EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5765
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quantization of Singular Systems using Fractional
Calculus

Eyad Hasan Hasan

Tafila Technical University, Faculty of Science, Applied Physics Department, P.O.Box: 179,
Tafila 66110, Jordan

Abstract. In this paper, we examined the theory of singular systems using fractional calculus.
We quantized these systems using the fractional WKB approximation. We applied the Hamil-
ton–Jacobi treatment for these systems. We obtained equations of motion. We constructed the
fractional Hamilton–Jacobi partial differential equations (FHJPDEs) to obtain the action func-
tions S. The action function enables us to obtain the wave function for these systems. We achieved
that the quantum results agree with the classical results. Finally, we examined two mathematical
examples to demonstrate the theory.

2020 Mathematics Subject Classifications: 34A08, 70H20, 70H45, 81Q05, 81Q20

Key Words and Phrases: Fractional WKB Quantization, Second-Order Singular Systems, Frac-
tional Calculus, Fractional Action Function

1. Introduction

The quantization of singular systems has been treated with more interest by Dirac’s
work for quantizing the gravitational field [1][2]. Following Dirac’s work, researchers de-
veloped the canonical method for investigating these systems [3][4][5][6], they used this
method to quantize these systems using path integral technique and WKB approxima-
tion [4][5][6]. A general theory has been investigated for quantizing higher-order singular
systems using WKB approximation by Hasan et al [6]. In this theory, researchers have
achieved that the quantum results approach the classical results.

In this paper, we would like to apply the fractional derivatives for this theory of
singular systems. The quantization of fractional singular Lagranians has been studied
for physical systems and fractional Lagrangians systems with second-order derivatives
have been treated with more interest and importance [7][8]. Researchers have investi-
gated Hamilton-Jacobi formalism for these systems within fractional derivatives. They
constructed the Euler-Lagrange equations and analyzed Hamilton’s equations [8]. More

DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5765

Email addresses: iyad973@yahoo.com, dr eyad2004@ttu.edu.jo (E. H. Hasan)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 2 of 16

recently, researchers have constructed formalism using the canonical method for quanti-
zation singular systems using different techniques as path integral approach and WKB
approximation for first-order derivatives [7]. The equations of motion are calculated in
fractional form as total differential equations. In addition, the solution of this set HJPDEs
in fractional form and the fractional Hamilton-Jacobi function or action function (HJF) S
are obtained.

In this paper, we would like to extend the work for fractional singular Lagrangians
systems with second-order derivatives and quantize these systems using a new approach
which it is called WKB approximation. We constructed a formalism for investigating
fractional singular Lagrangian systems and Hamilton-Jacobi formalism for second-order
derivatives. Besides, we constructed the fractional (HJF) S to build the appropriate wave
function to quantize the singular systems.

Now, we will define the most important formula of fractional calculus as Left Rie-
mann–Liouville derivatives [9].

aD
α
t f(t) =

1

Γ(n− α)

(
d

dt

)n ∫ t

a
(t− τ)n−α−1f(τ)dτ (1)

and the Right Riemann–Liouville derivative is given by:

tD
α
b f(t) =

1

Γ(n− α)

(
− d

dt

)n ∫ b

t
(τ − t)n−α−1f(τ)dτ (2)

and these derivatives have properties as follows:

aD
α
t f(t) =

(
d

dt

)α

f(t) (3)

tD
α
b f(t) =

(
− d

dt

)α

f(t) (4)

In this work, we aim to construct the formalism for quantizing singular Lagrangians
systems with second-order derivatives within the framework of fractional derivatives.

2. Fractional Singular Lagrangian and Hamilton-Jacobi Formalism with
Second-Order Derivatives

In this section, we will formulate the second-order singular Lagrangian within frame-
work of fractional calculus. We will start with fractional derivative Lagrangian is defined
by [7].

L = L(Dα−1qi, D
αqi, D

2αqi, t) (5)

Thus, the fractional of the Hessian matrix is defined as

Wij =
∂2L

∂D2αqi∂D2αqj
, i, j = 1, 2, . . . , N, i, j = 1, 2, . . . , N (6)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 3 of 16

If its rank isN , the Lagrangian is called regular otherwise the Lagrangian is singular

N −R,R < N.

