## Abstract

The behaviour of a neutral particle (atom, molecule) with an induced electric dipole moment in a region with a uniform effective magnetic field under the influence of the Kratzer potential (Kratzer 1920 *Z. Phys.* **3**, 289–307. (doi:10.1007/BF01327754)), and rotating effects is analysed. It is shown that the degeneracy of the Landau-type levels is broken and the angular frequency of the system acquires a new contribution that stems from the rotation effects. Moreover, in the search for bound state solutions, it is shown that the possible values of this angular frequency of the system are determined by the quantum numbers associated with the radial modes and the angular momentum, the angular velocity of the rotating frame and by the parameters associated with the Kratzer potential.

## 1. Introduction

Rotating effects have attracted a great deal of discussion in the literature, for instance, Landau & Lifshitz [1] pointed out a geometrical point of view where a coordinate transformation from a system at rest to a uniformly rotating frame yields an intriguing behaviour: the line element of the Minkowski space–time becomes not well defined for large distances, i.e. the coordinate system becomes singular at large distances. This singular behaviour at large distances is associated with the velocity of the particle would be greater than the velocity of the light. Recently, this spatial constraint has been explored in studies of the Dirac oscillator [2] and the Landau quantization for neutral particles [3]. Another context in which rotating effects have been investigated is in interferometry, where quantum effects associated with the geometric quantum phases have been observed. Examples of these quantum effects are the Sagnac effect [4–6], the Mashhoon effect [7] and the Aharonov–Carmi geometric phase [8]. In condensed matter physics, rotating effects have been investigated in Bose–Einstein condensation in ultracold diluted atomic gases [9], the quantum Hall effect [10], the Aharonov–Carmi geometric phase in *C*_{60} molecules [11,12], in quantum rings [13–15] and spintronics [16–18]. It is worth mentioning works with Dirac fields [19], scalar fields [20,21] and based on the coupling between the angular momentum and the angular velocity of the rotating frame [22–24].

The aim of this work is to investigate rotating effects on an atom (molecule) with an induced electric dipole moment subject to the Kratzer potential [25–27] in a region with a uniform effective magnetic field. In recent years, the quantum dynamics of a neutral particle with no permanent electric dipole moment in a region with a uniform effective magnetic field has been investigated in reference [28], where it is shown that an analogue of the Landau quantization [29] can be observed and it has interests in cold atom technology [30–37]. In [38], the Landau-type quantization for an atom (molecule) with an induced electric dipole moment was investigated in a two-dimensional quantum ring. In this work, by searching for bound state solutions, we show that the angular frequency of the system differs from the analogue of the cyclotron frequency obtained in reference [28], and the possible values of this angular frequency of the system are determined by the quantum numbers associated with the radial modes and the angular momentum, the angular velocity of the rotating frame and by the parameters associated with the Kratzer potential [25–27].

The structure of this paper is: in §2, we make a brief introduction of the quantum dynamics for a moving atom (molecule) with an induced electric dipole moment and the Landau quantization associated with it; thus, we analyse this Landau-type system subject to the Kratzer potential [25–27] in a rotating frame; in §3, we present our conclusions.

## 2. Rotating effects

In this section, we investigate the rotating effects on an atom (molecule) with an induced electric dipole moment that interacts with external fields subject to the Kratzer potential [25–27]. Let us begin by introducing the Landau system associated with an atom (molecule) with no permanent electric dipole moment. First of all, in the rest frame of the neutral particle or in the laboratory frame, the electric dipole moment of a moving neutral particle can be considered to be proportional to the electric field: ** d**=

*α*

**, where**

*E**α*is the dielectric polarizability of the atom (molecule) [39,40]. On the other hand, if the neutral particle is moving with a velocity

**(**

*v**v*≪

*c*), then it interacts with an electric field

**′ determined by the Lorentz transformation. By applying the Lorentz transformation of the electromagnetic field up to terms of order**

*E***′ must be replaced with**

*E***′=**

*E***+**

*E***×**

*v***, the fields**

*B***and**

*E***correspond to the electric and magnetic fields in the laboratory frame, respectively [40]. Hence, the dielectric polarizability of the atom (molecule) can be written as**

