# Geometric effects on the electronic structure of curved nanotubes and curved
graphene: the case of the helix, catenary, helicoid, and catenoid

J.D.M. de Lima  0000-0002-4296-6001 E. Gomes  0000-0002-3812-9331 F.F. da
Silva Filho  0000-0003-3832-787X F. Moraes  0000-0001-7045-054X
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
Rural de Pernambuco, Recife, PE, 52171-900, Brazil  R. Teixeira
0000-0003-0314-8142 Departamento de Matemática, Universidade Federal Rural de
Pernambuco, Recife, PE, 52171-900, Brazil

###### Abstract

Since electrons in a ballistic regime perceive a carbon nanotube or a graphene
layer structure as a continuous medium, we can use the study of the quantum
dynamics of one electron constrained to a curve or surface to obtain a
qualitative description of the conduction electrons’ behavior. The confinement
process of a quantum particle to a curve or surface leads us, in the so-called
“confining potential formalism” (CPF), to a geometry-induced potential (GIP)
in the effective Schrödinger equation. With these considerations, this work
aims to study in detail the consequences of constraining a quantum particle to
a helix, catenary, helicoid, or catenoid, exploring the relations between
these curves and surfaces using differential geometry. Initially, we use the
variational method to estimate the energy of the particle in its ground state,
and thus, we obtain better approximations with the use of the confluent Heun
function through numerical calculations. Thus, we conclude that a quantum
particle constrained to an infinite helix has its angular momentum quantized
due to the geometry of the curve, while in the cases of the catenary,
helicoid, and catenoid the particle can be found either in a single bound
state or in excited states which constitute a continuous energy band.
Additionally, we propose measurements of physical observables capable of
discriminating the topologies of the studied surfaces, in the context of
topological metrology.

Geometry-induced potential, Constrained quantum particle, Low-dimensional
carbon structures

††preprint: APS/123-QED

##  I Introduction

Due to the great advance of materials science, it became possible to
synthesize increasingly smaller carbon-based structures. Along with the
discovery of graphene, the first truly two-dimensional system, several
theoretical techniques have been proposed to describe the quantum dynamics of
its charge carriers. Many other carbon nanostructures have also been
synthesized with different shapes, such as fullerenes, nanotubes, nanocones,
and structures with non-trivial geometries [1]. In this sense, it is necessary
to understand the effects of the geometry on the quantum dynamics of a
particle in such curved structures.

One of the first formal approaches addressed to solve this problem was
proposed by H. Jensen and H. Koppe in 1971, which studied quantum particles
constrained to move between two parallel surfaces in the limit where the
inter-surface distance goes to zero [2]. In 1981, R.C.T. da Costa considered
that the constraint procedure can be realized through a smooth potential well
[3, 4]. In both cases, it was showed that it leads to a geometry-induced
potential (GIP) which acts upon the dynamics. If the particle is constrained
to a space curve, the GIP depends on its curvature. On the other hand, when
dealing with curved surfaces the GIP dependence is on its mean and Gaussian
curvatures. This is known [5] as “confining potential formalism” (CPF).
Experimental verification of geometric effects due to confinement were
verified first in photonic topological crystals [6] and in a peanut-shaped
$C_{60}$ polymer [7].

As the electronic transport in graphene is mesoscopically ballistic, such
electrons perceive the structure as a continuous medium [8]. This implies that
the study of the quantum dynamics of an electron confined to a surface (or a
curve) leads to a qualitative description of the behavior of electrons in the
ballistic regime in a graphene layer (or in a carbon nanotube) with
topological defects, such that the geometry has an important role in the
dynamics [9].

The study of curved structures can be essential for the understanding of their
properties and possible applications, especially for electronic devices. For
example, Ref. [10] suggests that p-n junctions can be induced by curvature in
a graphene bilayer, whose geometry is related to a catenoid [11]. In Ref. [12]
it was shown that a helicoidal ribbon can induce, due to its curvature, a
charge separation which results in a electrical field crossing the molecule.
Furthermore, the catenoid can be considered in the study of wormholes [13, 14]
and as an approximation to the geometries of ion-channel membrane proteins
[15, 16]. Similarly, the helix and the helicoid can also be considered as
approximations to the geometries of $\beta$-sheet proteins, DNA molecules [12,
15, 17], and helicenes [18], for example.