Now, we can define the momenta πi conjugate to the coordinates Dαqi as:

πa =
∂L

∂D2αqa
(7)

πµ =
∂L

∂D2aqµ
(8)

The rank of the Hessian matrix is N −R, therefore, Eq. (7) can be solved to obtain
N −R accelerations D2αqa in terms of Dα−1qi, D

αqi, πa and D2αqµ as follows:

D2αqa =Wa(D
α−1qi, D

αqi, πa, D
2αqµ) (9)

Substituting (9) in (8), we can obtain:

πµ = −Hπ
µ (D

α−1qi, D
αqi, ρa, πa) (10)

A similar expression for the momenta pµ can be obtained as:

pµ = −Hp
µ(D

α−1qi, D
αqi, pa, πa) (11)

and the momenta pi corresponding to the coordinate Dα−1qi can be written as:

pa =
∂L

∂Dαqa
− d

dt

(
∂L

∂D2αqa

)
(12a)

pµ =
∂L

∂Dαqµ
− d

dt

(
∂L

∂D2αqµ

)
(12b)

where a = 1, 2, . . . , N −R, µ = 1, . . . , R.
we can write equations (10) and (11) as follows

H ′p
µ (Dα−1qi, D

αqi, pi, πi) = pµ +Hp
µ = 0 (13a)

H ′π
µ (Dα−1qi, D

αqi, pi, πi) = πµ +Hπ
µ = 0 (13b)

Thus, equations (13) represent primary constraints [1][2].
The Hamiltonian formalism for higher-order derivatives has been studied by Ostro-

gradski [10].
He treated the derivatives as coordinates. Therefore, one can treat Dα−1qi and D

αqi
as coordinates. So, the Poisson bracket for second-order derivatives can be defined as:

{A,B} =
∂A

∂Dα−1qi

∂B

∂pi
− ∂A

∂pi

∂B

∂Dα−1qi
+

∂A

∂Dαqi

∂B

∂πi
− ∂A

∂πi

∂B

∂Dαqi



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 4 of 16

Where A and B are functions described in terms of canonical variablesDα−1qi, D
αqi, pi,

and πi.
Here, the generalized momenta pi and πi are conjugated to the generalized coordinates

Dα−1qi and D
αqi respectively. Thus, the fundamental Poisson brackets are:

{Dα−1qi, D
α−1qj} = {Dαqi, D

αqj} = 0 = {Dαqi, D
α−1qj} = {pi, πi}

{Dα−1qi, pj} ≡ δij and {Dαqi, πj} ≡ δij

where i, j = 1, . . . , N
It is well known from Dirac’s formalism that the number of degrees of freedom can be

reduced due to the constraints [1][2].
Thus, the Hamiltonian H◦ can be defined as

H◦ = −L(Dα−1qi, D
αqµ, D

2αqµ,Wa) + PaD
αqa + πaD

2αqa −DαqµH
p
µ −D2αqµH

π
µ (14)

where µ = 1, . . . , R and a = R+ 1, . . . , N .
Because of the nature of singular Lagrangian, the momenta pµ and πµ are not inde-

pendent of pa and πa.
Thus, the set of fractional FHJPDEs is written as [11].

H ′
◦(D

α−1qi, D
αqi,

∂S

∂Dα−1qa
,

∂S

∂Dα−1qµ
,

∂S

∂Dαqa
,

∂S

∂Dαqµ
) = p◦ +Hµ = 0 (15a)

H ′p
µ (Dα−1qi, D

αqi,
∂S

∂Dα−1qa
,

∂S

∂Dα−1qµ
,

∂S

∂Dαqa
,

∂S

∂Dαqµ
) = pµ +Hp

µ = 0 (15b)

H ′π
µ (Dα−1qi, D

αqi,
∂S

∂Dα−1qa
,

∂S

∂Dα−1qµ
,

∂S

∂Dαqa
,

∂S

∂Dαqµ
) = πµ +Hπ

µ = 0 (15c)

Here, the fractional Hamilton’s function can be written as

S = S(Dα−1qa, D
α−1qµ, D

αqa, D
αqµ, t) (16)

and we can define:

pa =
∂S

∂Dα−1qa
, pµ =

∂S

∂Dα−1qµ
, πa =

∂S

∂Dαqa
, πµ =

∂S

∂Dαqµ
, Po =

∂S

∂t

Thus, we can write the fractional equations of motion as follows [11]:



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 5 of 16

dDα−1qa =
∂H ′

◦
∂pa

dt+
∂H ′p

µ

∂pa
dDα−1qµ +

∂H ′π
µ

∂pa
dDαqµ (17a)

dDαqa =
∂H ′

◦
∂πa

dt+
∂H ′p

µ

∂πa
dDα−1qµ +

∂H ′π
µ

∂πa
dDαqµ (17b)

−dpi =
∂H ′

◦
∂Dα−1qi

dt+
∂H ′p

µ

∂Dα−1qi
dDα−1qµ +

∂H ′π
µ

∂Dα−1qi
dDαqµ (17c)

−dπi =
∂H ′

◦
∂Dαqi

dt+
∂H ′p

µ

∂Dαqi
dDα−1qµ +

∂H ′π
µ

∂Dαqi
dDαqµ (17d)

If the total derivative of equation (15) is zero [3],

dH ′
◦ = 0 ; dH ′p

µ = 0; dH ′π
µ = 0 (18)

This means that equations (17) are integrable, and the rank of Hessian matrix is
N −R. Because of constraints, the degrees of freedom are reduced from N to N −R,
thus, the canonical coordinates transform from

{Dα−1qi, pi, D
αqi, πi}

to
{Dα−1qa, Pa, D

αqa, πa}.

Thus, we can write Eqs. (15) as follows:

∂S

∂t
+H◦(D

α−1qi, D
αqi, ρa, πa) = 0 (19a)

∂S

∂Dα−1qµ
+Hp

µ(D
α−1qi, D

αqi, Pa, πa) = 0 (19b)

∂S

∂Dαqµ
+Hπ

µ (D
α−1qi, D

αqi, Pa, πa) = 0 (19c)

3. Hamilton-Jacobi Function and Quantization using WKB
approximation

Following refs. [6] for investigating the Hamilton-Jacobi function and quantization us-
ing WKB approach for higher-order singular Lagrangian systems, the fractional Hamilton-
Jacobi function can be written as:

S(Dα−1qa, D
α−1qµ, D

αqa, D
αqµ, t) = f(t) +Wa(D

α−1qa, Ea) +W ′
a(D

αqa, Ea, E
′
a)

+ fµ(D
α−1qµ) + f ′µ(D

αqµ) +A. (20)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 6 of 16

In this case, we would like to write a general solution for Eqs. (19) in a separable form
of Eq. (20).

Here, f(t) has the solution

f(t) = −
N−R∑
a=1

Eat,

where E′
a are representing as constants of integration; Dα−1qµ and Dαqµ are in-

dependent variables, and the remaining functions Wa(D
α−1qa, Ea), W

′
a(D

αqa, Ea, E
′
a),

fµ(D
α−1qµ), and f

′
µ(D

αqµ) are time-independent.
Now, we can use the canonical transformations [12] to get the motion equations as

follows:

ηa =
∂S

∂E′
a

(21a)

λa =
∂S

∂Ea
(21b)

pi =
∂S

∂Dα−1qi
(21c)

πi =
∂S

∂Dαqi
(21d)

ηa and λa are constants.
Because of constraints, the fractional wave function Ψ for these systems can be written

as [6].

Ψ(Dα−1qa, D
α−1qµ, D

αqa, D
αqµ, t) =

[
N−R∏
a=1

ψ0a(D
α−1qa)φ0a(D

αqa)

]

× exp

(
iS(Dα−1qa, D

α−1qµ, D
αqa, D

αqµ, t)

ℏ

)
. (22a)

ψ0a =
1√

P (Dα−1qa)
(22b)

φ0a =
1√

π(Dαqa)
(22c)

and the fractional wave function Eq. (22a) must satisfy the following conditions:

Ĥ ′
0Ψ = 0, Ĥ ′P

µ Ψ = 0, and Ĥ ′π
µ Ψ = 0 (23)

Here, the generalized coordinates and momenta are written in fractional form as op-
erators:



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 7 of 16

Dα−1qi → Dα−1qi (24a)

Dαqi → Dαqi (24b)

pi → p̂i =
ℏ
i

∂

∂Dα−1qi
(24c)

πi → π̂i =
ℏ
i

∂

∂Dαqi
(24d)

p0 −→ p̂0 =
ℏ
i

∂

∂t
(24e)

