Research Article | Open Access
B. C. Wang, E. C. Brenag, R. G. G. Amorim, V. C. Rispoli, S. C. Ulhoa, "Exact Noncommutative Two-Dimensional Hydrogen Atom", Advances in High Energy Physics, vol. 2021, Article ID 5562179, 5 pages, 2021. https://doi.org/10.1155/2021/5562179
Exact Noncommutative Two-Dimensional Hydrogen Atom
In this work, we present an exact analysis of the two-dimensional noncommutative hydrogen atom. In this study, the Levi-Civita transformation was used to perform the solution of the noncommutative Schrodinger equation for Coulomb potential. As an important result, we determine the energy levels for the considered system. Using the result obtained and experimental data, a bound on the noncommutativity parameter was obtained.
The concept of noncommutativity in physical theories was formally introduced by Snyder in 1947 [1–3]. In a seminal paper, Snyder stated that spatial coordinates would not commute with each other at small distances. In this sense, a new paradigm was proposed in which the space-time should be understood as a collection of tiny cells of minimum size, where there is no such idea of a point. So far, once the minimum size is reached, in the realm of some high-energy phenomenon, the position should be given by the noncommutative coordinate operators. As a direct consequence, it would be impossible to precisely measure the position of a particle. Over the last years, the interest of the scientific community on noncommutative geometry has increased due to works on nonabelian theories , gravitation [5–7], standard model [8–10], and quantum Hall effect . More recently, the discovery that the dynamics of an open string can be associated with noncommutative spaces has contributed to the latest revival of noncommutative theories . Noncommutative physics has a wide range of applications, from noncommutative geometry to corrections in classical systems due to noncommutative coordinates. Particularly, noncommutative geometry has been a promising approach to understanding some limit of quantum gravitation . Some models foresee a possible experimental measurement given the current advance in astrophysical techniques applied to black holes .
From the mathematical point of view, the simplest algebra obeyed by the coordinate operators is where is a skew-symmetric constant tensor called noncomutativity parameter. It is worth to point out that the mean values of the position operators do correspond to the actual position observed. Thus, it is said that such operators are Hermitian ones. It is well known in quantum mechanics that a noncommutative relation between two operators leads to a specific uncertainty relation; hence, the above expression yields which implies a set of uncertainty relations between position coordinates analogous to the Heisenberg uncertainty principle. Following the ideas introduced by Snyder, it is possible to associate the minimum size with a distance of the order of magnitude. Thus, the noncommutative effects turn out to be relevant at such scales. Usually, the noncommutativity is introduced by means of the Moyal product defined as  with a constant . Then, the usual product is replaced by the Moyal product in the classical Lagrangian density. In a similar perspective, the noncommutative quantum mechanics is introduced by imposing further commutation relations between the position coordinates themselves.
In this perspective, we aim to apply the ideas about noncommutativity of space to the hydrogen atom. The hydrogen atom is an electrically neutral atom with a positively charged proton and a negatively charged electron bounded to the nucleus. This system plays a significant role in quantum mechanics and field theory. There are many good reasons to address the hydrogen atom . As an example, the hydrogen atom with high-precision measurements in atomic transitions is one of the best laboratories to test quantum electrodynamics theory . Other applications of the hydrogen atom appear in many occasions, such as to examine the constancy of fine structure constant over a cosmological time scale . There are some approaches to treat the hydrogen atom in a non-commutative space. Such approaches are differentiated by the role of non-commutativity in a specific representation. In this context, it is interesting to note that corrections for Lamb’s displacement were obtained in a context of non-commutative quantum electrodynamics . Even non-commutative corrections for the hydrogen atom in a curved space were obtained . In this paper, we analyze the two-dimensional noncommutative hydrogen atom. A two-dimensional hydrogen atom can be defined as a system in which the motion of the electron around the proton is constrained to be planar. Then, in this work, we consider that this plane is noncommutative. As a practical example, a semiconductor quantum well under illumination is a quasi-two-dimensional system, in which photoexcited electrons and holes are essentially confined to a plane [20, 21]. There are many works that consider the hydrogen atom in a noncommutative context, but they present disagreement in results. Our method presents an approach using the Levi-Civita mapping, which allows an exact treatment.
This paper is organized as follows. In Section 2, we present the mathematical framework of the two-dimensional hydrogen atom. The Levi-Civita mapping is presented in Section 3. In Section 4, we obtain the solution and spectrum for the noncommutative hydrogen atom. Finally, in Section 5, we present our concluding remarks.
2. Mathematical Framework of Noncommutative Two-Dimensional Hydrogen Atom
The Hamiltonian that defines two-dimensional hydrogen atom is given by where and stands for the momentum of the electron in directions and , respectively; and represents the electron coordinates and the constant , where is the elementary charge and is the vacuum electric permittivity. To quantize the Hamiltonian given in Equation (4), as usual, the momentum operators are given by and , where and is the Planck constant.
