EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5828
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quantization of Dissipative Anisotropic Harmonic
Oscillators

Bashar. M. Al-khamiseh1, Yazen. M. Alawaideh1,∗, Hayat Issaadi2,
Muhammad Bilal3, Shajar Abbas4, Hala Ghannam5, Dumitru Baleanu6,7

1 MEU Research Unit, Middle East University, Amman, Jordan
2 USTHB University, Operational Research Department, BP 32 El-Alia,Bab-Ezzouar, 16111,
Algeria
3 Department of Physics, Shanghai University, Shanghai 200444, China
4 AS (Ayub Shah), Institute of Mathematics and Research Centre, Multan, Pakistan
5 Department of Basic Sciences, Middle East University, Amman 11831, Jordan
6 Department of Computer Science and Mathematics, Lebanese American University, Beirut,
Lebanon
7 Institute of Space Sciences – INFLPR Subsidiary, Magurele-Bucharest, Romania

Abstract. In this study, we developed a Hamiltonian framework for the anisotropic harmonic
oscillator and applied the Hamilton-Jacobi equations to analyze a dissipative system. By incorpo-
rating boundary conditions, we derived the action function. The system was quantized using three
methods: the WKB approximation, canonical quantization, and creation-annihilation operators.
The Heisenberg equations were reformulated using Poisson bracket quantum variables, and energy
levels and quantum states were obtained. All methods yielded consistent results, demonstrating
the model’s robustness and its significance for understanding dissipative quantum systems.

2020 Mathematics Subject Classifications: 54C08, 54C60

Key Words and Phrases: Poisson bracket, creation and annihilation operators, WKB ap-
proximation, staeckel boundary conditions ,Two-Dimensional Anisotropic, Harmonic Oscillator
dissipative.

1. Introduction

Describing microscopic systems presents a significant challenge, particularly in bridg-
ing classical and quantum representations through Hamiltonian operator quantization.
Among these systems, the damped harmonic oscillator is of particular interest due to its
inherent dissipative nature, which complicates its quantization. Canonical quantization
methods, including coherent states and time-dependent creation-annihilation operators,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5828

Email addresses: yazen.awaida@yahoo.com (Y. M. Alawaideh),

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



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 2 of 25

have been widely explored to address such systems [1, 2]. The quantum Bateman method
has also been extensively studied for dissipative systems, offering valuable insights [3–
5]. Additionally, fractional calculus techniques have been employed to derive Lagrangians
and Hamiltonians for non-conservative systems, further advancing the field [6? ]. Recent
studies have focused on time-dependent harmonic oscillators and particles under specific
potentials, emphasizing time-dependent invariants [7]. Efforts to address Hamilton-Jacobi
equations using electromagnetic field techniques have also been made, with methods such
as variable separation under Staeckel conditions and Newtonian formalism being applied
[8]. Despite these advancements, challenges remain in quantifying dissipative systems,
particularly when dissipation depends solely on coordinates [9]. This highlights the need
for innovative approaches to better understand and model such systems.

In this context, recent research has explored the application of fractional calculus in
modeling complex dynamical systems, such as the immune system’s response to HIV-tumor
interactions, providing a fractional framework to understand these interactions. Similarly,
numerical investigations using Caputo-Fabrizio fractional derivatives have been employed
to study the dynamical behavior of the Hepatitis B virus, offering insights into the complex
dynamics of viral infections. These studies demonstrate the versatility of fractional cal-
culus in addressing real-world dynamical systems, which aligns with the challenges faced
in quantizing dissipative systems. For instance, the fractional quantization of Podolsky
electrodynamics using the fractional Hamilton-Jacobi formulation highlights the potential
of fractional calculus in extending classical and quantum theories to systems with non-
local and memory dependent behaviors. This approach not only bridges the gap between
classical and quantum descriptions but also provides a robust framework for analyzing
systems with fractional dynamics, such as those encountered in biological and physical
systems [10–12].

Furthermore, the Madelung formalism has been applied to dissipating and decaying
systems, providing a deeper understanding of the quantum dynamics of such systems
[13]. The quantum Hamilton-Jacobi formalism has also been extensively studied, offering
a robust framework for analyzing quantum systems with dissipation[14]. Additionally,
generalized creation and annihilation operators have been developed using complex non-
linear Riccati equations, which are crucial for understanding the quantization of dissipa-
tive systems [15]. Recent advancements in dissipative fuzzy tracking control for nonlinear
networked systems with quantization have also contributed to the field, particularly in un-
derstanding the effects of quantization on dissipative systems [16]. Recent advancements
in dissipative fuzzy tracking control for nonlinear networked systems with quantization
have also contributed to the field, particularly in understanding the effects of quantization
on dissipative systems [17]. This research aims to develop a Hamiltonian framework to
describe the two-dimensional anisotropic harmonic oscillator with damping effects by uti-
lizing the Hamilton-Jacobi equations to analyze the dissipative system. To achieve this,
the researchers employed three different quantization methods: the WKB approximation
and creation and annihilation operators, and canonical quantization.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 3 of 25

The paper follows a structured approach, beginning with an Introduction that dis-
cusses the challenges of quantizing dissipative systems and reviews existing methods such
as canonical quantization, fractional calculus, and Hamilto Jacobi approaches. It then
develops the Hamiltonian and Lagrangian formulations for the two-dimensional dissipa-
tive harmonic oscillator, based on Bateman’s approach. The study applies the WKB
approximation to quantize the system and analyze its energy levels, followed by Canoni-
cal Quantization using the Nikiforov-Uvarov method, which solves the Schrödinger equa-
tion to derive energy eigenvalues and wave functions. The Separation of Variables in the
Hamilton-Jacobi Equation is examined using Staeckel boundary conditions to facilitate the
quantization process. Additionally, the paper reformulates the dissipative system through
Poisson Brackets, applying standard quantization rules to obtain Heisenberg equations and
expectation values. The study further explores the Applications of the Hamilton-Jacobi
Technique, demonstrating its use in understanding oscillatory dynamics, dissipative effects,
and chaos in quantum and relativistic systems. To illustrate the effects of dissipation on
quantum states, the paper presents graphical representations of energy levels and prob-
ability distributions. In the Future Directions, it proposes extending the model to three
dimensions and incorporating additional coupling terms to enhance the understanding of
anisotropic and dissipative quantum systems. The study also compares its findings with
Previous Research, emphasizing improvements in quantization techniques and theoretical
modeling. Finally, the Conclusions summarize key findings, including the consistency of
different quantization methods, the validation of the Hamilton-Jacobi formalism, and the
significance of dissipation in quantum mechanics.

