Advances in High Energy Physics

Volume 2014 (2014), Article ID 459345, 6 pages

http://dx.doi.org/10.1155/2014/459345

Research Article

Noncommutative Phase Space Schrödinger Equation with Minimal Length

H. Hassanabadi,¹ Z. Molaee,² and S. Zarrinkamar²

¹Department of Basic Sciences, Shahrood Branch, Islamic Azad University, Shahrood, Iran

²Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran

Received 23 October 2013; Revised 12 December 2013; Accepted 18 December 2013; Published 30 January 2014

Academic Editor: Elias C. Vagenas

Copyright © 2014 H. Hassanabadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We consider the Schrödinger equation under an external magnetic field in two-dimensional noncommutative phase space with an explicit minimal length relation. The eigenfunctions are reported in terms of the Jacobi polynomials, and the explicit form of energy eigenvalues is reported.

1. Introduction

Although the advent of Schrödinger equation dates back to many decades ago [1–3], the interest in the study of the equation has been renewed due to the noncommutative phase space commutations relations and the so-called minimal length (alternatively called generalized) uncertainty relation. The noncommutativity between space-time coordinates was first explained by Snyder [4] and provides us with motivating scenarios in string and M-theories [5, 6] as well as modern cosmology [7–9]. The application of the formulation is not limited to these fields and noncommutative theories are capable of explaining the IR/UV mixing and nonlocality [10], Lorentz violation [11], and new physics at very small scales [12, 13]. Extensive studies on the incorporation of ordinary quantum mechanics and classical mechanics with the noncommutative relations can be found in [14–16]. On the other hand, the modification of our ordinary uncertainty relation, which is inspired by the string theory, quantum gravity, and doubly special relativity [17–20], has become an appealing case of study in the recent years. In the minimal length formulation, the ordinary Heisenberg relation is modified by an ad hoc term and is written as , where is a small parameter determined from a fundamental theory such as string theory [21, 22]. Such a commutation relation corresponds to the following uncertainty relation: , which implies the existence of a minimal length [21, 22]. Recently, the universality of quantum gravity corrections was studied by Das and Vagenas. They indicated the existence of a minimum measurable length and the related generalized uncertainty principle (GUP) [23]. In another interesting paper, Ali et al. considered a GUP consistent with string theory, black hole physics, and doubly special relativity [24].

In the present work, we combine these topics within the framework of nonrelativistic Schrödinger equation. The organization of this work is as follows. In Section 2, we consider the Schrödinger equation in the presence of magnetic field in noncommutative phase space (NCPS). In Section 3, we study this problem with minimal length relations. In Section 4, we consider some special cases of the solutions to check the validity of the results.

2. The Two-Dimensional Schrödinger Equation in Noncommutative Phase Space

The free particle Schrödinger equation in two spatial dimensions reads

where is mass of the particle. Using the operator relations

and recalling that the vector potential is written as , where is magnetic field, (1) takes the form

In NCPS the position operators are satisfied in the following relations [25–27]:

where is an antisymmetric tensor and is of space dimension . The other commutation relations appear as

with the effective plank constant being

where , , , and are the noncommutative parameters. The NCPS parameters are related to the commutative space parameters via

By inserting (7) into (3), we obtain the equation in commutative space as

in which is angular momentum.

3. The Problem with a Minimal Length

For the sake of simplicity, we bring the problem into the momentum space. Recalling that , [28],

as well as writing

and , where , the Schrödinger equation appears as

or, via , in the form

A transformation of the form brings (12) into

By simplifying (13), we obtain the following equation:

which, via the change of variable , takes the more familiar form

with

Equation (15) possesses the eigenfunctions

and its energy eigenvalues can be derived from

which yields

4. Special Cases

Let us now check the special cases. First, when the minimal length is absent, that is, , the energy relation (19) reduces to

The energy has plotted the energy eigenvalues versus in Figure 1. The energy relation in the special cases of and , that is, for vanishing NCPS parameters, respectively, yields

Figure 1: The energy eigenvalues versus .

We have depicted in Figures 2 and 3 the energy versus . Finally, when both the noncommutative and minimal length parameters are absent, that is, when , and also , the energy spectrum takes the form

Figure 2: The energy eigenvalues versus .

Figure 3: The energy eigenvalues versus .

This is the known form of energy in [29–31]. In Figure 4, we have shown the energy values versus .

Figure 4: The energy eigenvalues versus .

5. Conclusion

In this study, we considered the Schrödinger equation incorporated with two recent trends in the field: the noncommutative formulation and the minimal length uncertainty relation. After two novel transformations, we transformed the equation into a familiar form and reported the solutions in terms of the Jacobi polynomials. Our results, in the special cases of the engaged parameters, yield the previous results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to give their sincere gratitude to the referees for their technical comments on the paper.

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