Non-relativistic quantum effects on the harmonic oscillator in a spacetime with a distortion of a vertical line into a vertical spiral
Eur. Phys. J. C (2019) 79:657
https://doi.org/10.1140/epjc/s10052-019-7166-9
Regular Article - Theoretical Physics
Non-relativistic quantum effects on the harmonic oscillator
in a spacetime with a distortion of a vertical line into a vertical
spiral
W. C. F. da Silva1, K. Bakke1,a , R. L. L. Vitória2,b
1 Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, PB 58051-900, Brazil
2 Departamento de Física e Química, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, Vitoria, ES 29060-900, Brazil
Received: 1 April 2019 / Accepted: 23 July 2019 / Published online: 7 August 2019
© The Author(s) 2019
Abstract Non-relativistic quantum effects of the topology
of the spacetime with the distortion of a vertical line into a
vertical spiral on the harmonic oscillator are investigated. By
searching for analytical solutions to the Schrödinger equation
in this topological defect background, it is shown that the
topology of the spacetime modifies the spectrum of energy
of the harmonic oscillator. Besides, it is shown that there
exists an Aharonov–Bohm-type effect for bound states.
1 Introduction
Topological defect spacetimes have brought a great attention in recent decades. The best example of a topological
defect spacetime is the cosmic string spacetime [1–4]. It has
a singularity determined by the curvature concentrated on
its symmetry axis. It is known as a conical singularity [5].
Interesting works [6–17] have explored this topological characteristic of the cosmic string with the purpose of searching
for analogues of the Aharonov–Bohm effect [18,19]. It is
worth clarifying that Peshkin and Tonomura [20] showed
that if a quantum particle is confined to move in a circular
ring of radius R and there is a solenoid (extremely long) of
radius a < R, then, the angular momentum quantum number
is modified by leff = l − e/2π (where is the magnetic
flux through the solenoid and e is the electric charge). In this
way, the spectrum of energy becomes determined by leff even
though no interaction between the particle and the magnetic
field inside the solenoid exists. This influence of the magnetic flux on the energy levels is known as the Aharonov–
Bohm effect for bound states. Hence, an analogue effect of the
Aharonov–Bohm effect that we have mentioned corresponds
to the influence of the topological defect on the spectrum
a e-mail:
b e-mail:
of energy even though no interaction between the quantum
particle and the topological defect exists. Another interest
is the topological defect spacetimes associated with torsion,
since Aharonov–Bohm-type effects have been reported in
the literature. Examples of Aharonov–Bohm-type effects are
given by relativistic quantum particles in the spacetime with
a space-like dislocation [21–25], the chiral conical spacetime
[26–28] and the spacetime with a spiral dislocation [29,30].
Another perspective of searching for Aharonov–Bohmtype effects due to backgrounds of topological defect spacetimes was given in the non-relativistic limit. By dealing with
the non-relativistic limit of the Klein–Gordon equation and
the Dirac equation in topological defect spacetimes, therefore, it has been shown the connection with the elastic theory
in solids [31,32]. Since then, the spatial part of the line element of a topological defect spacetime describes defects in
solids, such as, disclinations and dislocations. In particular,
examples of dislocations are the screw dislocation and the
spiral dislocation [33,34]. Quantum effects associated with
a screw dislocation have been studied in quantum rings [35–
37], electrons subject to the deformed Kratzer potential [38],
electron gas in a cylindrical shell [39], and with an electron in
a uniform magnetic field [40–43]. On the other hand, quantum effects associated with a spiral dislocation have been
studied with geometric quantum phases [29] and the harmonic oscillator [44]. Other interesting works that follow
this line of research are [45–49].
In this work, we follow this line of study that searches
for Aharonov–Bohm-type effects in the non-relativistic limit.
We analyse non-relativistic quantum effects on the harmonic
oscillator in the spacetime with the distortion of a vertical line
into a vertical spiral. Then, we show that analytical solutions
to the Schrödinger equation for the harmonic oscillator in this
background can be obtained. Due to the effects of the topology of the spacetime, the spectrum of energy of the harmonic
123
657 Page 2 of 5
Eur. Phys. J. C (2019) 79:657
oscillator is modified. Besides, we discuss the appearance of
an Aharonov–Bohm-type effect for bound states.
This paper is structured as follows: in Sect. 2, we introduce the line element of the spacetime with the distortion of a
vertical line into a vertical spiral. Then, we analyse the topological effects on the harmonic oscillator by showing that
an analogue of the Aharonov–Bohm effect for bound states
exists; in Sect. 3, we present our conclusions.
space coordinates. Henceforth, we work with the units: c = 1
and h̄ = 1.
Thereby, with the line element (1), the time-independent
Schrödinger equation (2) becomes
∂ψ
r
1 ∂ 2ψ
+ 2
Eψ = −
2
2m ∂r 2
r − β ∂r
1
∂
∂ 2
∂ 2ψ
+ 2
−
β
ψ+
2
∂ϕ
∂z
∂z 2
r −β
1
+ mω2 r 2 ψ.
2
2 Spacetime with a distortion of a vertical line into a
vertical spiral
Let us start by considering a generalization of a topological
defect in gravitation. From Ref. [34], we can write the line
element of a spacetime with a distortion of a vertical line into
a vertical spiral as
ds 2 = −c2 dt 2 + dr 2 + r 2 dϕ 2
+2β dϕ dz + dz 2 ,
(1)
where 0 < r < ∞, 0 ≤ ϕ ≤ 2π and −∞ < z < ∞. The
parameter β is a constant that characterizes the torsion field
(dislocation). As mentioned in the introduction, there is a
connection with the elastic theory in solids [31,32] when we
consider the spatial part of the line element of a topological
defect spacetime to describe a defect in solids. As shown
in Ref. [34],
the parameter β is related to the Burger vector
b (β ∝ b). Thus, the spatial part of the line element (1)
describes a kind of screw dislocation, since it has the Burger
vector perpendicular to the plane z = 0. Besides, the value
of the parameter β in a solid is of the order of the interatomic
displacement, hence, it can be defined in the range 0 < β <
1. Therefore, in terms of the elastic theory in solids, this
screw dislocation corresponds to the distortion of a vertical
line into a vertical spiral [34]. Note that it differs from the
screw dislocation worked in Refs. [35–37,40,45].
Since our aim is to search for Aharonov–Bohm-type
effects in the non-relativistic limit. In particular, we are going
to analyse the topological effects of the the distortion of a vertical line into a vertical spiral on the two-dimensional harmonic oscillator. For this purpose, let us follow Refs. [40–
42], then, the time-independent Schrödinger equation for the
harmonic oscillator in (...truncated)