In the noncommutative perspective, we define the following position operators: in which is the noncommutativity parameter in Cartesian coordinates. We notice that as expected.
However, the treatment of the Hamiltonian given in Equation (4) is difficult because of the operators in the denominator of the potential energy term. For this reason, in the next section, we present a transformation that puts the system in a more suitable way.
3. Levi-Civita Mapping
The Levi-Civita (also known as Bohlin) transformation is a parabolic coordinate mapping that is capable of converting the planar Coulomb problem into a two-dimensional harmonic oscillator [22–25]. It is a surjection defined by
Given Equation (7), it is immediate to conclude that and, by inversion,
As a direct consequence of Equation (10), the momentum operators can be rewritten in this new coordinate system as
It should be noted that the Levi-Civita mapping is a canonical transformation .
Finally, the hypersurface defined by is given by
Equation (15) is the main result of this section and is the one to be used from now on.
4. Analysis of Two-Dimensional Hydrogen Atom
Applying the following set of operators in Equation (15), we obtain the modified Schrodinger equation as where is the potential, is the noncommutativity parameter in parabolic coordinates, , , and . It is crucial to note that the order of is , due to Equation (6).
Performing the change of variable , we finally obtain
It should be noted that Equation (20) is a special case of Kummer’s differential equation [26, 27]. Therefore, its solution can be written in terms of a linear combination of Kummer’s confluent hypergeometric functions [26, 27]. However, in this paper, the solution will be written in terms of Laguerre polynomials; that is why we note that Equation (20) has the following form: which is the Laguerre differential equation. If is an integer , the solution of Laguerre’s equation is given by Laguerre polynomials . It could also be noted that Laguerre polynomials can be defined through confluent hypergeometric functions [26, 27]. Finally, we obtain the solution
The energy levels can be determined from with . Using the condition given in Equation (23), the spectrum can be calculated as where . Solving Equation (24), we obtain the spectrum of noncommutative two-dimensional hydrogen atom
Considering and using the binomial series , we calculate the following approximation:
Notice that in the limit , we obtain the same result of the usual two-dimensional hydrogen atom given in the literature [16, 17]. Notice also that the first-order term in does not contribute to the energy of this system.
Then, the noncommutative correction, , for the energy is given by
The result given in Equation (27) can be used to estimate the bound on the noncommutativity parameter . The experimental value for frequency transition in the hydrogen atom is . The uncertainty in this experimental value can fix the upper bound on the parameter . In this sense, the theoretical value for the error in transition , denoted by , is given by
Using the fact that in the two-dimensional case, the energy is four times the energy of the three-dimensional case, then , where is the Planck constant. So, we have
Performing all the calculations, we obtain . In this case, the bound of the noncommutativity parameter is . Using the definition of the length scale factor, , that is, the length scale where the noncommutative effects of the space will be relevant, we found for the considered case , which is about one hundred times smaller than the proton radius .
It is interesting to note that the noncommutative relationship depends on the coordinates adopted, given that the dimension of the noncommutative parameter itself changes with such a choice. Thus, we denote by the parameter with dimension of squared distance, which establishes a noncommutative relationship between the Cartesian coordinates. On the other hand, denotes the parameter that establishes this same relationship for parabolic coordinates. There is certainly a nontrivial algebraic relationship between these parameters. But as we are already in an approximate regime, we are guided by dimensional analysis to establish a limit for . Such a limit will be a good approximation for the real order of magnitude of this parameter.
5. Concluding Remarks
Using the Levi-Civita mapping, we treated the nontrivial problem of the noncommutative hydrogen atom. As a result, we obtain the solution of the Schrodinger equation for this system and calculate the energy levels. Using the spectrum obtained and experimental data, we estimated the noncommutativity parameter , which has an order of magnitude of , and the noncommutative effects will be relevant to a length smaller than . This result has the same order of magnitude obtained in , in which the authors studied the hydrogen atom in rotationally invariant noncommutative space. In this way, they founded the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity. The upper bound of the parameter of noncommutativity is estimated on the basis of the experimental results for 1s to 2s transition frequence. The calculations for obtaining the corrected energy levels were performed using a perturbative method while in our research, the calculations of energy levels are performed exactly. Our results are in agreement with the literature [28, 30], bearing in mind that there is no prediction of energy dependence with a noncommutativity parameter linearly. It is interesting to write the parameter of noncommutativity in natural units in order to compare it with possible experimental data. Therefore, it is necessary to divide by , which results to . It is important to say that the 2D atom is not a real model; it is more of a toy model that enables us to make some conclusions about the system. It is interesting to notice that the model does not allow to verify a transition between states and , because , which appears in the energy, is an odd integer. In the sequence, we intend to analyze the noncommutative three-dimensional hydrogen atom. In addition, in order to obtain better accuracy in estimating of noncommutative parameter, we intend to analyze the noncommutative Zeeman and Stark effects.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was partially supported by CAPES and CNPq of Brazil.
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