4. Examples

Example 1: Fractional Regular Lagrangian

L =
1

2

(
(D2αq)2 − (Dαq)2

)
(25)

The momenta are:

p = −Dαq −D3αq (26a)

π = D2αq (26b)

The Hamiltonian H0 is written as

H◦ = pDαq +
1

2
π2 +

1

2
(Dαq)2 (27)

The corresponding set of fractional HJPDE are:

H ′
◦ = p◦ +H◦ = p◦ + pDαq +

1

2
π2 +

1

2
(Dαq)2 (28)

We note in this example that there are no primary constraints [1][2].
The motion equations are written as

dDα−1q =
∂H ′

◦
∂p

dt = Dαqdt (29a)

dDαq =
∂H ′

◦
∂π

dt = πdt (29b)

−dp = ∂H ′
◦

∂Dα−1q
dt = 0 (29c)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 8 of 16

−dπ =
∂H ′

◦
∂Dαq

dt = (p+Dαq)dt (29d)

The HJPDE, Eq. (19a), reads

H ′
◦ = p◦ +H◦ =

∂S

∂t
+Dαq

∂S

∂Dα−1q
+

1

2

(
∂S

∂Dαq

)2

+
1

2
(Dαq)2 = 0 (30)

Substituting Eq. (20) into (30), we have:

∂f

∂t
+Dαq

∂W

∂Dα−1q
+

1

2

(
∂W

∂Dαq

)2

+
1

2
(Dαq)2 = 0 (31)

Since H0 is time-independent, we can write f(t) = −E′t. Eq. (31) can then be written
as

−E′ +Dαq
∂W

∂Dα−1q
+

1

2

(
∂W

∂Dαq

)2

+
1

2
(Dαq)2 = 0 (32)

From Eq. (32), the function W depends only on Dα−1q and W ′ depends only on Dαq.
This means that

∂W

∂Dα−1q
= E (33a)

W = Dα−1qE (33b)

Substituting Eq. (33b) into (32), one can obtain:

−E′ +DαqE +
1

2

(
∂W ′

∂Dαq

)2

+
1

2
(Dαq)2 = 0 (34)

This equation gives

W ′(Dαq, E,E′) =

∫ √
2E′ + E2 − (Dαq + E)2 dDαq (35)

Thus, one can obtain the Hamilton-Jacobi function as:

S(Dα−1q,Dαq, E,E′) = −E′t+Dα−1qE +

∫ √
2E′ + E2 − (Dαq + E)2 dDαq +A (36)

By using Eqs. (21a, 21b), we can obtain the solutions for the coordinates.

η =
∂S

∂E′ = −t+
∫

dDαq√
2E′ + E2 − (Dαq + E)2

(37a)

λ =
∂S

∂E
= Dα−1q +

∫
[E − (Dαq + E)]√

2E′ + E2 − (Dαq + E)2
dDαq (37b)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 9 of 16

The generalized momenta can be determined by using Eqs. (21c, 21d).

p =
∂S

∂Dα−1q
= E (38a)

π =
∂S

∂Dαq
=

√
2E′ + E2 − (Dαq + E)2 (38b)

Thus, the wave function is calculated as:

Ψ(Dα−1q,Dαq, t) =
[
ψ01(D

α−1q)φ01(D
αq)

]
exp

(
iS(Dα−1q,Dαq, t)

ℏ

)
(39)

Where

ψ01 =
1√

p(Da−1q)
= [E]−

1
2 (40a)

φ01 =
1√

π(Dαq)
=

[
2E′ + E2 − (Dαq + E)2

]− 1
4 (40b)

and the function S is calculated by Eq. (36).
Now, we can apply the fractional HJPDE, Eq. (28), to the function Ψ, we represent

the generalized coordinates and momenta as operators:

Ĥ ′
◦Ψ =

[
ℏ
i

∂

∂t
+Dαq

ℏ
i

∂

∂Dα−1q
− ℏ2

2

∂2

∂(Dαq)2
+

1

2
(Dαq)2

]
Ψ (41)

After some algebra, we have

ℏ
i

∂

∂t
Ψ = −E′Ψ (42a)

ℏ
i

∂

∂Dα−1q
Ψ = EΨ (42b)

