = 5/3. 2.3. Coupling and external potential Now, we consider the presence of an external potential in each Schrödinger equation, i.e., V11 (x, t ) = −u and V22 (x, t ) = −u with u = const. Then, we analyze, from the mathematical point of view, the consequences which arise for the solutions when the condition for the probability conservation is not verified. For the first case, the set of Schrödinger equations in the Fourier space can be written as
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) − uψ¯ 1 ( p, t ) + vψ¯ 2 ( p, t ) , ∂t 1
(38)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) − uψ¯ 2 ( p, t ) + vψ¯ 1 ( p, t ) . ∂t 2
(39)
628
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
1.2
1.0
0.8
0.6
0.4
0.2
0.0 -2
0
2
4
x Fig. 2. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ1 = 6/5 and μ2 = 5/3. We consider, for simplicity, the initial condition ϕ1(2) (x ) =
2/ πσ12(2)
1/4
2 2 e−(x−x1(2) ) /σ1(2) , Dμ1 = 1, Dμ2 = 1, u = 2, v = 1, x1 = 2, x2 = 1, σ12 = 0.1, σ22 = 2, and t = 0.2.
The solution for this set of equations is given by
¯ v,u ( p, t ) − iv ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−i(Dμ2 | p|
t
dt e−i(Dμ2 | p|
) − iv
μ2 +u t
t
) ¯ v,u ( p, t − t ) ,
μ2 +u t
0
dt e−i(Dμ2 | p|
) ψ¯ 1 ( p, t − t ) ,
(41)
| p|μ2 +u )(t −t )
(42)
μ2 −v t
0
(40)
where
¯ v,u ( p, t ) = e−i(Dμ1 | p|
∞
(−v2 )n
n=1
×
) +
μ1 +u t
t
dt t n (t − t )n−1 e−i(Dμ1 | p|
0
) e (
μ1 +u t −i D μ
2
.
Now, performing the inverse of Fourier transform, we obtain that
ψ1 (x, t ) =
∞ −∞
×
ψ2 (x, t ) =
dx ϕ1 (x )v,u (x − x , t ) − iv ∞
d x
−∞ ∞
−∞
∞
−∞
t
dt eivt
0
dx ϕ˜ 2 (x )Gμ2 (x − x , t )v,u (x − x , t − t ) ,
(43)
dx ϕ2 (x )Gμ2 (x − x , t )eivt
−iv
t 0
dt eiv(t −t )
∞ −∞
d x G μ2 ( x − x , t − t ) ψ 1 ( x , t ) .
(44)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
629
Fig. 3. Behavior of 1/|ψ 1 (0, t)|2 and 1/|ψ 2 (0, t)|2 versus x for μ1 = μ2 = 3/2. We consider, for simplicity, the initial condition ϕ1 (x ) = 2/ πσ12
1/4
2 2 e−x /σ1 ,
ϕ2 (x ) = 0, Dμ = 1, u = v = 1, and σ12 = 0.1.
with
v,u (x, t ) = Gμ1 (x, t )e−iut + ×e−iut
t
dt
0
∞
(−v2 )n (n ) (1 + n ) n=1
∞
−∞
dx t n (t − t )n−1 Gμ1 (x − x , t )Gμ2 (x , t − t ) .