2. Dissipative Anisotropic Harmonic Oscillators: Hamiltonian and
Lagrangian Formalis

In this section, we use the Lagrangian proposed by Bateman to describe the two-
dimensional dissipative system. This Lagrangian provides a framework for describing
dissipation in the system, but quantization is carried out using other methods such as the
WKB approximation and canonical quantization .The Lagrangian used in Eq. (1) was
originally proposed by Bateman [18] and later utilized in various studies, including [19],
to model dissipative two dimensional anisotropic harmonic oscillators

L =
2∑
i=1

(
1

2
mq2i − 2π2mv2i q

2
i )e

λit (1)

In accordance with (1), the equations of motion can be expressed as:

q̈1 + λ1q1 + 4π2v21q1 = 0

q̈2 + λ2q1 + 4π2v22q2 = 0



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 4 of 25

This transformation clearly delineates the dissipative nature of the two-dimensional anisotropic
harmonic oscillator.

xi =

2∑
i=1

qie
λit

2

For i = 1, 2, the Lagrangian is transformed as follows based on the preceding equation:

L =

2∑
i=1

1

2
mx2i +

1

8
mλ2ix

2
i −

1

4
mλiẋixi − 2π2mv2i x

2
i

When the terms ẋi and xi in the Lagrangian don’t affect the equations of motion Fi (where

(dFi
dt = 0 = ẋixi)) it suggests Fi =

x2i
2 . Consequently, we obtain the following equivalent

Lagrangian:

L =

2∑
i=1

1

2
mx2i +

1

8
mλ2ix

2
i − 2π2mv2i x

2
i

Here, the subscript (i) means sum over (x1 = x, x2 = y).
The Lagrangian describes the dissipative, two-dimensional anisotropic harmonic oscillator.
The equations of motion can be derived from this lagrangian.

ẍ+ (w2
x −

λ2x
4
)x = 0 (2)

ÿ + (w2
y −

λ2y
4
)y = 0 (3)

A significant finding is that the damped oscillator’s angular frequency assumes values√
(w2

x −
λ2x
4 ) and

√
(w2

y −
λ2y
4 ) where (wx, wy) represents the angular frequencies of the

conservative system. To revert to the basic harmonic oscillator, one sets the damping
coefficient (λy, λx) to zero in the equations. The Hamiltonian (H) can be expressed as a
function of coordinates (x, y) and conjugate momenta (Px, Py).

H = Pxẋ+ Pyẏ − L

The generalized momenta are defined as follows [20]:

Px =
∂L

∂x
= mẋ and Py =

∂L

∂y
= mẏ

The following is the Hamiltonian’s expression in terms of the time derivatives:

H =
P 2
x

2m
+
P 2
y

2m
− 1

4
mλ2xx

2 − 1

4
mλ2yy

2 + 2π2mv2xx
2 ++2π2mv2yy

2

The current study’s conclusions and Rabei et al.’s analysis differ for a number of reasons.
Firstly, the current work focused on scattering using a number of approaches, including
Hamilton-Jacobi and variable separation, with a focus on dissipative. Secondly, Rabei et



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 5 of 25

al. [21] utilized the WKB for constrained systems. Hence, the Hamilton–Jacobi equation
assumes the form

1

2m
((
∂S

∂x
)2 + 4π2m2v2xx

2 − 1

2
m2λ2xx

2 + (
∂S

∂y
)2 + 4π2m2v2yy

2 − 1

2
m2λ2yy

2) +
∂S

∂t
= 0 (4)

Where ∂S
∂x and ∂S

∂y represent the canonical momenta Px and Py , respectively. By substi-
tuting these into the equation, it can be rewritten as follows:

1

2m
((Px)

2 + 4π2m2v2xx
2 − 1

2
m2λ2xx

2 + (Py)
2 + 4π2m2v2yy

2 − 1

2
m2λ2yy

2) +
∂S

∂t
= 0

The given Hamiltonian is expressed as:

H =
1

2m
(Px)

2 + 4π2m2v2xx
2 − 1

2
m2λ2xx

2 +
1

2m
(Py)

2 + 4π2m2v2yy
2 − 1

2
m2λ2yy

2

By comparing the two expressions, we observe that the kinetic energy terms (Px)2

2m +
(Py)2

2m
are identical in both equations. For the potential energy terms, dividing the terms in the
Hamilton-Jacobi equation by 2m results in 4π2m2v2xx

2− 1
2m

2λ2xx
2+4π2m2v2yy

2− 1
2m

2λ2yy
2

, which matches the corresponding terms in the Hamiltonian. Additionally, the time-
dependent term ∂S

∂t in the Hamilton-Jacobi equation is consistent with the role of time in
classical dynamics, even though it does not explicitly appear in the Hamiltonian. Conse-
quently, the Hamilton-Jacobi equation and the given Hamiltonian are equivalent, as their
kinetic, potential, and temporal components align perfectly. The principal Hamiltonian
function S can be expressed as follows:

S(x, y, α, αy, t) =Wx(x, α, αy) +Wy(y, α, αy)− αt This leads to
∂S

∂t
= −α (5)

The partial derivatives of the function W with respect to x and y are defined by the
formulas Px = ∂W

∂x and Py = ∂W
∂y , respectively The Hamilton-Jacobi equation (HJE) is

rewritten in terms of the functions and by inserting equation 2 into equation 3:

1

2m
(

[
∂Wx

∂x

]2
+ 4π2m2v2xx

2 − 1

2
m2λ2xx

2 +

[
∂Wy

∂y

]2
+ 4π2m2v2yy

2 − 1

2
m2λ2yy

2) = α

By separating the variables, we can get the following results:

1

2m
(

[
∂Wx

∂x

]2
+ 4π2m2v2xx

2 − 1

2
m2λ2xx

2) = α. (6)

