Fractional WKB Approximation
Fractional WKB approximation
FRACTIONAL WKB APPROXIMATION 
Eqab M. Rabei٭ and Ibrahim M.A.Altarazi 
Department of physics, Mu'tah University, Al-karak, Jordan 
Sami I. Muslih 
Department of physics, Al-Azhar University, Gaza, Palestine 
Dumitru Baleanu 
Department of Mathematics and Computer Science, Faculty of Arts and 
Science, Çankaya University 
Institute of Space Sciences, P.O.BOX, MG-23, R 76900, 
Magurele-Bucharest, Romania 
Abstract 
Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems 
is investigated in this paper using the fractional calculus. In the fractional 
case  the wave function is constructed such that the phase factor is the same 
as the Hamilton's principle function "S". To demonstrate our proposed 
approach two examples are investigated in details. 
Keywords: fractional derivative, fractional WKB approximation 
Hamilton's principle function. 
eqabrabei@yahoo.com٭
I. Introduction 
Fractional calculus is a branch of mathematics that deal with a 
generalization of well-known operations of differentiations and 
integrations to arbitrary non-integer order, which can be real non-integer or 
even imaginary number. 
Nowadays physicists have used this powerful tool to deal with some 
problems which were not solvable in the classical sense. Therefore, the  
fractional calculus became one of the most powerful and widely useful 
tools in describing and explaining some physical complex systems. 
Recently, the Euler-Lagrange equations has been presented for 
unconstrained and constrained fractional variational problems  [1 and other 
references]. This technique enable us to solve some  problems including 
describing the behavior of non-conservative systems developed by Riewe 
[2], where he used the fractional derivative to construct the Lagrangian and 
Hamiltonian for non-conservative systems. 
From these reasons in  [3] was developed a general formula for the potential 
of any arbitrary force conservative or not conservative, which leads directly 
to the consideration of dissipative effect in Lagrangian and Hamiltonian 
formulation. Also, the canonical quantization of non-conservative systems 
has been carried out in [4].  
Starting from a Lagrangian containing a fractional derivative, the fractional 
Hamiltonian is achieved in [5]. In addition, the passage from Hamiltonian 
containing fractional derivatives to the fractional Hamilton-Jaccobi is 
achieved by Rabei et.al [6]. The equations of motion are obtained in a 
similar manner to the usual mechanics. 
All these outstanding results using the fractional derivative make us 
concentrate on another branch of quntam physics. WKB approximation [7, 
8, 9, 10,14]. In this paper we are mainly interested to construct the solution 
of Schroödinger equation in an exponential form (Griffith 1995) starting 
from fractional Hamilton-Jaccobi equation and how it leads naturally to this 
semi-classical approximation namely fractional WKB.  
     The purpose of this paper is to find the solution of Schrödinger equation  
for some systems that have a fractional behavior in their Lagrangians and 
obey the WKB approximation assumptions. 
The plan of this paper is as follows: In section II the derivation of 
generalized Hamilton-Jaccobi partial differential equation which given in 
[6] is briefly reviewed. In section III the fractional WKB approximation is 
derived. In Section IV some examples with the fractional WKB technique 
is reported. Section V is dedicated to conclusions. 
II. Basic Tools 
The left and right Reimann-Loville fractional derivative are defined as 
follows [3] 
The left Riemann-Liouville fractional derivative is given by 
αα dfx
xa ∫ −−−⎟⎠
= )()(
)( 1                         (1) 
The right Riemann-Liouville fractional derivative has the form 
ββ dfx
bx ∫ −−−⎟⎠
= )()(
)( 1                     (2) 
Here α, β are the order of derivation such that n-1≤α <n, n-1≤β<n, and they 
are  not zero. 
If α is an integer, these derivatives are defined in usual sense as 
)()( xf
xfDxa
⎛=                                                                    (3-a) 
)()( xf
xfDbx
⎛−=                                                                (3-b) 
                                                                         