The description of regular curves and surfaces is very well established by
differential geometry. Two of the simplest regular curves that we can study in
this context are the helix and the catenary. The surface formed by
continuously varying the radius of the helix is called helicoid, while the
surface of revolution associated with the catenary is called catenoid. As the
helicoid and the catenoid are locally isometric and both of them are minimal
surfaces, their properties have been reported in detail by several authors
[19, 20, 21].

The confinement of a quantum particle to a helicoid or catenoid with the CPF
has been the subject of some studies. V. Atanasov, R. Dandoloff, and A. Saxena
applied the CPF to the helicoid in a work published in 2009, where they showed
that the constrained particle is subject to an effective radial potential that
can be attractive or repulsive, depending on the angular momentum [12]. For
the catenoid, the CPF was used in 2010 by R. Dandoloff, A. Saxena, and B.
Jensen to understand the GIP in a two-dimensional section of a wormhole [22].
Recently, in 2020, J.E.G. Silva et al. used the electron confinement to the
catenoid to study the electronic properties of bilayer graphene connected by a
bridge [11]. Also in 2020, M.C.R. Ribeiro Jr. et al. studied the confinement
problem on the helicoid considering the presence of an harmonic oscillator
potential and anisotropic mass [23].

With the above motivations, we study the Schrödinger equation for a quantum
particle constrained to a helix, catenary, helicoid, or catenoid, using the
CPF. We intend here to understand the particularities of the confinement
problem in each one of them. In section II we present a brief review of the
differential geometry of curves and surfaces, while the CPF is described in
section III. In section IV we solve the problem for the helix, and we
introduce the problems for the catenary, helicoid, and catenoid (solving the
angular part for the surfaces). In section V we estimate the ground state
energy using the variational method for the cases of the catenary, helicoid,
and catenoid, besides we present solutions in terms of the confluent Heun
functions. In section VI we discuss the solutions for the ground state.
Finally, topological aspects are discussed in section VII and our conclusions
and perspectives for future works are presented in Section VIII.

##  II Overview of differential geometry

In this section we do a brief presentation of some concepts of differential
geometry that are necessary for the understanding of this text. More details
can be found in Ref. [21]. The procedure that we will follow in this section
is entirely focused on developing a mathematical support to be used in the
discussion about the CPF.

###  II.1 Curves

Let $\boldsymbol{\alpha}$ be a regular curve in $\mathbb{R}^{3}$ whose
parametrization is given by

$\boldsymbol{\alpha}(t)=\left(x(t),y(t),z(t)\right),$ (1)

where $t$ is a parameter in an open interval $T\subset\mathbb{R}$. Thus, the
arc length of $\boldsymbol{\alpha}$, from some arbitrary $t_{0}\in T$, is
defined by

$s(t)=\int_{t_{0}}^{t}\left|\boldsymbol{\alpha}^{\prime}(\tau)\right|d\tau.$
(2)

When it is possible to obtain $s(t)$ and its inverse, $s^{-1}(t)$, we can find
an arc length parametrization by making
$\boldsymbol{\gamma}(s)=\boldsymbol{\alpha}\circ s^{-1}$, which is convenient
since $|\boldsymbol{\gamma}^{\prime}(s)|=1,\,\forall\,s$. In this
parametrization, the curvature of the curve is given by

$\displaystyle\kappa(s)=\left|\boldsymbol{\gamma}^{\prime\prime}(s)\right|,$
(3)

where $\kappa(s)$ is always positive [21, 19].

The helix (see Fig. 1) with radius $b>0$ and pitch (vertical distance between
each loop) given by $2\pi|a|$, for $a\in\mathbb{R}$, is a regular space curve
parametrized by

$\boldsymbol{\alpha}(\phi)=\left(b\cos(\phi),b\sin(\phi),a\phi\right),$ (4)

where $\phi\in\mathbb{R}$ is the curve parameter. Consequently, its arc length
is $s=\sqrt{a^{2}+b^{2}}\phi$ and, after obtaining $\boldsymbol{\gamma}(s)$,
we get from equation (3) the constant curvature

$\kappa(s)=\dfrac{b}{a^{2}+b^{2}}.$ (5)

Figure 1: Three-dimensional representation of the helix.