1

2m
(

[
∂Wy

∂y

]2
+ 4π2m2v2yy

2 − 1

2
m2λ2yy

2) = αy (7)

By rewriting equations 4 and 5, we get:

∂Wy

∂y
=

√
2mαy − 4π2m2v2yy

2 +
1

2
m2λ2yy

2 (8)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 6 of 25

∂Wx

∂x
=

√
2mα− 2mαy − 4π2m2v2xy

2 +
1

2
m2λ2xx

2 (9)

Equations 6 and 7 can be solved by integrating the equation to obtain:

Wy′ =

∫ y
′

0

√
(2mαy − 4π2m2v2yy

2 +
1

2
m2λ2yy

2)dy (10)

Wx′ =

∫ x
′

0

√
(2mα− 2mαy − 4π2m2v2xy

2 +
1

2
m2λ2xx

2)dx (11)

Substituting equation 1 and equation 11 in equation

S(x, y, α, αy, t) =Wx(x, y, α) +Wy(x, y, α)− αt

we obtain:

S(x, y, α, αy, t) =

∫ x
′

0

√
(2mα− 2mαy − 4π2m2v2xy

2 +
1

2
m2λ2xx

2)dx+

∫ y
′

0

√
(2mαy − 4π2m2v2yy

2 +
1

2
m2λ2yy

2)dy − αit (12)

By differentiating the previous equation with respect to αi, we obtain:

By′ =
∂S

∂αy′
=

∫ y
′

0

m

2mαy − 4π2m2v2yy
2 + 1

2m
2λ2yy

2)
dy−

sin−1

(√(
2mπ2v2

y
′− 1

4
mλ2

y
′

α
y
′

)
y
′

)
√

4π2v2
y′

−
sin−1

(√(
2mπ2v2

x
′− 1

4
mλ2

x
′

α−α
y
′

)
x

′

)
√

4π2v2
x′

(13)

βx′ + t =

∫ x
′

0

m

2mα− 2mαy − 4π2m2v2xx
2 + 1

2m
2λ2xx

2)
dx =

sin−1

(√(
2mπ2v2

x
′− 1

4
mλ2

x
′

α−α
y
′

)
x

′

)
√
4π2v2

x′

(14)

After integration, equation 14 was subjected to necessary algebraic simplifications, leading
to the expression shown in equations 15.

x =

√√√√( α− αy

2mπ2v2
x′
− 1

4mλ
2
x′

)
sin
[√

4π2v2
x′
(βx′ + t)

]
(15.a)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 7 of 25

y =

√√√√( α− αy

2mπ2v2
y′
− 1

4mλ
2
y′

)
sin
[√

4π2vy′2(βy′ + t)
]

(15.b)

3. Quantization of a Dissipative Two-Dimensional Anisotropic
Harmonic Oscillator Using the WKB Approximation

We start our formalism by studying the Schrödinger equation, denoted as:

Hψ = ih
∂ψ

∂t

For our dissipative Two-Dimensional Anisotropic Harmonic Oscillator, the following for-
mulation is proposed.

(
h

i

∂

t
− P 2

x

2m
−
P 2
y

2m
− 2π2mv2xx

2 − 2π2mv2yy
2 +

1

4
mλ2xx

2 +
1

4
mλ2yy

2) = 0

Where {
P̂x = h

i
∂
∂x

P̂y =
h
i
∂
∂y

That is (
h
i
∂
∂t −

h2

2m
∂2ψ
∂x2

− h2

2m
∂2ψ
∂y2

−2π2mv2xx
2ψ

−2π2mv2yy
2ψ +1

4mλ
2
xx

2 +1
4mλ

2
yy

2

)
= 0

Using the WKB approximation, we take ψ as follows:

ψ =
1√
PxPy

e
iS
h

By substituting ψ and performing the required derivatives, we obtain:−α+
1

2m

 ih∂
2S
∂x2

+ ∂S
∂x

∂S
∂x+ ih∂

2S
∂y2

+ ∂S
∂y

∂S
∂y

+4π2m2v2xx
2 +4π2m2v2yy

2 + y2

−1
2mλ

2
xx

2 − 1
2mλ

2
yy

2


ψ = 0 (16)

Taking the semi-classical limit k −→ 0 and remembering that Px = ∂Wx
∂x = ∂S

∂x and

Py =
∂Wy

∂y = ∂S
∂y , we get :

1

2m

((
∂Wx

∂x

)2

+

(
∂Wy

∂y

)2

+ 4π2m2v2xx
2 + 4π2m2v2yy

2 − 1

2
mλ2xx

2 − 1

2
mλ2yy

2

)
= α

(17)
Which is the classical HJE. So, we have satisfied the Schrodinger equation and quantized
the Two-Dimensional Anisotropic Harmonic Oscillator dissipative. The WKB method
demonstrated compatibility between quantum solutions and the classical Hamilton-Jacobi
equation, confirming the reliability of the semiclassical approach in determining energy
levels.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 8 of 25

4. Canonical Quantization of a Damped Two-Dimensional Anisotropic
Harmonic Oscillator with the Nikiforov-Uvarov Approach

The Schrödinger equation governing the wave function , for a two-dimensional anisotropic
harmonic oscillator with dissipation is given by

2∑
i=1

(
− h2

2m

∂2ψ

∂x2i

)
− 1

2
4π2m2v2i ψ +

1

4
mλ2ix

2
iψ − Eiψ = 0 (−∞ < xi < +∞)

We will employ the Nikiforov-Uvarov (NU) method to solve this eigenvalue problem, as
outlined in [21]. Here, ψ(xi) must be bounded and satisfy the normalization condition.∫ +∞

−∞
|ψ(xi)|2dx = 1

We introduce the dimensionless variables ζ and ϵ, representing xi and Ei , respectively.
This equation describes a two-dimensional anisotropic harmonic oscillator under damping.
After substituting with according to the relation;

2∑
i=1

ζi =
2∑
i=1

axi where ζi =

√
π

mw
xi

And replacing Ei with;

Ei =

2∑
i=1

hwϵi

We obtain the following differential equation:

2∑
i=1

ψi + (2ϵi − ζ2i )ψi = 0

This equation is of the hypergeometric type, denoted by primes to signify differentiation
with respect to [22].