(45)
Fig. 2 shows the behavior of the solutions obtained for this case of a constant external potential. Fig. 3 shows the behavior of 1/|ψ 1 (0, t)|2 and |ψ 2 (0, t)|2 as a measure of the spreading of each distribution. In this figure, it is also interesting to note, after the initial transient, the oscillations between the ψ 1 (x, t) and ψ 2 (x, t) and, in addition, the feature of the spreading of these distributions growing with tμ/2 , which reveal a non-usual spreading. Let us discuss some mathematical results concerning the cases in which the conserving probability is not a necessary condition. This means a relaxation on the conditions of the hermiticity of the Lagrangean defined by Eq. (1). In this sense, we consider the situation characterized by V11 (x, t ) = −u, V22 (x, t ) = −v, V21 (x, t ) = u, and V12 = v, where u and v are arbitrary constants. This scenario in a diffusive context may characterize a reaction diffusion process, where 1 2, i.e., a reversible reaction. The set of Schrödringer equations, for this case, in the Fourier space are
i
∂ ¯ ψ ( p, t ) = Dμ1 | p|μ1 ψ¯ 1 ( p, t ) − uψ¯ 1 ( p, t ) + vψ¯ 2 ( p, t ) , ∂t 1
(46)
i
∂ ¯ ψ ( p, t ) = Dμ2 | p|μ2 ψ¯ 2 ( p, t ) − vψ¯ 2 ( p, t ) + uψ¯ 1 ( p, t ) . ∂t 2
(47)
Their correspondent solutions read as
¯ v,u ( p, t ) − iv ψ¯ 1 ( p, t ) = ϕ¯ 1 ( p)
ψ¯ 2 ( p, t ) = ϕ¯ 2 ( p)e−i(Dμ2 | p|
t
dt e−i(Dμ2 | p|
0
) − iv
μ2 −v t
t 0
) ¯ v,u ( p, t − t ) ,
μ2 −v t
dt e−i(Dμ2 | p|
) ψ¯ 1 ( p, t − t ) ,
μ2 −v t
(48)
(49)
630
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
Fig. 4. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ = 3/2. We consider, for simplicity, the initial condition ϕ1(2) (x ) = 2/ πσ12(2) Dμ = 1, v = 2u = 2, x1 = −1, x2 = 3, σ12 = 0.1, σ22 = 1.5, and t = 0.2.
1/4
2 2 e−(x−x1(2) ) /σ1(2) ,
where
¯ v,u ( p, t ) = e−i(Dμ1 | p|
∞
(−uv )n
n=1
×
) +
μ1 −u t
t 0
dt t n (t − t )n−1 e−i(Dμ1 | p|
) e (
μ1 −u t −i D μ
2
| p|μ2 −v )(t −t )
.
(50)
By performing the inverse of Fourier transform, we obtain that
ψ1 (x, t ) =
t dx ϕ1 (x )v,u (x − x , t ) − iv dt eivt −∞ 0 ∞ ∞ × d x dx ϕ2 (x )Gμ2 (x − x , t )v,u (x − x , t − t ) , ∞
−∞
ψ2 (x, t ) =
∞ −∞
dxϕ2 (x )Gμ2 (x − x , t )eivt − iv
with
v,u (x, t ) = Gμ1 (x, t )eiut + ×
(51)
−∞
∞
t
dt eiv(t −t )
∞ −∞
0
(−uv )n eivt ( n ) (1 + n ) n=1
t
dt ei(u−v )t
d x G μ2 ( x − x , t − t ) ψ 1 ( x , t ) ,
(52)
0
∞
dx t n (t − t )n−1 Gμ1 (x − x , t )Gμ2 (x , t − t ) .
(53)
−∞
For the case μ1 = μ2 = μ the kinetic term of both Schrödinger equations are equal; consequently, the coupling terms have a pronounced effect on the spreading of the system. For this case, after performing some calculations, it is possible to write the previous solutions for ψ 1 (x, t) and ψ 2 (x, t) as follows
ψ1 (x, t ) =
v
∞
dx
ϕ1 ( x ) + ϕ2 ( x ) Gμ ( x − x , t )
u + v −∞ 1 i(u+v )t ∞ + e dx uϕ1 (x ) − vϕ2 (x ) Gμ (x − x , t ) , u+v −∞
(54)
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
631
a 1.2 μ1 = 3/2
1.0 0.8 0.6 0.4 0.2 0.0 -4
-2
0
2
4
6
2
4
6
x
b 0.8
0.6
0.4
μ2 = 3/2
0.2
0.0 -4
-2
0
x Fig. 5. Behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| versus x for μ1 = 3/2 and μ2 = 2. We consider, for simplicity, the initial condition ϕ1(2) (x ) =
2/ πσ12(2)
1/4
2 2 e−(x−x1(2) ) /σ1(2) , Dμ = 1, v = 2u = 2, x1 = −1, x2 = 3, σ12 = 0.1, σ22 = 1.5, and t = 0.2.