*B***=**

*d**α*(

**+**

*E***×**

*v***) (SI units). Thereby, the Lagrangian of the system must be written in terms of the electric**

*B***′=**

*E***+**

*E***×**

*v***as**

*B**V*is a scalar potential,

*m*=

*M*+

*αB*

^{2}is the effective mass of the system and

*M*is the mass of the neutral particle [39]. Let us work by assuming that

*B*

^{2}=

*constant*, hence, the Hamiltonian operator that describes this system is [28,39,41,42,] (with the units

**and**

*E***are the electric and magnetic fields in the laboratory frame, respectively. According to reference [43], the term**

*B**αE*

^{2}given in equation (2.1) is very small compared with the kinetic energy of the atoms, therefore, we can neglect it without loss of generality from now on.

Furthermore, in reference [28], it is shown that a field configuration of crossed magnetic and electric fields can give rise to an analogue of the Landau quantization [29] for a neutral particle (atom, molecule) with an induced electric dipole moment, where this field configuration is given by
*χ* is a constant related to the uniform volume charge density, *B*_{0} is a constant and *z*-direction. It is well known in the literature that the Landau quantization [29] takes place when the motion of an electrically charged particle in a plane perpendicular to a uniform magnetic field acquires distinct orbits, and the energy levels of this system become discrete and infinitely degenerate. It is important in studies of two-dimensional surfaces [44–46], the quantum Hall effect [47] and Bose–Einstein condensation [48,49]. With this field configuration (2.3), thus, we have an effective vector potential given by *B*_{eff}. It is worth pointing out that a field configuration of crossed electric and magnetic fields has attracted interest in studies of the hydrogen atom [50–55], large electric dipole moments [56], atoms and molecules in strong magnetic field [57–59] and the quasi-Landau behaviour in atomic systems [60,61].

Let us now consider the Landau-type system for an atom with no permanent electric dipole moment to be subject to the Kratzer potential [25–27]:
*D* and *a* are constants. It has attracted great interest in studies of molecules [62–64]. Further, if this system rotates with a constant angular velocity *μ*=2*Da* and *τ*^{2}=*Da*^{2}.

Note that the Hamiltonian operator of the right-hand side of equation (2.6) commutes with the operators *l*=0,±1,±2,…,*k* is a constant and *f*(*ρ*) is a function of the radial coordinate. Henceforth, we assume that *k*=0 in order to reduce the system to a planar system. By substituting this particular solution into equation (2.6) and by performing a change of variables given by *ω* was defined in reference [28] as being the cyclotron frequency of the Landau quantization associated with an atom with an induced electric dipole moment.

We proceed with the analysis of the asymptotic behaviour of the possible solutions to equation (2.7). The asymptotic behaviour is determined for *f*(*r*) in terms of an unknown function *H*(*r*) as follows [71–74]
*H*(*r*) is a solution to the following second-order differential equation
*ν*=*β*/*mϖ*−2−2|*γ*|. The second-order differential equation (2.10) is called the biconfluent Heun function [74], and the function *H*(*r*) is also the biconfluent Heun function [74]: *H*(*r*)=*H*_{B}(2|*γ*|,0,*β*/*mϖ*,2*ϑ*,−*r*).

Henceforth, let us use the Frobenius method [75] in order to write the solution to equation (2.10) as a power series expansion around the origin: *a*_{1}=−(*ϑ*/(1+2|*γ*|))*a*_{0}.

Let us start with *a*_{0}=1, then, from equation (2.11), we can obtain other coefficients of the power series expansion *a*_{1}, *a*_{2} and *a*_{3} are given by

We focus on achieving bound state solutions, therefore, we need to impose that the biconfluent Heun series becomes a polynomial of degree *n*. In this way, we guarantee that the function *f*(*r*) becomes well behaved at the origin and vanishes at *n* by imposing that [42,71,76]
*n*=1,2,3,…. By analysing the condition *ν*=2*n*, we obtain a general expression for the energy levels
*n*=1,2,3,… is the quantum number associated with the radial modes, *l*=0,±1,±2,… is the angular momentum quantum number and *ω* is called the cyclotron frequency of the Landau-type system [28] which is defined in equation (2.8). In particular, we have in equation (2.14) the coupling between the angular momentum quantum number *l* and the angular velocity *Ω* that corresponds to the Page–Werner *et al.* term [22–24].