The catenary (see Fig. 2) is a regular plane curve parametrized by

$\boldsymbol{\alpha}(v)=(v,a\cosh(v/a)),$ (6)

for $v\in\mathbb{R}$. Here, the constant $a>0$ is associated with the catenary
opening. From Eq. (2) we get $s=a\sinh(v/a)$, such that (6) gives us

$\boldsymbol{\gamma}(s)=\left(a\operatorname{arcsinh}(s/a),\sqrt{a^{2}+s^{2}}\right),$
(7)

and

$\kappa(s)=\dfrac{a}{a^{2}+s^{2}}.$ (8)

Figure 2: Two-dimensional representation of a catenary.

###  II.2 Surfaces

Let $S$ be a regular surface embedded in $\mathbb{R}^{3}$ whose
parametrization is

$\mathbf{r}(q_{1},q_{2})=\left(x(q_{1},q_{2}),y(q_{1},q_{2}),z(q_{1},q_{2})\right),$
(9)

where $q_{1}$ and $q_{2}$ are parameters in an open set
$U\subset\mathbb{R}^{2}$. According to Ref. [21], the regularity condition
implies that $\mathbf{r}_{1}=\dfrac{\partial\mathbf{r}}{\partial q_{1}}$ and
$\mathbf{r}_{2}=\dfrac{\partial\mathbf{r}}{\partial q_{2}}$ are linearly
independent, i.e., $\mathbf{r}_{1}\times\mathbf{r}_{2}\neq 0$ at all points of
$S$, which allows us to define the unit vector normal to the surface,
$\mathbf{N}=\dfrac{\mathbf{r}_{1}\times\mathbf{r}_{2}}{|\mathbf{r}_{1}\times\mathbf{r}_{2}|}$.

The metric tensor of the surface $S$ is given by

$\mathbf{g}=\begin{pmatrix}g_{11}&g_{12}\\\ g_{21}&g_{22}\end{pmatrix},$ (10)

where $g_{ij}=\mathbf{r}_{i}\cdot\mathbf{r}_{j}$ (for $i,j=1,2$) are the
coefficients of the first fundamental form. Thus, denoting by $g^{ij}$ the
elements of the inverse matrix $\mathbf{g}^{-1}$, we get

$\mathbf{g}^{-1}=\begin{pmatrix}g^{11}&g^{12}\\\
g^{21}&g^{22}\end{pmatrix}=\dfrac{\operatorname{adj}(\mathbf{g})}{\det(\mathbf{g})}=\dfrac{1}{g}\begin{pmatrix}g_{22}&-g_{12}\\\
-g_{21}&g_{11}\end{pmatrix},$ (11)

for $g=\det(\mathbf{g})$, the determinant of the metric tensor. Further, for
$\mathbf{r}_{ij}=\dfrac{\partial^{2}\mathbf{r}}{\partial q_{i}\partial
q_{j}}$, the coefficients of the second fundamental form are defined as
$h_{ij}=\mathbf{N}\cdot\mathbf{r}_{ij}$. Thus, the mean and Gaussian
curvatures are, respectively,

$\displaystyle M$
$\displaystyle=\dfrac{1}{2}\dfrac{g_{11}h_{22}-2g_{12}h_{12}+g_{22}h_{11}}{g_{11}g_{22}-g_{12}^{2}},$
(12) $\displaystyle K$
$\displaystyle=\dfrac{h_{11}h_{22}-h_{12}^{2}}{g_{11}g_{22}-g_{12}^{2}}.$ (13)

For the helicoid, we consider $q_{1}=\phi\in\mathbb{R}$ and
$q_{2}=u\in\mathbb{R}$, which are coordinates on $S$. The parametrization of
the helicoid (see Fig. 3) is given by