ζσ(ζi) = 1

To prove the equation ζσ(ζi) = 1, at the point ζi the equation can be solved as follows:

σ(ζi) =
1

ζi

Thus, σ(ζi) =
1
ζi
, the equation becomes:

ζi.
1

ζi
= 1

This proves the validity of the equation ζ.σ(ζi) = 1 Additionally, the following relations
hold:

τ̂(ζi) = 0, and σ̂(ζi) = 2ϵi − ζ2i (18)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 9 of 25

In the given situation, with ρ̂(ζi) = 1, the requirement for
√
ρ̂(ζi)ψ(ζi) to be integrable

squares arises directly from the normalization conditions. By following the previously
outlined method, we can transform the equation governing into a hypergeometric one,
and thus we get:

σ(ζi)η
′′
i + η(ζi)η

′
i + λiηi = 0 (19)

From the above equation, we express ψ(ζi) =
∑2

1 ϕ(ζi)η(ζi) as a summation. To achieve
this, we utilize the condition outlined in equation 18, where ϕ(ζi) satisfies the equation.

ϕ
′

ϕ
=
π(ζi)

σ(ζi)
(20)

The polynomial π(ζi) is represented by the expression
∑2

1±
√
ki − 2ϵi + ζ2i , where the

constant ki is chosen to result in a double zero, specifically set as ki = 2ϵi. This expression
gives rise to two potential polynomials, which are π(ζi) = ±ζi. Therefore, we obtain:

τ(ζi) = τ̂(ζi) + 2π(ζi) (21)

The existence of a negative derivative indicates that certain conditions must be met for
the function τ(ζi). These conditions are satisfied when we take τ(ζi) = 2ζi. Under these
conditions, we get:

τ(ζi) = −ζi ϕ(ζi) = e
−ζ2i
2

λi = 2ϵ2 − 1 ρ(ζi) = e−ζ
2
i

The values of the energy eigenvalue ϵ = ϵni = nx + ny +
1
2 hence s(λi) are determined by

an equation based on τ and σ, which is used to rearrange to achieve equilibrium. The
expression for the energy eigenvalues (ϵ) is:

Enxny = Ex + Ey = (2πnxvx + 2πnyvy +
2π(vx + vy)

2
)h

where nx = 0, 1, 2, · · · and ny = 0, 1, 2, · · ·

. The eigenfunctions ηnr(x, y) are defined as : ηnr(x, y) = Bnre
x2+y2(−1)n+r ∂

n+r

∂r∂n e
−x2−y2

and involve Hermite functions Hnr(x, y). Similarly, the wave functions ψnm(x, y) are
derived from the eigenfunctions, with the form

ψnm(x, y) = Cnre
m(wxx2+wyy2)

2k Hnr(
mwx
h

x,
mwy
h

y) =

(m
πk

) 1
2
(wx)

1
2 (wy)

1
2 e−

m(wxx2+wyy2)

2k Hnr(
mwx
h

x,
mwy
h

y) where Cnr =
(mw
πk

) 1
2

(22)

Accurate energy eigenvalues were derived using the NU method, with results consistent
across various approaches, demonstrating the robustness of the methodologies employed.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 10 of 25

4.1. The continuity equation and the probability current

To derive the continuity equation for the dissipative two-dimensional anisotropic har-
monic oscillator, we start from the Schrödinger equation of the system, taking into account
the dissipation effects included in the Hamiltonian. The continuity equation expresses the
conservation of probability in the quantum system, linking the change in probability den-
sity with the probability current.

The Schrödinger equation for the system

The Schrödinger equation for the wave function ψ(x, y, t) is given as follows:

i

h

∂ψ

∂t
= Hψ

Where H is the Hamiltonian of the system. In our case, the Hamiltonian for the two-
dimensional dissipative system is given as follows:

H =
P 2
x

2m
+
P 2
y

2m
+ V (x, y) +

1

4
mλ2x +

1

4
mλ2y

Here, V (x, y) = 2π2m(v2xx
2 − v2yy

2) represents the potential energy, and the terms 1
4mλ

2
x

and 1
4mλ

2
y account for the dissipation effects.

Probability Density
The probability density ρ(x, y, t) is defined as:

ρ(x, y, t) = |ψ(x, y, t)|2ψ(x, y, t) ∗ ψ(x, y, t)

Where ψ is the complex conjugate of the wave function ψ∗.
Derivation of the Continuity Equation with Dissipation:
We start by multiplying the Schrödinger equation by ψ∗ and subtracting the complex
conjugate of the Schrödinger equation multiplied by ψ :

ψ ∗
(
ih
∂ψ

∂t

)
− ψ

(
ih
∂ψ∗
∂t

)
= ψ ∗Hψ − ψHψ∗

After simplification, we obtain:

ih

(
ψ ∗ ∂ψ

∂t
+ ψ

∂ψ∗
∂t

)
= ψ ∗Hψ − ψHψ∗

The left-hand side can be written as:

ih
∂(ψ ∗ ψ)

∂t
= ih

∂ρ

∂t

The right-hand side can be simplified using the Hamiltonian:

ψ ∗Hψ − ψHψ∗ = ψ ∗
(
P 2
x

2m
+ P 2

y 2m+ V (x, y) +
1

4
mλ2x +

1

4
mλ2y

)
ψ−



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 11 of 25

ψ

(
P 2
x

2m
+ P 2

y 2m+ V (x, y) +
1

4
mλ2x +

1

4
mλ2y

)
ψ∗

Since V (x, y) is a real potential, it cancels out, leaving:

ψ∗
(
P 2
x

2m + P 2
y 2m+ V (x, y) + 1

4mλ
2
x +

1
4mλ

2
y

)
ψ−ψ

(
P 2
x

2m + P 2
y 2m+ V (x, y) + 1

4mλ
2
x +

1
4mλ

2
y

)
ψ∗

Using the definition of the momentum operators P̂x = h
i
∂
∂x and P̂y =

h
i
∂
∂y , we get :

ψ ∗
(
− h2

2m
∂2

∂x2
− h2

2m
∂2

∂y2
+ 1

4mλ
2
x +

1
4mλ

2
y

)
ψ − ψ

(
− k2

2m
∂2

∂x2
− k2

2m
∂2

∂y2
+ 1

4mλ
2
x +

1
4mλ

2
y

)
ψ∗

After simplification, we obtain:

− h2

2m

(
ψ ∗ ∂2ψ

∂x2
− ψ ∂2ψ∗

∂x2

)
− k2

2m

(
ψ ∗ ∂2ψ

∂y2
− ψ ∂2ψ∗

∂y2

)
+ 1

4mλ
2
x(ψ∗ψ−ψψ∗)+ 1

4mλ
2
y(ψ∗ψ−ψψ∗)

The terms involving λx and λy vanish because ψ ∗ψ = ψψ∗. Thus, the equation simplifies
to:
− h2

2m
∂
∂x

(
ψ ∗ ∂ψ

∂x − ψ ∂ψ∗
∂x

)
− h2

2m
∂
(
ψ ∗ ∂ψ

∂y − ψ ∂ψ∗
∂y

)
We notice that the expression inside the parentheses is the probability current in the x
and y directions:

Jx =
h

2mi

(
ψ ∗ ∂ψ

∂x
− ψ

∂ψ∗
∂x

)
Jy =

h

2mi

(
ψ ∗ ∂ψ

∂y
− ψ

∂ψ∗
∂y

)
Therefore, the equation can be written as:

ih
∂ρ

∂t
= − h2

2m

(
∂Jx
∂x

+
∂Jy
∂y

)
Dividing both sides by ih, we obtain:

∂ρ

∂t
+
∂Jx
∂x

+
∂Jy
∂y

= 0

This is the continuity equation, which expresses the conservation of probability in the
quantum system, even in the presence of dissipation. The inclusion of dissipative terms
in the Hamiltonian, as demonstrated by Serhan et al. [18], ensures that the continuity
equation remains valid for systems with energy loss due to damping or environmental
interactions.
Probability Current
The probability current J is given by:

J = (Jx, Jy)

Where:

Jx =
h

2mi

(
ψ ∗ ∂ψ

∂x
− ψ

∂ψ∗
∂x

)
Jy =

h

2mi

(
ψ ∗ ∂ψ

∂y
− ψ

∂ψ∗
∂y

)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 12 of 25

The continuity equation expresses the conservation of probability in the quantum sys-
tem, linking the change in probability density to the probability current. For the two-
dimensional system with dissipation, the continuity equation is:

∂ρ

∂t
+▽J = 0 where ▽ J =

∂Jx
∂x

+
∂Jy
∂y

(23)

5. Separation of variables in the Hamilton-Jacobi equation using
Staeckel boundary conditions

Consider the motion of a two-dimensional anisotropic harmonic oscillator with a dis-
sipative potential V = 2π2m(v2xx

2 + v2yy
2). The Hamilton-Jacobi equation is given by:

H = T + V =
1

2m

((
P 2
x + 4π2m2v2xx

2 − 1

2
mλ2xx

2

)
+

(
P 2
y + 4π2m2v2yy

2 − 1

2
mλ2yy

2

))
(24)

Comparing this equation with H = 1
2(P − a)T−1(P − a) + V (q) , we obtain:

T−1 =

(
1
m 0
0 1

m

)
On the other hand, the Staeckel conditions require the fulfillment of two main criteria:

(i) Diagonal Matrix Condition:The inverse of the kinetic energy matrix T−1 must
be diagonal. In this case, this condition is satisfied, as:

(T−1)ii =
1

Tii
=

(
1
m 0
0 1

m

)

(ii) Separable Potential Condition: The potential V (q) must be separable in the
coordinates x and y, meaning it can be expressed as:

(ϕ−1)1j =
1

Tjj
=

(
1
m

1
m

0 1
m

)
From the given potential:

V (q) =
Vi(qi)

Tii
=

(
ψ1(x)

m
+ ψ1(y)m

)
If the Staeckel conditions are satisfied, then Hamilton’s characteristic function is
completely separable:

W (q) =
∑
i

Wi(qi)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 13 of 25

Inserting from equation 23 into equation (H(q, ∂W∂q ) +
∂S0
∂t = 0) and using the defi-

nition of momentum p = ∂W
∂q we obtain (more details see Appendix A):

1

2m

([
∂Wx

∂x

]2
+ 4π2m2v2xx

2 − 1

2
λ2xx

2 +

[
∂Wy

∂y

]2
+ 4π2m2v2yy

2 − 1

2
λ2yy

2

)
= α

(25)

6. Hamiltonian Dynamics of a Dissipative Two-Dimensional
Anisotropic Oscillator Using Poisson Brackets

The formulation of Poisson brackets of Hamiltons equations of motion are then
calculated as:

ẋ = {x,H} =

x, 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)

 =

{
x,
P 2
x

2m

}
=
Px
m

(26)
After rewriting the previous equation in terms of ẏ,we get :

ẏ = {y,H} =

y, 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)

 =

{
y,
P 2
y

2m

}
=
Py
m

(27)
If the Poisson brackets satisfy the canonical relations:

{x, Px} = {y, Py} = 1

Then the time evolution of the momentum Px is derived as:

Ṗx = {Px, H} =

Px, 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)


=

{
Px, 4π

2m2v2xx
2 − 1

4
mλ2xx

2

}
= −4π2mv2xx+

1

4
mλ2xx (28)

Likewise, equation (34) may be expressed in Ṗy form as follows:

Ṗy = {Py, H} =

Py, 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)


=

{
Py, 4π

2m2v2yy
2 − 1

4
mλ2yy

2

}
= −4π2mv2yy +

1

4
mλ2yx (29)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 14 of 25

Given the complete alignment with equations 2 and 3, it follows that the Poisson
bracket relations [x, Px] = ih imply the corresponding commutation relations [x, x] =
[y, y] = [Px, Px] = [Py, Py] = 0. This alignment ensures the consistency between
the classical and quantum descriptions of the system. Building on this foundation.
This approach is consistent with the methodology outlined in [23], where similar
techniques were employed to analyze dissipative quantum systems

d

dt
⟨x⟩ = i

h

〈 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)

 , x

〉

=
i

h

〈[
P 2
x

2m
,x

]〉
=

〈
Px
m

〉
(30)

We can write equation 26 in terms of , we get:

d

dt
⟨y⟩ = i

h

〈 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)

