Non-inertial eﬀects on a non-relativistic quantum harmonic oscillator in the presence of a screw dislocation L. C. N. Santos1F. M. da Silva2yC. E. Mota3zand V. B. Bezerra1x

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Non-inertial eﬀects on a non-relativistic quantum harmonic

oscillator in the presence of a screw dislocation

L. C. N. Santos,1, ∗F. M. da Silva,2, †C. E. Mota,3, ‡and V. B. Bezerra1, §

1Departamento de Física, CCEN-Universidade Federal da Paraíba; C.P. 5008,

CEP 58.051-970, João Pessoa, PB, Brazil

2Núcleo Cosmo–ufes & Departamento de Física,

Universidade Federal do Espírito Santo, Av. Fernando Ferrari,

540, CEP 29.075-910, Vitória, ES, Brazil

3Departamento de Física, CFM-Universidade Federal de Santa Catarina; C.P. 476,

CEP 88.040-900, Florianópolis, SC, Brazil

Abstract

We investigate non-inertial eﬀects induced by a rotating frame on a non-relativistic quantum

harmonic oscillator as well as of the topology associated to a screw dislocation, which corresponds

to a distortion of a vertical line into a vertical spiral. To do this, we obtain the analytical solutions of

the time-independent Schrödinger equation for this harmonic oscillator potential in this background.

The expressions for the energy spectrum are obtained and the solutions for four quantum states,

namely n= 0,1,2and 3, are analysed. Our results show that the presence of the topological defect

(screw dislocation) as well the fact that we are analysing the system from the point of view of

a rotating frame, changes the solutions of Schrödinger equation and the corresponding spectrum.

Now these quantities depend on the angular velocity of the rotating frame, Ω, and also on the

parameter β, which codiﬁes the presence of the screw dislocation. Particularly, with respect to the

energy spectrum of the system the changing is such that when Ωincreases, the energy can increase

or decrease depending on the values we assign to the eigenvalues of the angular and linear momenta.

Additionally, we observe that the values of the parameter βthat characterizes the screw dislocation

causes a shift in the energy spectrum.

PACS numbers: 03.65.Ge, 03.65.–w, 03.65.Pm, 04.20.Gz

∗luis.santos@ufsc.br

†franmdasilva@gmail.com

‡clesio200915@hotmail.com

§valdirbarbosa.bezerra@gmail.com

arXiv:2210.02559v1 [quant-ph] 4 Oct 2022

I. INTRODUCTION

Like cracks that form when water freezes out into ice, topological defects as cosmic strings

[1, 2] have an analogue interpretation which is related to phase transitions occurred in the

early universe [3]. It is believed that these defects can modify the trajectory of test particles

such that the energy spectrum of quantum systems can carry a dependence on the defect

parameters, which characterize the spacetimes associated to them [4–10]. In this context,

particles under the inﬂuence of potentials that have been studied in relativistic and non-

relativistic quantum mechanics in the ﬂat spacetime with a trivial topology, can be analysed

in a scenario which corresponds to spacetimes with diﬀerent geometries as well as topologies,

as for example, the geometries associated to disclinations and dislocations [11–13]. Indeed,

this is a fruitful ﬁeld of research and applications have been found in several works in the

background spacetime generated to a cosmic string [6–10, 14]. In which concerns similar

research in the geometry associated to dislocations, several works have been performed.

Among these we can mention [15–17]. Particularly, in [16] the authors considered the eﬀects

of the topology of the same dislocation we considering, also on an harmonic oscillator, in

the framework of non-relativistic quantum mechanics, but without taking into account the

angular momentum contribution to the energy spectrum arising from a system as viewed

by an observer placed in a rotating frame. In the literature, the combined eﬀects of the

topology associated with any type of topological defect and of the rotation of a frame was

already considered [18–24]. In this work, we also consider these combined eﬀects, taking

as a system to be analysed the same considered in [16], by adding the non-inertial eﬀects

arising from a rotating frame with angular velocity Ω.

Following this line of research, some well-known systems were analysed by taking into

account the Klein-Gordon and Dirac equations in a generalized spacetime, providing useful

insights about diﬀerent systems [11]. The eﬀects of dislocations on quantum systems have

been extensively studied, by taking into account several systems. For instance, a system

involving scattering of one electron by a screw dislocation with an internal magnetic ﬂux,

was considered in [15], one and two-dimensional quantum rings in the presence of a single

screw dislocation [25] and quantum rings under non-inertial eﬀects [21].

On the other hand, non-inertial eﬀects due to rotating frames is another subject of study

with several interesting results. The inclusion of these eﬀects in relativistic wave equations

can be done through a coordinate transformation. The systems analysed with this scheme

include the Dirac oscillator [19], solutions of the DKP equation [26], and the relation be-

tween the ground state energy of a scalar ﬁeld and non-inertial eﬀects [18, 20]. In the

non-relativistic case, the inﬂuence of a rotating frame is codiﬁed through a redeﬁnition of

Hamiltonian by assuming a coupling with the angular momentum operator, which is equiv-

alent to introduce a rotating frame. Therefore, it is worth investigating the non-relativistic

wave equations in the geometry associated with a screw dislocation, which corresponds to a

distortion of a vertical line into a vertical spiral taking into account as well the non-inertial

eﬀects due to a rotating frame. In this work, we obtain the solutions of the Schrödinger

equation with a harmonic oscillator potential in the background under consideration, viewed

from a rotating frame and discuss in details how to express these solutions in terms of well-

known functions. The energy spectrum is also obtained and some particular results are

discussed.

The paper is organized as follow: In Section II we describe the line element associated

to a screw dislocation and how to include non-inertial eﬀects which is done by taking into

account an additional term in the Hamiltonian, by assuming that this procedure is equivalent

to introduce a rotating frame. In Section III, we obtain the time-independent Schrödinger

equation with a harmonic oscillator potential in the background under consideration. In

Section IV, we ﬁnd the solutions in terms of the conﬂuent Heun function. The energy

spectrum of the quantum oscillator is also obtained. Finally, in Section V the conclusions

and some remarks are presented.

II. THE GEOMETRY OF A SCREW DISLOCATION AND EFFECT DUE TO A

ROTATING FRAME

We are committed to investigate the inﬂuence of topological defects on non-relativistic

particles that experience a scalar potential Vs, so that the general expression for the Hamil-

tonian operator for this system, using the units ~= 1, is given by

H0=−1

pg(3) ∂i(pg(3)gij ∂j) + Vs,(1)

where g(3) is the determinant of gij and gij is the contravariant metric tensor, the inverse of

gij . Note that the indices iand jrepresent only the spatial coordinates.

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Non-inertialeectsonanon-relativisticquantumharmonicoscillatorinthepresenceofascrewdislocationL.C.N.Santos,1,F.M.daSilva,2,yC.E.Mota,3,zandV.B.Bezerra1,x1DepartamentodeFísica,CCEN-UniversidadeFederaldaParaíba;C.P.5008,CEP58.051-970,JoãoPessoa,PB,Brazil2NúcleoCosmoufes&DepartamentodeFísica,Universi...

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Non-inertial eﬀects on a non-relativistic quantum harmonic oscillator in the presence of a screw dislocation L. C. N. Santos1F. M. da Silva2yC. E. Mota3zand V. B. Bezerra1x

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