Hamilton formalism with fractional derivative was proposed in [5] namely 
),,,(),,,( tqDqDqLqDpqDptppqH bttabtta
αβα −+=  ,               (4) 
where L represents the fractional Lagrangian obtained by replacing the 
classical derivatives with the corresponding fractional ones [5]. 
Hamilton's equations of motion are obtained as follows [5] 
; ;qDp
 ;qDp
β pDpD
btta +=∂
           (5) 
In  [6] based on the sequential derivatives the fractional Hamilton-Jacobi 
partial differential equation is obtained. The Hamilton-Jacobi function in 
configuration space is written in a similar manner to the usual mechanics 
by using the Reimann-Loville fractional derivative.  In [6] the following 
generating function  is used, where α 
and β are bigger or equal to 1. Thus, the new Hamiltonian is expressed as  
StPPqDqDFF btta ==
−− ),,,,( 112 βα
),,,('),,,( tQDQDQLQDPQDPtPPQK bttabtta
αβα −+=         (7)                      
It is concluded that, the following relation relates the two Hamiltonians 
KQDPQDPHqDpqDp bttabtta +−+=−+
α                    (8) 
According to reference [6] the function F is proposed as   
),,,,( 11 tPPqDqDSF btta βα
βα −−= QDPQDP Btta
11 −− −− ββ
α  ,                    (9) 
                                                           
The function S is called Hamilton's principle function. 
                    
       Therefore, requiring that the transformed Hamiltonian K  shall be zero 
the Hamilton-Jacobi equation is satisfied. In other words Q, Pα, Pβ are 
constants. 
H                                                                                          (10) 
Since Q, Pα, Pβ are constants, The Hamilton’s principle function is written 
),,,,( 21
11 tEEqDqDSS btta
−−= βα                                                            (11) 
where 
1EP =α               2EP =β
If the Hamiltonian is explicitly independent of time, then S can be written 
as follows 
                                 (12) ),,(),(),( 212
1 tEEfEqDWEqDWS btta ++=
−− βα
W represents the Hamilton's characteristic function; therefore, the 
following equations of motion are obtained in [6] as: 
αα                         qD
ββ                                 (13) 
11 λα =
QDta       2
21 λβ =
QDbt                            (14) 
Here λ1 ,λ2 are constants. 
 III. Fractional WKB approximation 
The outstanding result regarding the meaning of the state function ψ and its 
relationship to Hamilton's principle function S enables us to write the 
exponential solution of Schrödinger equation [13]. 
exp),(
tqψ                                   (15) 
                    
The phase of state function obeys the same mathematical equation, as does 
Hamilton's principle function S. The physical significance of S in classical 
mechanics is that it represents the generator of trajectories [12] for 
fractional systems; the fractional Hamilton's principle function is become 
the phase of the state function ψ. One can write the solution of Schrödinger 
equation under the postulated constrains by the WKB approximation and 
using the fractional Hamilton's principle function eq (12).  Thus we 
propose the fractional state function  as: 
⎛= −−−− tqDqDS
tqDqD bttabtta ,,exp),,(
1111 βαβαψ
       (16)                 
From the quantization using WKB approximation [7,8,9,10,14] a general 
solution of Schrödinger equation is obtained using the expansion for S and 
then using the transformation to the N-dimensional system as: 
exp)(
ψψ                                                      (17)                      
where 
iio qp
                                                                                (18) 
                                                                                       