 , x

〉

=
i

h

〈[
P 2
y

2m
, y

]〉
=

〈
Py
m

〉
(31)

The equation 31 agree with equation 27.

d

dt
⟨Px⟩ =

i

h

〈 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)

 , Px

〉

=
i

h

〈[
2π2mv2xx

2, Px
]〉

= −4π2mv2x ⟨x⟩+
1

4
mλ2x ⟨x⟩ (32)

And

d

dt
⟨Py⟩ =

i

h

〈 1

2m

 (P 2
x +4π2m2v2xx

2 −1
4mλ

2
xx

2)
+

(P 2
y +4π2m2v2yy

2 −1
4mλ

2
yy

2)

 , Py

〉

=
i

h

〈[
2π2mv2yy

2, Py
]〉

= −4π2mv2y ⟨x⟩+
1

4
mλ2y ⟨x⟩ (33)

ẍ =
Ṗx
m

therefore ẍ+ (42x −
1

4
λ2x)x = 0 (34)

ÿ =
Ṗy
m

which means that ÿ + (42y −
1

4
λ2y)y = 0 (35)

These equations align with the results obtained through other quantization methods.
The formulation is further extended using creation and annihilation operators to
derive the energy eigenvalues and wave functions, as detailed in the Appendix B.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 15 of 25

7. Applications of the Hamilton-Jacobi Technique in Analyzing
Harmonic Oscillator Dynamics

This section discusses how we can use the Hamilton-Jacobi approach to study the
harmonic oscillator, showing its flexibility and effectiveness in different situations. It
explains how this method helps us understand the relationship between the Hamilton-
Jacobi equation of motion and the i-factor of an oscillator, which matches the sys-
tem’s physical features well. This confirms that the Hamilton-Jacobi approach works
well and proves that the fractional oscillator template is useful for describing strongly
damped vibrations. Additionally, we can apply this approach to see how dissipative
processes affect other oscillating physical systems, like the ordered Hamilton-Jacobi
harmonic oscillator. Another important point is exploring how classical chaos af-
fects the harmonic oscillator, giving us insights into realistic systems in classical,
quantum, and relativistic domains. While Newtonian physics suggests the harmonic
oscillator is integrable no matter what, this approach lets us explore chaotic behavior
in relativistic, one dimensionally driven oscillators, helping us understand complex
dynamical systems better.

8. Discussion of the Figures: The Impact of Dissipation on the
Quantum Oscillator

In this section, we analyze the effect of dissipation on the quantum two-dimensional
anisotropic oscillator using mathematical analysis and graphical representations that
illustrate the relationship between various dynamic parameters. Graphical represen-
tations have been provided to demonstrate how energy and probability distributions
are influenced by dissipation. The figures have been plotted with enhanced clarity
to ensure a better understanding of the results.

8.1. Relationship Between Energy and Dynamic Parameters

As shown in Figure (1), the total energy of the dissipative quantum oscillator is
analyzed as a function of a key parameter related to thermal and quantum effects.
The results indicate that increasing this parameter leads to a rise in total energy
due to quantum contributions. Additionally, at low temperatures, the dissipation
effect becomes more pronounced, causing a significant deviation from classical be-
havior. This highlights the fundamental role of dissipation in shaping the quantum
dynamics of the system. At low temperatures, the effect of dissipation becomes more
pronounced, leading to a significant deviation from classical behavior. This indicates
that dissipation plays a fundamental role in shaping the quantum dynamics of the
system, especially under conditions where quantum effects dominate. This devia-
tion can be explained by the fact that dissipation leads to continuous energy loss,
affecting the quantum energy distribution and altering the system’s states. In Figure



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 16 of 25

0 2 4 6 8 10
0

2

4

6

8

10

α

β
U
,β
E
,β
E

Figure 1: Illustrates how dissipation affects the energy probability distribution in a quantum system, leading to
energy transfer and changes in the system’s states due to interaction with the environment.

(2), we analyze the impact of the damping coefficient on energy. The graph clearly
illustrates that increasing the damping coefficient results in higher internal energy
levels, confirming that dissipation directly influences the quantum energy spectrum
and alters the system’s dynamics.

0 0.5 1 1.5 2
0.5

1

1.5

γ/ωp

β
U
,β
E
,β
E

Figure 2: The dimensionless energy

This increase in energy can be explained by the fact that dissipation leads to addi-
tional interactions between the system and its surrounding environment, increasing
the system’s internal energy. This aligns with the theory that dissipation alters the
quantum energy distribution and leads to the emergence of new energy states.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 17 of 25

8.2. Probability Distribution and the Effect of Dissipation

Figure (3) presents the probability distribution as an indicator of the quantum in-
teraction with the surrounding environment. It is observed that dissipation shifts
the probability distribution towards higher energy values, reflecting the energy loss
induced by system-environment interactions. This shift in probability distribution
indicates that dissipation not only reduces the total energy of the system but also
alters the distribution of quantum states. This can be explained by the fact that dis-
sipation causes energy to transfer from the system to the surrounding environment,
changing the probability distribution of quantum states.

0 1 2 3 4 5
0

0.2

0.4

0.6

0.8

1

1.2

1.4

ω/ω0

ω
0
P
I
=
U
,E
(ω
/ω

0
)

Figure 3: The distribution functions for the dissipative oscillator, showing the functions as a function of the
rescaled heat-bath frequency. The parameters are set to specific value

Moreover, Figures (4a, 4b, 4c) illustrate the effect of a magnetic field on the energy
distribution in a three dimensional quantum oscillator. The interaction between
damping and the magnetic field leads to the emergence of multiple peaks in the
probability distribution, indicating resonance effects caused by the interplay between
quantum energy levels and dissipation. The appearance of multiple peaks in the
probability distribution reflects resonance effects that occur when the system’s fre-
quencies match those of the surrounding environment. This suggests that dissipation
and the magnetic field work together to modify the quantum energy distribution,
leading to the emergence of new energy states and changes in the system’s dynamics.

