Page 77 - IJOCTA-15-1

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.1, pp.71-81 (2025)
https://doi.org/10.36922/ijocta.1586

RESEARCH ARTICLE

Conformable Schr¨odinger equation in D-dimensional space

Eqab.M.Rabei 1,2* , Mohamed Ghaleb Al-Masaeed 3,4 , Sami I. Muslih 5,6 , Dumitru Baleanu 7,8

1 Physics Department, Faculty of Science, Al al-Bayt University, P.O. Box 130040, Mafraq 25113, Jordan
2 Science Department, Faculty of Science Jerash Private University, Jerash, Jordan
3
Ministry of Education, Jordan
4
Ministry of Education and Higher Education, Qatar
5
Al-Azhar University-Gaza, Gaza, Palestine
6
Department of Physics, University of Illinois at Urbana Champaign, Urbana, IL, 61801, USA
7
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
8
Institute of Space Sciences, Magurele–Bucharest, Romania
eqabrabei@gmail.com, moh.almssaeed@gmail.com, sami.muslih@siu.edu, dumitru.baleanu@lau.edu.lb
ARTICLE INFO ABSTRACT
Article History: In this paper, we investigate a time-dependent conformable Schr¨odinger equa-
Received: 8 April 2024 tion of order 0 < β ≤ 1, in fractional space domains of space dimension,
Accepted: 8 November 2024 0 < D s ≤ 3. We examine a specific example within the realm of free particle
Available Online: 24 January 2025 conformable Schr¨odinger wave mechanics, focusing on both N-Polar and N-
Keywords: Cartesian coordinates systems. We find that the conformable quantities align
Conformable derivative with the regular counterparts when β = 1.
Schr¨odinger equation
D-dimensional space
Fractional space
AMS Classification 2010:
81Q35; 34A08; 26A33; 37K10; 35R11

1. Introduction the actual confining structure, with the non-
integer dimension serving as a measure of its
The concept of fractional dimension was intro- confinement or anisotropy. Furthermore, re-
duced by mathematician Felix Hausdorff in 1918. searchers across various scientific and technolog-
This concept gained significant importance, espe- ical domains have dedicated considerable efforts
cially after Mandelbrot’s groundbreaking discov- to exploring this study. 4–11
1
ery of fractal geometry, where he applied frac-
tional dimensionality to elucidate the connections Many physical systems have been investi-
between fractional and integer dimensions using gated using methodologies involving fractional-
the scaling method, specifically, dimensional space. Research on critical phenom-
ena (see, for example, 12) fractal structures, 13–15
α and the modeling semiconductor heterostructure
π 2 |x| α−1
α
d x = α dx, 0 < α ≤ 1. (1) systems 16,17 usually takes into consideration the-
Γ( )
2 oretical schemes dealing with non-integer space
dimensionalities. Fractional Dimensions refer to
Physically, the confinement of low-dimensional the general concept of non-integer dimensions
systems has been successfully described using in various mathematical contexts, whereas Frac-
fractional dimensional space. This approach by, tal Dimensions are a specific type of fractional
He 2,3 involves substituting an effective space for dimension used to measure the complexity and
*Corresponding Author
71