In our case, S behaves like a 2-dimensional problem with two distinct 
momenta. Thus, 
                                                             (19) qDq ta
−≡ α αPP ˆ1≡
                                                                                (20) qDq bt
−≡ β βPP ˆ2 ≡
And the momenta are defined as operators. Therefore, we can propose the 
wave function ψ of the fractional system in the following form  
= −−−− tEEqDqDS
tqDqD bttabtta ,,,,exp
),,( 21
1111 βα
h       (21) 
and the momenta operators in the form 
−− qDi
ˆ,ˆ ββαα
                                     (22) 
We conclude that (21) is the solution of Schrödinger equation for any given 
fractional systems. If α and β both are equal to unity, then we will return to 
the usual classical solution of Schrödinger equation, also we can notice 
how the probability is inversely proportional to the momentum 
IV. Examples  
IV. a) Example 1: 
As a first model let us consider the following fractional Lagrangian, 
( ) ( )21
qDqDL tt
βα +=                                       (23) 
The fractional Hamilton-Jacobi equation for this fractional Lagrangian can 
be calculated as: 
( ) ( ) .0
1 22 =
PP βα                                          (24) 
where 
              qD
=   ;     qD
=    
Making use of equation (13), the fractional Hamilton-Jacobi equation (24) 
becomes: 
βα                            (25) 
Taking into account 
           t
−=                                                                (26) 
If we apply (26) on a wave function it gives: 
)( 21 EEEt
+−≡−=
                                               (27) 
By using the fact that E is the total energy of the system and taking into 
account (27) we obtain 
βα           (28) 
Thus, both sides of (28) should be zero, and we obtain  
qDEWqDEW tt
011 2,2
−− == βα               (29) 
By using  (12) and (21) we obtain 
−+= −−−− tEqDEqDE
tqDqD tttt
0 22exp
),,( βα
                                                                                                          (30) 
Which represents the wave function of the following Hamiltonian: 
( ) ( )22
βα PPH +=                                                          (31) 
Let us deal now with the momenta as operators of the form (22), and 
applying these operators on the wave function, one obtain the following 
momenta eigenvalues 
ψψψψ βα 21 2ˆ2ˆ EPEP ==                    (32) 
Then, 
                  21 2ˆ2ˆ EPEP == βα                   (33) 
It’s the same as the classical solution. Also, when applying the energy 
operator it gives the energy eigenvalues:  
( ) ( ) ψψψ βα 22 ˆ2
PPH +=         ( )ψ21 EE +=                               (34) 
as in the classical case. 
IVb.)Example 2: 
As a second example let us consider the following fractional Lagrangian 
( ) ( ) 210
qqDqDqDqDL tttt ++++=
                                     (35) 
The corresponding fractional Hamilton is calculated as follows 
( ) ( ) 222
qPPH −−+−= βα                                                              (36) 
Thus, the fractional Hamilton-Jacobi equation becomes 
( ) ( ) 0
1 222 =
+−−+−
qPP βα                                                       (37)                      
The fractional Hamilton's principle function is calculated as,  
( ) ( ) tEEqDEqDEqS tt )(1212 211121012 +−++++= −− βα                      (38) 
As a result the wave function can be written in the form 
( ) ( )( )⎟
−++++
tEqDEqDEq
tqDqD
1212exp
(39) 
   To identify the influence of the operators let us test the effect of the 
momenta 
−− qDi
         (40) 
Using the characteristic equations, it can be shown that  
12ˆ,12ˆ 21
2 +=++= EPEqP βα              (41) 
The result shown in (41 ) is  the same classical solution. When applying the 
energy operator it will give the energy eigenvalues  
( ) ( ) ψψψψ βα 222 2
qPPH −−+−=                                        (42) 
( ) ( ) ψψψψ βαβα 222 2
1ˆˆˆˆ
qPPPP −++−+=        
( ) ψ}
222222
EEqEq
−++++−
+++++=
                                                                   
Then we get 
        ψψ EH =                                                                (43) 
which is exactly the total energy as the case for the classical systems. 
V. Conclusions  
We use the generating function "S" of the Hamilton-Jaccobi equation in its 
fractional form to be the phase factor of the wave function describing some 
potentials valid for the assumptions suggested by the WKB approximation       
The proof of our results arises from the new proposed concepts of the 
momentum and energy operators, that they give the same eigenvalues 
producing the ordinary results achieved by the classical approach. 
Giving the same eigenvalues that means this form of fractional operator 
also eigen, valid, and useful in effecting on a state functions. 
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	Abstract