Next, let us analyse the condition *a*_{n+1}=0 given in equation (2.13). For this purpose, let us consider the cyclotron frequency *ω* [28] that can be adjusted in such a way that the condition *a*_{n+1}=0 can be satisfied. This is possible, because we can adjust either the intensity of the magnetic field *B*_{0} or the intensity of the electric field through the parameter *χ* associated with the uniform volume charge density [42]. With this assumption, we have that both conditions imposed in equation (2.13) are satisfied and a polynomial solution to the function *H*(*r*) is obtained. As an example, let us take *n*=1 and label *ω*=*ω*_{n,l}. For *n*=1, we have the ground state of the system, then *a*_{n+1}=*a*_{2}=0, and thus the possible values of the cyclotron frequency associated with the ground state of the system are
*ω*_{1,l} given in equation (2.15) yields *ϖ*>0, and thus the asymptotic behaviour of the radial wave function when

Hence, this example shows us that only specific values of the angular frequency *ω* are allowed and depend on the quantum numbers {*n*,*l*} of the system and the angular velocity of the rotating frame. From the quantum mechanics point of view, this relation of the angular frequency *ω* to the quantum numbers of the system {*n*,*l*} and the angular velocity of the rotating frame is a quantum effect that stems from the influence of the rotating effects and the Kratzer potential on the Landau-type quantization associated with the neutral particle (atom, molecule) with an induced electric dipole moment. Observe that, by substituting (2.15) into equation (2.14), we have

Finally, let us rewrite the energy levels (2.14) in a general form as

Hence, the energy levels (2.18) are obtained from the effects of the Kratzer potential and rotating effects on the Landau-type system for a neutral particle (atom, molecule) with an induced electric dipole moment. Note that the energy levels are modified in contrast to that obtained in reference [28] for the Landau quantization, where the ground state of the system becomes determined by the quantum number *n*=1 instead of the quantum number *n*=0, and the degeneracy of the Landau levels is broken as we can see in equations (2.14)–(2.16). By comparing with the cyclotron frequency of the Landau-type quantization *ω*=*αχB*_{0}/*m* given in reference [28], we have an angular frequency given by *ω* are allowed in order that a polynomial solution to the function *H*(*r*) can be obtained, where the allowed values depend on the quantum numbers {*n*,*l*}, the angular velocity of the rotating frame and the parameters associated with the Kratzer potential as we can see in equation (2.15) for the ground state of the system. Moreover, observe that the last term of equation (2.18) corresponds to the coupling between the angular momentum quantum number *l* and the angular velocity *Ω* which is called as the Page–Werner *et al.* term [22–24]. Further, by taking

## 3. Conclusion

We have discussed the effects of rotation and the Kratzer potential [25–27] on the Landau quantization associated with neutral particle (atom, molecule) with a induced electric dipole moment. We have seen that the angular frequency of the system acquires a new contribution that stems from the rotation effects. Besides, the Landau-type levels obtained in reference [28] are modified, where the degeneracy of the Landau-type levels is broken and the ground state of the system becomes determined by the quantum number *n*=1 instead of the quantum number *n*=0. Moreover, a quantum effect characterized by the dependence of the cyclotron frequency of the Landau-type quantization on the quantum numbers of the system and the angular velocity of the rotating frame is obtained, which means that only some specific values of the cyclotron frequency *ω* are allowed in order to achieve bound state solutions. A new contribution to the analogue of the Landau levels arises from the coupling between the angular velocity of the rotating frame and the angular momentum, which is called the Page–Werner *et al.* term [22–24]. Finally, in the limit

## Authors' contributions

A.B.O. and K.B. conceived the mathematical model, interpreted the results and wrote the paper. A.B.O. carried out most of the calculations in consultation with K.B. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

A.B.O. thanks CAPES and K.B. thanks CNPq (grant no. 300860/2013-7) for financial support.

- Received December 16, 2015.
- Accepted May 10, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.