$\mathbf{r}(\phi,u)=\left(u\cos(\phi),u\sin(\phi),a\phi\right),$ (14)

where $a$ is a real constant. As we can see, for each fixed $u$ we get a
different helix. From (14), the coefficients of the first fundamental form are
$g_{11}=a^{2}+u^{2}$, $g_{12}=g_{21}=0$, and $g_{22}=1$, which give us the
metric

$ds^{2}=(a^{2}+u^{2})d\phi^{2}+du^{2}.$ (15)

In the same way, the coefficients of the second fundamental form are
$h_{11}=0$, $h_{12}=h_{21}=\dfrac{a}{\sqrt{a^{2}+u^{2}}}$, and $h_{22}=0$.
Thus, from Eqs. (12) and (13), we get

$M=0\quad\text{and}\quad K=-\dfrac{a^{2}}{(a^{2}+u^{2})^{2}}.$ (16)

Figure 3: Three-dimensional graph of the helicoid with $0<\phi<2\pi$.

The catenoid (see Fig. 4) is a surface generated by the revolution of a
catenary. Making $q_{1}=\phi\in(0,2\pi)$ and $q_{2}=u\in\mathbb{R}$, a
parametrization for the catenoid is given by

$\displaystyle\mathbf{r}(\phi,u)=\Big{(}\sqrt{a^{2}+u^{2}}\cos(\phi),\sqrt{a^{2}+u^{2}}\sin(\phi),$
(17) $\displaystyle a\operatorname{arcsinh}(u/a$ $\displaystyle)\Big{)},$

where $a$, the “throat radius”, is a positive constant. Thus, from (17), we
get $g_{11}=a^{2}+u^{2}$, $g_{12}=g_{21}=0$, and $g_{22}=1$, which give us the
same metric in Eq. (15). The coefficients of the second fundamental form are
$h_{11}=-a$, $h_{12}=h_{21}=0$, and $h_{22}=\dfrac{a}{a^{2}+u^{2}}$. Since the
helicoid and the catenoid have the same coefficients of the first fundamental
form in these parametrizations, they are locally isometric. In other words, it
is possible to deform a catenoid onto a helicoid with $0<\phi<2\pi$ while
maintaining the same metric. It follows from Gauss’s Theorema Egregium that
they must have the same Gaussian curvature. In fact, from Eqs. (12) and (13),
we get for the catenoid the same mean and Gaussian curvatures of the helicoid,
which are given by Eqs. (16) (cf. [21, 19]). The regions of same color in
Figs. 3 and 4 indicate corresponding regions on the respective surfaces. In
other words, the colored helicoid of Fig. 3 becomes the catenoid of Fig. 4
under a suitable deformation.

Figure 4: Three-dimensional graph of the catenoid. The color indicates the
corresponding regions of the helicoid in Fig. 3.

##  III Confining potential formalism

In order to maintain a particle constrained to a surface $S$, it is proposed
that the particle is under the action of forces orthogonal to $S$ in all its
points, such that the “spreading” of the wave function in the transverse
direction is “squeezed” [2, 3, 4]. Therefore, the proposal consists of
considering a potential that is always constant on $S$ but increases sharply
in the normal direction, in such a way that the confinement occurs at the
limit where an infinite potential keeps the particle bound to $S$ [24, 25]. An
analogous procedure is done for curves. Thus, a GIP appears in the effective
Schrödinger equation of a particle constrained to a curve or to a surface as a
consequence of the confinement.

###  III.1 Particle constrained to a curve

Let $\boldsymbol{\gamma}$ be a regular curve (embedded in ${\mathbb{R}}^{3}$)
which is parametrized by its arc length, $s$, and has curvature
$\kappa=\kappa(s)$. As shown in Refs. [2, 3], a quantum particle of mass $m$,
constrained to $\boldsymbol{\gamma}$, is subject to a GIP

$\displaystyle V_{C}(s)=-\dfrac{\hbar^{2}}{8m}\kappa^{2}.$ (18)

From Eq. (18) we get the time-independent Schrödinger equation

$-\dfrac{\hbar^{2}}{2m}\dfrac{d^{2}\psi}{ds^{2}}-\dfrac{\hbar^{2}}{8m}\kappa^{2}\psi=E\psi,$
(19)

where $\psi=\psi(s)$ is the wave function and $E$ is the energy of the
particle. The attractive potential indicates that the particle may be found in
bound states due to geometric effects.

