Spherically symmetric potential in noncommutative spacetime with a compactified extra dimensions
Abstract
The Schrödinger equation of the spherically symmetrical quantum models such as the hydrogen atom problem seems to be analytically nonsolvable in higher dimensions. When we try compactifying one or several dimensions this question can maybe solved. This paper presents a study of the spherically symmetrical quantum models on noncommutative spacetime with compactified extra dimensions. We provide analytically the resulting spectrum of the hydrogen atom and Yukawa problem in \(4+1\) dimensional noncommutative spacetime in the first order approximation of the noncommutative parameter. The case of higher dimensions \(D\ge 4\) is also discussed.
Keywords
Extra Dimension Radial Equation Yukawa Potential Noncommutative Space Angular Momentum Quantum Number1 Introduction
Our paper is organized as follows. In Sect. 2, we focus on the hydrogen atom in \((D+1)\)dimensional noncommutative space with noncompactified extra dimension. We discuss the particular case where \(D= 4\) in which the solution of the spectral problem can be solved. The Yukawa potential is also discussed in this section. In Sect. 3 the same problem is solved with compactified extra dimensions. The discussion and conclusion are given in Sect. 4.
2 Hydrogen atom in noncommutative space with noncompactified extra dimension
3 Hydrogen atom in noncommutative space with compactified extra dimension
Remark 1

Our result shows that the energy spectrum (31) does not depend on the NC parameter \(\theta \) if we consider the first order approximation in this parameter. The solution of the eigenvalue problem of the hydrogen atom with one compactified dimension is solved numerically in [17, 18] (see also [19, 20] in the case where no dimensions are compactified). Due to the fact that \(\lim _{\theta \rightarrow 0} E_{nl}=E_{nl}\), the expression (31) can be considered as a solution of the hydrogen atom in \(4+1\)dimensional spacetime for both the NC^{1} and the commutative cases, where one dimension is compactified.
 The quantitycorresponds to the reduced dimension energy spectrum and is discussed in the introduction of our paper.$$\begin{aligned} E_{nl}'=\frac{\hbar ^2\zeta ^2}{2m(2l + 1 +\sqrt{1+4\lambda _4})^2} \end{aligned}$$(32)
4 Discussion and conclusion
Footnotes
 1.
This energy is valid where the first order approximation in \(\theta \) is considered.
Notes
Acknowledgments
DOS research at MaxPlanck Institute is supported by the Alexander von Humboldt foundation. The authors are grateful to the referee for his useful comments that allowed them to improve the paper.
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