− ℏ2

2

∂

∂(Dαq)
Ψ =

[
− 5ℏ2

8
(Dαq + E)2(2E′ + E2 − (Dαq + E)2)−2

− ℏ2

4
(2E′ + E2 − (Dαq + E)2)−1 +

1

2
(2E′ + E2 − (Dαq + E)2)

]
Ψ. (42c)

Thus, Eq. (41) becomes

Ĥ ′
◦Ψ =

− E′ +DaqE − 5ℏ2

8
(Daq + E)2(2E′ + E2 − (Daq + E)2)−2−

ℏ2

4
(2E′ + E2 − (Daq + E)2)−1 +

1

2
(2E′ + E2 − (Daq + E)2) +

1

2
(Daq)2

Ψ

(43)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 10 of 16

Taking the semiclassical limit ℏ → 0 in Eq. (43), we obtain

Ĥ ′
◦Ψ =

[
−E′ +DαqE +

1

2

(
2E′ + E2 − (Dαq + E)2

)
+

1

2
(Dαq)2

]
Ψ = 0 (44)

Example 2: We will discuss the following mathematical singular Lagrangian with two
primary first-class constraints

L =
1

2

(
(D2αq1)

2 + (D2αq2)
2
)
+Dαq3D

2αq3 +Dαq3D
α−1q3 +Dα−1q2D

αq2 (45)

The corresponding generalized momenta, Eqs. (7,8) and (12) are:

p1 = −D3αq1 (46a)

p2 = Dα−1q2 −D3αq2 (46b)

p3 = Dα−1q3 = −Hp
3 (46c)

π1 = D2αq1 (46d)

π2 = D2αq2 (46e)

π3 = Dαq3 = −Hπ
3 (46f)

Here, Equations (46c) and (46f) can be written as:

Ĥp
3 = p3 −Dα−1q3 = 0 (47a)

Ĥπ
3 = π3 −Dαq3 = 0 (47b)

and represent as primary constraints [1][2].
We calculate the Hamiltonian H0 as

H◦ = p1D
αq1 + (p2 −Dα−1q2)D

αq2 +
1

2
(π21 + π22) (48)

The set of fractional HJPDEs, Eqs. (15), reads:

H ′
◦ = p◦ +H◦ = p1D

αq1 + (p2 −Dα−1q2)D
αq2 +

1

2
(π21 + π22) (49a)

H ′p
3 = p3 −Dα−1q3 = 0. (49b)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 11 of 16

H ′π
3 = π3 −Dαq3 = 0. (49c)

Here, the Poisson brackets

{H ′p
3 , H

′
◦} = 0, {H ′π

3 , H
′
◦} = 0 and {H ′p

3 , H
′π
3 } = 0

The set of fractional HJPDEs, Eqs. (19), reads:

H ′
◦ = p◦ +H◦ =

∂S

∂t
+Dαq1

∂S

∂Dα−1q1

+Dαq2

(
∂S

∂Dα−1q2
−Dα−1q2

)
+

1

2

(
∂S

∂Dαq1

)2

+
1

2

(
∂S

∂Dαq2

)2

= 0. (50a)

H ′p
3 =

∂S

∂Da−1q3
−Da−1q3 = 0. (50b)

H ′π
3 =

∂S

∂Daq3
−Daq3 = 0. (50c)

The function S Eq. (20) can be written as

S(Da−1q1, D
a−1q2, D

a−1q3, D
aq1D

aq2, D
aq3, t) = f(t)+W1(D

a−1q1, E1)+W2(D
a−1q2, E2)

+W ′
1(D

aq1, E1, E
′
1) +W ′

2(D
aq2, E2, E

′
2) + f3(D

a−1q3) + f ′3(D
aq3) +A (51)

Since H0 is time-independent and the coordinates Dα−1q3 and Dαq3 are treated as
independent variables, one can write

f(t) = −
(
E′

1 + E′
2

)
t

Substituting S into Eq. (50a), we have

−E′
1+D

αq1
∂W1

∂Dα−1q1
+
1

2

(
∂W1

∂Dαq1

)2

−E′
2+D

αq2

(
∂W2

∂Dα−1q2
−Dαq2

)
+
1

2

(
∂W2

∂Dαq2

)2

= 0

(52)
We note that W1 depends only on Dα−1q1 and W2 depends only on Dα−1q2. We can

then write

∂W1

∂Dα−1q1
= E1

so that



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 12 of 16

W1 = Dα−1q1E1 (53a)

and

∂W2

∂Dα−1q2
−Dα−1q2 = E2

so that

W2 = Dα−1q2E2 +
1

2
(Dα−1q2)