###  III.2 Particle constrained to a surface

Let $S$ be a regular surface embedded in ${\mathbb{R}}^{3}$ with parameters
$q_{1}$ and $q_{2}$. Following the CPF, a quantum particle of mass $m$,
constrained to $S$, is subjected to a GIP

$V_{S}(q_{1},q_{2})=-\dfrac{\hbar^{2}}{2m}\left(M^{2}-K\right),$ (20)

where $M$ and $K$ are defined in (12) and (13), respectively [24, 4, 3]. Thus,
from (20) we arrive at the time-independent Schrödinger equation

$-\dfrac{\hbar^{2}}{2m}\Delta_{g}\chi-\dfrac{\hbar^{2}}{2m}\left(M^{2}-K\right)\chi=E\chi,$
(21)

where $\chi=\chi(q_{1},q_{2})$ is the wave function and $\Delta_{g}$ is the
Laplace-Beltrami operator which is given by [26]

$\Delta_{g}\chi=\sum_{i,j=1}^{2}\dfrac{1}{\sqrt{g}}\dfrac{\partial}{\partial
q_{i}}\left(\sqrt{g}g^{ij}\dfrac{\partial\chi}{\partial q_{j}}\right).$ (22)

Isometric surfaces have the same Gaussian curvature but their respective mean
curvatures differ [3, 21], except in the case of minimal surfaces, which have
$M=0$. This is the case of the object of our study in this article, the
helicoid and the catenoid, which have the same GIPs, given by Eq. (20).

##  IV Schrödinger equations

In this Section, we apply Eq. (19) to the helix and the catenary, and Eq. (21)
to the helicoid and the catenoid. In addition, we introduce the respective
boundary conditions for the wave function in both cases and discuss their
implications.

###  IV.1 Helix

From Eqs. (5) and (18), we get the GIP that acts on a particle constrained to
a helix,

$V_{C}=-\dfrac{\hbar^{2}}{8m}\dfrac{b^{2}}{(a^{2}+b^{2})^{2}},$ (23)

which is constant. Thus, recalling that $s=\sqrt{a^{2}+b^{2}}\phi$ and using
Eqs. (19) and (23), we write

$\dfrac{d^{2}\psi}{d\phi^{2}}+l^{2}\psi=0,$ (24)

where

$l^{2}=\dfrac{b^{2}}{4(a^{2}+b^{2})}+\dfrac{2m(a^{2}+b^{2})}{\hbar^{2}}E.$
(25)

Now, let us discuss the boundary conditions for Eq. (24) when we have an
infinite helix ($\phi\in\mathbb{R}$). Let $p$ be an arbitrary point on the
helix and $B(p)\subset\mathbb{R}^{3}$ be a ball of radius $\varepsilon_{B}$
centered on $p$. Within that ball, the particle perceives the surrounding
environment in the same way at any point of the helix, because $B$ is flat,
and the helix has constant curvature and torsion. Thus, the wave function is
distributed in the same way in the surrounding of $\boldsymbol{\alpha}(\phi)$
and $\boldsymbol{\alpha}(\phi+2\pi)$. Therefore, we have the appropriate
boundary condition $\psi(\phi)=\psi(\phi+2\pi)$. This immediately leads to the
quantization of the angular momentum,
$\hat{L}_{\phi}=-i\hbar\dfrac{d}{d\phi}$. In other words, if $\psi$ is a
eigenfunction of $\hat{L}_{\phi}$, we have the eigenvalues $l\hbar$, for
$l=0,\pm 1,\pm 2,\dots$, such that Eq. (24) is retrieved in terms of
$\hat{L}_{\phi}^{2}$. The quantized energy is obtained from Eq. (25):

$E_{l}=\dfrac{l^{2}\hbar^{2}}{2m(a^{2}+b^{2})}-\frac{\hbar^{2}b^{2}}{8m(a^{2}+b^{2})^{2}}.$
(26)