9. Future Directions in the Study of Dissipative Forces in
Lagrangian Mechanics

In future studies, this model can be expanded to three dimensions to investigate
the impact of the additional dimension on the dynamics of the dissipative system.
Furthermore, a coupling term can be introduced to analyze its effect on the system,



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 18 of 25

0 2 4 6 8
0

0.2

0.4

0.6

0.8

1

ω/ω0

ω
0
P
I
=
U
,E
(ω
/ω

0
)

Figure 4: wc = 0.5w0

0 2 4 6 8
0

0.5

1

1.5

2

2.5

3

3.5

ω/ω0

ω
0
P
I
=
U
,E
(ω
/ω

0
)

Figure 5: wc = 0.5w0

particularly in cases where the anisotropy is strong or when there are additional inter-
actions between the dimensions. While the current study provides a comprehensive
framework for understanding the two dimensional anisotropic harmonic oscillator
under dissipative effects, there remain opportunities for further exploration. One
potential avenue is the extension of this model to three dimensions, which could offer
deeper insights into the dynamics of dissipative systems. Additionally, the inclusion
of coupling terms, particularly in the context of anisotropic behavior, represents
an important area for future investigation. Although coupling terms were excluded
in the present work under the assumption of weak anisotropy, future studies could
examine their impact under different conditions and assess their role in complex
dissipative systems. Such developments would enrich the theoretical understanding
and practical applicability of the proposed framework.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 19 of 25

10. Current vs. Previous Research on Quantizing Dissipative

This study focuses on the quantization of dissipative mechanical systems using meth-
ods such as the WKB approximation, the Hamilton-Jacobi equation, the Staeckel
boundary conditions, and canonical quantization. The following comparison high-
lights the primary differences between this study and previous research:
Canonical Quantization: Previous studies, including those by Dekker, Suzuki,
and Majima, focused on modeling time-dependent damping variables. In contrast,
this study combines canonical quantization with the WKB approximation, result-
ing in a more comprehensive and accurate quantum model for dissipative systems.
Non-Conservative Systems: Riewe’s work with fractional calculus helped de-
velop new models for non conservative systems. This study takes these ideas further
by applying them to quantum dissipative systems using modern techniques, such
as the Hamilton-Jacobi equation, to provide a clearer and more accurate quantum
description.
WKB Approximation: The WKB approximation has been widely used in previ-
ous studies to analyze dissipative systems. This study extends its use to quantum
dissipative systems, improving the precision of energy level predictions and the ac-
curacy of state quantization.
Creation and Annihilation Operators: The integration of dissipation into the
quantum framework through the use of creation and annihilation operators results
in more accurate predictions for energy levels and quantum states. By explicitly in-
corporating dissipative effects—such as energy loss due to damping or environmental
interactions—the model becomes more realistic,

closely reflecting the physical behavior of the system. This refinement ensures that
the derived energy levels and wave functions are more precise, as they account for
the impact of dissipation on the system’s dynamics. As a result, the theoretical pre-
dictions align more closely with experimental observations, significantly enhancing
the reliability and applicability of the framework to real-world dissipative quantum
systems.

11. Conclusions

In this paper, we investigated the dissipative two-dimensional anisotropic harmonic
oscillator using the Hamilton Jacobi formalism. We obtained the Lagrangian and
Hamiltonian equations, solved the Hamilton-Jacobi equation, and quantified the sys-
tem using the WKB approximation and canonical quantization approaches. We for-
mulated the equations of motion and the Heisenberg equations by isolating variables
and using Poisson brackets and commutation relations. Using creation and annihi-
lation operations, we were able to derive energy levels and eigenstates that nearly
matched the results of canonical quantization. In exceptional situations, for one
dimension only, our results are in precise accord with other quantization schemes as



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 20 of 25

described in [24]. This study contributes to quantum mechanics by modeling damped
oscillatory systems, an area that has been insufficiently studied. A new Schrödinger
equation for the two-dimensional anisotropic oscillator enhances the understanding
of quantum dynamics. By integrating techniques like the WKB approximation and
canonical quantization, the study consistently derives energy levels and eigenstates,
providing insights into dissipative effects. The results show strong consistency. The
results show strong consistency, reinforcing the proposed model’s credibility and its
applicability to similar systems, thus advancing the understanding of dissipative sys-
tems in quantum mechanics. Key findings of this study include:
Hamilton-Jacobi Equation Solutions: The Hamilton-Jacobi equation was solved
using the separation of variables method under Staeckel boundary conditions, lead-
ing to explicit expressions for the action function.
Consistent Quantization Approaches: The system was quantized using three
different methods—WKB approximation, canonical quantization, and creation-annihilation
operators. Remarkably, all three methods yielded consistent results, reinforcing the
robustness of the proposed model.
Comparison with Classical Results: In the special case of a one-dimensional
system, the obtained quantum results align precisely with classical predictions, ver-
ifying the reliability of the quantization framework.

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B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 22 of 25

Appendix A: Detailed Derivations and Solution Steps

In this appendix, we present an analysis of the Hamilton-Jacobi equation for the
motion of a two-dimensional harmonic oscillator using Staeckel boundary conditions
for separation of variables. The Hamiltonian equation for the dissipative system is
derived and analyzed using the Hamilton-Jacobi function W (q) to separate the vari-
ables in the x and y directions. After applying Staeckel conditions, precise math-
ematical solutions are obtained, enabling a deeper understanding of the system’s
behavior and the calculation of energy levels and quantum states using creation and
annihilation operators.

1
2m

((
∂Wx
∂x

)2
+ 4π2m2v2xx

2 +
(
∂Wy

∂y

)2
+ 4π2m2v2yy

2

)
= α (A.1)

By employing variable separation in equation (A.1) above in terms of and Wy, re-

sulting in: 1
2m

((
∂Wy

∂y

)2
+ 4π2m2v2yy

2

)
− 1

2m
2λ2y2 = αy (A.2)

1

2m

((
∂Wx

∂x

)2

+ 4π2m2v2xx
2

)
− 1

2
m2λ2x2 + αy = αx (A.3)

After integration, we may express equations (A.2) and (A.3) as follows:

Wy′ =

∫ y
′

0

√
(2mαy − 4π2m2v2yy

2 +
1

2
m2λ2yy

2)dy (A.4)

Wx′ =

∫ x
′

0

√
(2mα− 2mαy − 4π2m2v2xy

2 +
1

2
m2λ2xx

2)dx (A.5)

Substituting equations. (A.4) and (A.5) into the equation Wy′ + Wx′ = W , we
obtain:

W =

∫ y
′

0

√
(2mαy − 4π2m2v2yy

2 +
1

2
m2λ2yy

2)dy+

∫ x
′

0

√
(2mα− 2mαy − 4π2m2v2xy

2 +
1

2
m2λ2xx

2)dx

With these result, the Hamilton-Jacobi function S(q, α, t) =W (q, α)− αt becomes:

S(q, α, t) =

∫ y
′

0

√
(2mαy − 4π2m2v2yy

2 +
1

2
m2λ2yy

2)dy+

∫ x
′

0

√
(2mα− 2mαy − 4π2m2v2xy

2 +
1

2
m2λ2xx

2)dx− αt

Taking the derivative with respect to αi for equation (A.6), one gets:

By′ =
∂S

∂αy′
=

∫ y
′

0

m

2mαy − 4π2m2v2yy
2 + 1

2m
2λ2yy

2)
dy−



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 23 of 25

sin−1

(√(
2mπ2v2

y
′− 1

4
mλ2

y
′

α
y
′

)
y
′

)
√

4π2v2
y′

−
sin−1

(√(
2mπ2v2

x
′− 1

4
mλ2

x
′

α−α
y
′

)
x

′

)
√
4π2v2

x′

(A.7)

βx′ + t =

∫ x
′

0

m

2mα− 2mαy − 4π2m2v2xx
2 + 1

2m
2λ2xx

2)
dx =

sin−1

(√(
2mπ2v2

x
′− 1

4
mλ2

x
′

α−α
y
′

)
x

′

)
√
4π2v2

x′

(A.8)

Equation (A.8) allows us to formulate the result as

βx′ + t =

sin−1

(√(
2mπ2v2

x
′− 1

4
mλ2

x
′

α−α
y
′

)
x

′

)
√
4π2v2

x′

After some algebraic work, we obtain:

x
′
=

√√√√( α− αy

2mπ2v2
x′
− 1

4mλ
2
x′

)
sin
[√

4π2v2
x′
(βx′ + t)

]
(A.9)

The above equation is exactly the same as the equation that has been derived by
(equation (15.a)) in the same classical solution. The formulation enabled a math-
ematically rigorous description of the system dynamics, accurately deriving energy
levels and quantum states through creation and annihilation operators, highlighting
the strength of this method. The creation and annihilation operators provide a pow-
erful framework for quantizing the dissipative two-dimensional anisotropic harmonic
oscillator. By expressing position and momentum in terms of these operators, the
Hamiltonian is simplified, enabling the systematic derivation of energy eigenvalues
and eigenfunctions. The energy levels are expressed in terms of quantum numbers
nx and ny , while the wavefunctions are constructed using Hermite polynomials.
This approach ensures consistency with classical mechanics and other quantization
methods, such as the WKB approximation and canonical quantization, demonstrat-
ing the robustness and versatility of the operator formalism in analyzing dissipative
quantum systems.



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 24 of 25

Appendix B: Additional Derivations and Quantization

In this appendix, we provide further derivations concerning the quantum treatment
of the dissipative twodimensional anisotropic harmonic oscillator. We discuss the
application of Heisenberg’s equations and the standard rules of quantization to de-
rive the equations of motion for the system in its quantum form. Additionally, a
thorough analysis of the creation and annihilation operators is presented, which are
instrumental in deriving the energy eigenvalues and wave functions. The goal of
this section is to offer deeper insights into the quantum methodology employed and
to illustrate the consistent derivation of both classical and quantum results. The
commutation relations are as follows:

ẍ =
Ṗx
m

therefore ẍ+ (42x −
1

4
λ2x)x = 0 (B.1)

ÿ =
Ṗy
m

which means that ÿ + (42y −
1

4
λ2y)y = 0 (B.2)

Given the complete alignment with equations (2) and (3), it follows that {x, Px} =
{y, Py} = 1 implies [x, Px] = [y, Py] = ih We can now proceed to introduce the pairs
associated with creation and annihilation [23].

a+ =

2∑
i=1

√
mwi
2h

(xj − i
Pj
mwj

a+ =

2∑
i=1

√
mwi
2h

(xj + i
Pj
mwj

(B.3)

Then, solving for xj and Pj , we have:

xj =
2∑
i=1

√
h

mwj
(ai + a+j ) Pj = −i

2∑
i=1

(ai − a+j )a (B.4)

It is easy to verify that [xj , Pj ] = ik implies [a, a+] = 1. Once we compute H =
Pxẋ+ Pyẏ − L, we obtain:

H =

2∑
i=1

hwj(a
+
j aj +

1

2
) (B.5)

The energy eigenvalues are given by:

E = En = hwj(2πnxvx + 2yvy +
1

2
2π(vx + vy) (B.6)

Where nx = 1, 2, · · · and ny = 1, 2, · · ·
The ground-state wave function ψ0(x, y)) obeys a ψ0(x, y)) = 0 where

ψ0(x, y)) = ϕ0(x)υ0(x) (B.7)



B. M. Al-khamiseh et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5828 25 of 25

The ground-state wave function ψ0(x, y)), expressed as the product of functions
ϕ0(x)υ0(x) , satisfies the condition ψ0(x, y)) = 0 , resulting in:

ϕ0(x) =
(mwx
πk

) 1
4
e

−mwxx2

2h (B.8)

υ0(x) =
(mwx
πk

) 1
4
e

−mwyy2

2h (B.9)

When we replace equation (B.7) and (B.8) with equation (B.6), we get:

ψ0(x, y)) =
(m
πh

) 1
2
(wx)

1
4 (wy)

1
4 e

−m(wyy2+wxx2)

2h (B.10)

Hermite polynomials allow us to express the eigenfunctions for every given state n
as follows:

Hnm(x, y) = Bnm(−1)n+mex
2+y2 ∂

m+n

n∂m
e−x

2−y2 (B.11)

ψnm(x, y)) =
(m
πh

) 1
2
(wx)

1
4 (wy)

1
4 e

−m(wyy2+wxx2)

2h Hnm

(nwx
h
x,
mwy
h

y
)

(B.12)

These expressions, particularly ( ), validate the findings presented in equation 22,
indicating their consistency and concordance with the obtained results