2 (53b)

Substituting Eqs. (53) into (52), we get:

−E′
1 +Dαq1E1 +

1

2

(
∂W ′

1

∂Dαq1

)2

− E′
2 +Dαq2E2 +

1

2

(
∂W ′

2

∂Dαq2

)2

= 0 (54)

Separation of variables in this equation yields

1

2

(
∂W ′

1

∂Dαq1

)2

+Dαq1E1 − E′
1 = 0 (55a)

1

2

(
∂W ′

2

∂Dαq2

)2

+Dαq2E2 − E′
2 = 0 (55b)

We solve the above two Eqs. (55) to obtain:

W ′
1(D

αq1, E1, E
′
1) =

∫ √
2E′

1 − 2Dαq1E1 dD
αq1 (56a)

W ′
2(D

αq2, E2, E
′
2) =

∫ √
2E′

2 − 2Dαq2E2 dD
αq2 (56b)

Using Eq. (50b), we find:

f3(D
α−1q3) =

1

2
(Dα−1q3)

2

and using Eq. (50c), we find:

f ′3(D
αq3) =

1

2
(Dαq3)

2

Thus, the function S can be written as

S(Dα−1q1, D
α−1q2, D

α−1q3, D
αq1, D

αq2, D
αq3, t) = (−E′

1−E′
2)t+D

α−1q1E1+
1

2
(Dα−1q2)

2

+

∫ √
2E′

1 − 2Dαq1E1 dD
αq1 +

∫ √
2E′

2 − 2Dαq2E2 dD
αq2



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 13 of 16

+
1

2
(Dα−1q3)

2 +
1

2
(Dαq3)

2 +A. (57)

Now, we can use the transformations Eqs. (21a, 21b) to obtain the solutions for the
coordinates as:

η1 =
∂S

∂E′
1

= −t+
∫

dDaq1√
2E′

1 − 2Daq1E1

(58a)

η2 =
∂S

∂E′
2

= −t+
∫

dDaq2√
2E′

2 − 2Daq2E2

(58b)

λ1 =
∂S

∂E1
= Dα−1q1 +

∫
Dαq1√

2E′
1 − 2Dαq1E1

dDαq1 (58c)

λ2 =
∂S

∂E2
= Dα−1q2 +

∫
Dαq2√

2E′
2 − 2Dαq2E2

dDαq2 (58d)

Using Eqs. (21c, 21d) to find the generalized momenta:

p1 =
∂S

∂Dα−1q1
= E1 (59a)

p2 =
∂S

∂Dα−1q2
= E2 +Dα−1q2 (59b)

p3 =
∂S

∂Dα−1q3
= Dα−1q3 (59c)

π1 =
∂S

∂Dαq1
=

√
2E′

1 − 2Dαq1E1 (59d)

π2 =
∂S

∂Dαq2
=

√
2E′

2 − 2Dαq2E2 (59e)

π3 =
∂S

∂Dαq3
= Dαq3 (59f)

where Dα−1q3 and Dαq3 are arbitrary parameters.
Also, we can determine the equations of motion using Eqs. (17).
Now, our purpose is to quantize our singular system. We can write the fractional wave

function Eq. (22a) for this example as:

Ψ(Dα−1q1, D
α−1q2, D

α−1q3, D
αq1, D

αq2, D
αq3, t) =

[
ψ01(D

α−1q1)ψ02(D
α−1q2)

]
× [φ01(D

αq1)φ02(D
αq2)] exp

(
iS

ℏ

)
(60)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 14 of 16

Where

ψ01(D
α−1q1) =

1√
P1(Dα−1q1)

= [E1]
− 1

2 (61a)

ψ02(D
α−1q2) =

1√
P2(Dα−1q2)

= [E2 +Dα−1q2]
− 1

2 (61b)

φ01(D
αq1) =

1√
π1(Dαq1)

=
[
2E′

1 − 2Dαq1E1

]− 1
4 (61c)

φ02(D
αq2) =

1√
π2(Dαq2)