ψ2 (x, t ) =
∞ u dx ϕ1 (x ) + ϕ2 (x ) Gμ (x − x , t ) u + v −∞ 1 i(u+v )t ∞ − e dx uϕ1 (x ) − vϕ2 (x ) Gμ (x − x , t ) . u+v −∞
(55)
Fig. 4 illustrates the behavior of |ψ 1 (x, t)| and |ψ 2 (x, t)| on the x coordinate at the instant of time t = 0.2 assuming μ = 3/2. In Fig. 5, we perform a comparison between the fractional case (μ1 = μ2 = 3/2 ) and usual (μ1 = μ2 = 2) cases. From this case, it is also possible to consider the situation for which v = 0 with u = 0 which yields
ψ1 (x, t ) = eiut
ψ2 (x, t ) =
∞
−∞
∞ −∞
dx ϕ1 (x )Gμ (x − x , t ) ,
dx ϕ2 (x )G (x − x , t ) + 1 − eiut
(56)
∞ −∞
dx ϕ1 (x )Gμ (x − x , t ) .
(57)
These equations show an oscillation between the components 1 and 2 when Im[u] = 0. It is interesting to note that by assuming, for this case, Im[u] > 0, we may observe an irreversible transition from the component 1 to component 2, i.e., P11 → P22 , with the presence of interference, i.e., P12 and P21 , during the transition process. Thus, this case directly implies that, there is a transfer of the density probability from one component to another, similar to an irreversible reaction process, i.e., 1 → 2.
632
E.K. Lenzi, A.S.M. de Castro and R.S. Mendes / Applied Mathematics and Computation 346 (2019) 622–632
3. Discussion and conclusions We analyzed a set of coupled Schrödinger equations in the context of fractional quantum mechanics, i.e., the spatial differential operators were extended to non-integer order by incorporating spatial fractional derivatives in the same way as the Riez–Wely ones [26]. The solutions for these equations were obtained by taking different scenarios into account. The first one considers a localized coupling between the Schrödinger equations. Next, in the second scenario, this coupling was extended by considering the coupling potential between these equations as constant; thus, the interplay between the wave functions is not localized as in the previous case. In the last scenario, the third case, we considered the presence of external potentials in each fractional Schrödinger equation. For these cases, we obtained solutions in terms of the H - Fox functions and generalized Mittag–Leffler functions. They have a long tailed behavior which is asymptotic characterized by power laws, in contrast to the usual case usually characterized by a Gaussian behavior. Furthermore, it was possible to observe non-exponential relaxations connected to the Mittag–Leffler functions present in these solutions. Acknowledgments E.K.L. thanks the CNPq for the partial financial support under the Grant no. 303642/2014-9. A.S.M. de Castro acknowledges the financial support Grant no. 453835/2014-7 from the Brazilian agency CNPq. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, second ed., World Scientific, Sigapore, 2013. J.J. Sakurai, Modern Quantum Mechanic, Addison-Wesley Publishing Company, 1994. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62 (20 0 0) 3135–3145. N. Laskin, Fractional Schrödinger equation, Phys. Rev. E 66 (2002) 056108. N.N. Achar, B.T. Yale, J.W. Hanneken, Time fractional Schrödinger equation revisited, Adv. Math. Phys. 2013 (2013) 290216. J. Klafter, S.C. Lim, R. Metzer, Fractional Dynamics, Higher Education Press, Beijing, 2010. V.E. Tarasov, Review of some promising fractional physical models, Int. J. Mod. Phys. 27 (2013) 1330 0 05. Z. Xiao, W. Chaozhen, L. Yingming, L. Maokang, Fractional corresponding operator in quantum mechanics and applications: a uniform fractional Schrödinger equation in form and fractional quantization methods, Ann. Phys. 350 (2014) 124–136. S. Longhi, Fractional Schrödinger equation in optics, Opt. Let. 40 (2015) 1117–1120. Y. Zhang, H. Zhong, M.R. Belic´ , Y. Zhu, W. Zhong, Y. Zhang, D.N. Christodoulides, M. Xiao, PT symmetry in a fractional Schrödinger equation, Las. Phot. Rev. 10 (2016) 526–531. Diwaker, B. Panda, A. Chakraborty, Exact solution of Schrödinger equation for two state problem with time dependent coupling, Physica A 442 (2016) 380–387. Y. Zhang, R. Wang, H. Zhong, J. Zhang, Milivoj, R. Belic´ , Y. Zhang, Optical Bloch oscillation and zener tunneling in the fractional Schrödinger equation, Sci. Rep. 7 (2017) 17872. T. Sandev, I. Petreska, E.K. Lenzi, Generalized time-dependent Schrödinger equation in two dimensions under constraints, J. Math. Phys. 59 (2018) 012104. E.K. Lenzi, B.F. de Oliveira, L.R.d. Silva, L.R. Evangelista, Solutions for a Schrödinger equation with a nonlocal term, J. Math. Phys. 49 (2008) 032108. S.I. Muslih, Solutions of a particle with fractional δ -potential in a fractional dimensional space, Int. J. Theor. Phys. 49 (2010) 2095–2104. E.K. Lenzi, H.V. Ribeiro, H. Mukai, R.S. Mendes, Continuous-time random walk as a guide to fractional Schröinger equation, J. Math. Phys. 51 (2010) 092102. M. Jeng, S.L.Y. Xu, E. Hawkins, J.M. Schwarz, On the non-locality of the fractional Schrödinger equation, J. Math. Phys. 51 (2010) 062102. H. Ertik, D. Demirhan, H. Sirin, F. Büyukkilic̎, Time fractional development of quantum systems, J. Math. Phys. 51 (2010) 082102. E.C. de Oliveira, J. Vaz Jr., Tunneling in fractional quantum mechanics, J. Phys. A Math. Theor. 44 (2011) 185303. X. Jiang, H. Qi, M. Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J. Math. Phys. 52 (2011) 042105. J. Dong, Green’s function for the time-dependent scattering problem in the fractional quantum mechanics, J. Math. Phys. 52 (2011) 042103. S.S¸ . Bayn, On the consistency of the solutions of the space fractional Schrödinger equation, J. Math. Phys. 53 (2012) 042105. S.S¸ . Bayn, Time fractional Schrödinger equation: Fox’s h-functions and the effective potential, J. Math. Phys. 54 (2013) 012103. Y. Luchko, Fractional Schrödinger equation for a particle moving in a potential well, J. Math. Phys. 54 (2013) 012111. L.R. Evangelista, E.K. Lenzi, Fractional Diffusion Equations and Anomalous Diffusion, Cambridge University Press, Cambridge, 2018. T. Sandev, I. Petreska, E.K. Lenzi, Time-dependent Schrödinger-like equation with nonlocal term, J. Math. Phys. 55 (2014) 092105. X. Jiang, H. Qi, M. Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J. Math. Phys. 52 (2011) 042105. R.B. Gerber, Semiclassical approximations for scattering by nonlocal potentials. I the effective-mass method, J. Chem. Phys. 58 (1973) 4936–4948. A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2009. F. Mainardi, G. Pagnini, R.K. Saxena, Fox h functions in fractional diffusion, J. Comput. Appl. Math. 178 (2005) 321–331.