The corresponding propagating modes are degenerate, since they can move either
“up” or “down” on the helix, depending on the sign of $l$. The delta-function
normalized wave functions are given by

$\psi_{l}(\phi)=\frac{1}{\sqrt{2\pi}}e^{il\phi}.$ (27)

For a finite helix with $0<\phi<\phi_{0}$, for some fixed $\phi_{0}$, we have
the boundary conditions $\psi(0)=\psi(\phi_{0})=0$, such that we get
$l=\dfrac{j\pi}{\phi_{0}}$, therefore, the solutions of Eq. (24) are

$\psi_{j}(\phi)=\sqrt{\dfrac{2}{\phi_{0}}}\sin\left(\dfrac{j\pi}{\phi_{0}}\phi\right),$
(28)

where $j=1,2,3,\dots$ is a quantum number. Thus, the possible energies are
given by

$E_{j}=\dfrac{j^{2}\pi^{2}\hbar^{2}}{2m(a^{2}+b^{2}){\phi_{0}}^{2}}-\dfrac{\hbar^{2}b^{2}}{8m(a^{2}+b^{2})^{2}}.$
(29)

Note that both Eqs. (26) and (29) are similar to those obtained for a particle
in an infinite potential well with the geometric contribution from the GIP
(23).

###  IV.2 Catenary

From Eqs. (8) and (18) we get

$V_{C}(s)=-\dfrac{\hbar^{2}}{8m}\dfrac{a^{2}}{(a^{2}+s^{2})^{2}},$ (30)

which is the GIP for a particle constrained to the catenary (see Fig. 5).
Thus, from (19) we get the Schrödinger equation

$-\dfrac{\hbar^{2}}{2m}\dfrac{d^{2}\psi}{ds^{2}}-\dfrac{\hbar^{2}}{8m}\dfrac{a^{2}}{(a^{2}+s^{2})^{2}}\psi=E\psi,$
(31)

whose solutions for bound states obey the boundary conditions:
$\displaystyle{\lim_{s\rightarrow\pm\infty}\psi=0}$. Due to the presence of
the GIP, finding the eigenfunctions and eigenvalues of (31) is not a simple
task. For now, it is more convenient developing the CPF to the helicoid and
catenoid, thus, we will return to the problem of confinement to a catenary in
section V.

Figure 5: GIP for a particle constrained to a catenary considering
$\hbar=2m=a=1$.

###  IV.3 Helicoid and catenoid

Substituting (16) into (20) we can obtain the GIP for both helicoid and
catenoid surfaces,

$V_{S}(u)=-\dfrac{\hbar^{2}}{2m}\dfrac{a^{2}}{(a^{2}+u^{2})^{2}},$ (32)

which is independent of $\phi$ and has the same behavior as the potential in
Fig. 5 (up to the multiplicative factor $1/4$). From Eqs. (10) and (11), we
find $g^{11}=\dfrac{1}{a^{2}+u^{2}},$ $g^{12}=g^{21}=0$, $g^{22}=1$, and
$g=a^{2}+u^{2}$, consequently, the Laplace-Beltrami operator is

$\Delta_{g}=\dfrac{1}{a^{2}+u^{2}}\dfrac{\partial^{2}}{\partial\phi^{2}}+\dfrac{\partial^{2}}{\partial
u^{2}}+\dfrac{u}{a^{2}+u^{2}}\dfrac{\partial}{\partial u},$ (33)

from (22). Using Eqs. (32) and (33) in Eq. (21), we get the Schrödinger
equation

$\begin{split}-\dfrac{\hbar^{2}}{2m}\bigg{(}\dfrac{1}{a^{2}+u^{2}}\dfrac{\partial^{2}\chi}{\partial\phi^{2}}+\dfrac{\partial^{2}\chi}{\partial
u^{2}}+\dfrac{u}{a^{2}+u^{2}}\dfrac{\partial\chi}{\partial u}\bigg{)}&\\\
-\dfrac{\hbar^{2}}{2m}\dfrac{a^{2}}{(a^{2}+u^{2})^{2}}\chi&=E\chi,\end{split}$
(34)

for the wave function $\chi=\chi(\phi,u)$.