=
[
2E′

2 − 2Dαq2E2

]− 1
4 (61d)

Thus, the function S can be given by Eq. (57).
Now, the fractional HJPDEs, Eqs. (49) can be applied to the wave function Ψ, after

representing the generalized coordinates and momenta as operators:

Ĥ ′
◦Ψ =

[
ℏ
i

∂

∂t
+Dαq1

ℏ
i

∂

∂Dα−1q1
+Dαq2

(
ℏ
i

∂

∂Dα−1q2
−Dα−1q2

)
−ℏ2

2

∂2

∂(Dαq1)2
− ℏ2

2

∂2

∂(Dαq2)2

]
Ψ (62a)

Ĥ ′p
3 Ψ =

[
ℏ
i

∂

∂Dα−1q3
−Dα−1q3

]
Ψ (62b)

Ĥ ′π
3 Ψ =

[
ℏ
i

∂

∂Dαq3
−Dαq3

]
Ψ (62c)

After some algebra, we have

ℏ
i

∂

∂t
Ψ = (−E1 − E2)Ψ (63a)

ℏ
i

∂

∂Dα−1q1
Ψ = E1Ψ (63b)

ℏ
i

∂

∂Dα−1q2
Ψ =

[
(Dα−1q2 + E2)−

ℏ
2i
(Dα−1q2 + E2)

−1

]
Ψ (63c)

−ℏ2

2

∂

∂(Dαq1)2
Ψ =

[
−5ℏ2

8
E2

1(2E
′
1 − 2Dαq1E1)

−2 +
1

2
(2E′

1 − 2Dαq1E1)

]
Ψ (63d)

−ℏ2

2

∂

∂(Dαq2)2
Ψ =

[
−5ℏ2

8
E2

2(2E
′
2 − 2Dαq2E2)

−2 +
1

2
(2E′

2 − 2Dαq2E2)

]
Ψ (63e)



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 15 of 16

ℏ
i

∂

∂Dα−1q3
Ψ = Dα−1q3Ψ (63f)

ℏ
i

∂

∂Dαq3
Ψ = Dαq3Ψ (63g)

Substituting the results of Eqs. (63) in Eqs. (62), we obtain:

Ĥ ′
◦Ψ =


− E′

1 − E′
2 +Daq1E1 +Daq2

[
(E2 +Da−1q2)−

ℏ
2i
(E2 +Da−1q2)

−1 −Da−1q2

]
−

5ℏ2

8
E2

1(2E
′
1 − 2Daq1E1)

−2 +
1

2
(2E′

1 − 2Daq1E1)−
5ℏ2

8
E2

2(2E
′
2 − 2Daq2E2)

−2)+

1

2
(2E′

2 − 2Daq2E2)

Ψ

(64a)

Ĥ ′p
3 Ψ =

[
ℏ
i

∂

∂Da−1q3
−Da−1q3

]
Ψ =

[
Da−1q3 −Da−1q3

]
Ψ = 0. (64b)

Ĥ ′π
3 Ψ =

[
ℏ
i

∂

∂Daq3
−Daq3

]
Ψ = [Daq3 −Daq3] Ψ = 0. (64c)

Taking the limit ℏ → 0 in Eq. (64a), we get:

Ĥ ′
◦Ψ =

− E′
1 − E′

2 +Daq1E1 +Daq2
[
E2 +Da−1q2 −Da−1q2

]
+

1

2
(2E′

1 − 2Daq1E1) +
1

2
(2E′

2 − 2Daq2E2)

Ψ = 0. (65)

5. Conclusion

In our work, we have extended a general theory for singular Lagrangian systems using
fractional calculus. In this work, we solved the set of the fractional HJPDEs for these
systems. In this paper, the canonical method is used for these systems to obtain fractional
HJPDEs. We determined the fractional function S to obtain the equations of motion.
Also, function S enables us to determine the appropriate fractional wave function for
these systems. We achieved that the constraints in singular systems become conditions on
fractional wave function. These conditions are achieved in semiclassical limit. Also, in this
limit, Schrödinger equation is satisfied. In other words, we have approved that the quan-
tum results agree with the classical results. Finally, we have examined two mathematical
examples.



E. H. Hasan / Eur. J. Pure Appl. Math, 18 (2) (2025), 5765 16 of 16

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