With the element of area given by $dA=\sqrt{g}d\phi du$, the probability of
finding the particle in a given region of the surface is

$\begin{split}P_{A}&=\iint|\chi(\phi,u)|^{2}(a^{2}+u^{2})^{1/2}d\phi du\\\
&=\iint|\chi_{g}(\phi,u)|^{2}d\phi du,\end{split}$ (35)

where (see [27, 11]) $\chi_{g}(\phi,u)=(a^{2}+u^{2})^{1/4}\chi(\phi,u)$. Thus,
Eq. (34) becomes

$\begin{split}-\dfrac{\hbar^{2}}{2m}\bigg{(}\dfrac{1}{a^{2}+u^{2}}\dfrac{\partial^{2}\chi_{g}}{\partial\phi^{2}}+\dfrac{\partial^{2}\chi_{g}}{\partial
u^{2}}\bigg{)}&\\\
-\dfrac{\hbar^{2}}{2m}\bigg{(}\dfrac{a^{2}}{4(a^{2}+u^{2})^{2}}+\dfrac{1}{4(a^{2}+u^{2})}\bigg{)}\chi_{g}&=E\chi_{g}.\end{split}$
(36)

Considering a separation of variables for the wave function,
$\chi_{g}(\phi,u)=\Phi(\phi)\psi(u)$, we write from (36) an angular equation,

$\dfrac{d^{2}\Phi}{d\phi^{2}}+l^{2}\Phi=0,$ (37)

and a radial equation,

$\begin{split}-\dfrac{\hbar^{2}}{2m}\dfrac{d^{2}\psi}{du^{2}}-\dfrac{\hbar^{2}}{2m}\bigg{(}&\dfrac{a^{2}}{4(a^{2}+u^{2})^{2}}\\\
&+\dfrac{1-4l^{2}}{4(a^{2}+u^{2})}\bigg{)}\psi=E\psi,\end{split}$ (38)

where $l^{2}$ is a separation constant associated with the angular momentum of
the particle.

As we can see, Eq. (37) admits the normalized solutions

$\Phi_{l}(\phi)=\dfrac{1}{\sqrt{2\pi}}e^{il\phi}$ (39)

for the catenoid. Thus, in this specific case, it is clear that the
parametrization of the catenoid is periodic in $\phi$, because we have
$\mathbf{r}(\phi+2\pi,u)=\mathbf{r}(\phi,u)$, such that the angular wave
function must also have periodicity $2\pi$, i.e.,
$\Phi_{l}(\phi+2\pi)=\Phi_{l}(\phi)$, which implies that the angular momentum
quantum number $l$ assumes only integer values ($l=0,\pm 1,\pm 2,\dots$).

The angular wave functions in the case of the helicoid behave similarly to the
solutions found in the case of the helix. Due to the geometry of the infinite
helicoid ($\phi\in\mathbb{R}$), equation (39) is obtained for $l=0,\pm 1,\pm
2,\dots$, exactly as in the case of the catenoid. On the other hand, for a
finite helicoid at $0<\phi<\phi_{0}$, we find the wave functions

$\Phi_{j}(\phi)=\sqrt{\dfrac{2}{\phi_{0}}}\sin\left(\dfrac{j\pi}{\phi_{0}}\phi\right),$
(40)

for $l=\dfrac{j\pi}{\phi_{0}}$, where $j=1,2,3,\dots$ is the quantum number.
Thus, despite the well-known isometry between a catenoid and a helicoid
limited by $0<\phi<2\pi$, the respective wave functions differ due to the
boundary conditions employed. Thereby, in section V we will look for solutions
of the radial equation (38) for possible bound states, i.e., considering the
usual boundary condition $\displaystyle{\lim_{u\rightarrow\pm\infty}\psi=0}$.

##  V Radial Wave Functions

As we can see, Eq. (38) can be conveniently rewritten as

$-\dfrac{d^{2}\psi}{d\tilde{u}^{2}}-\left(\dfrac{1}{4(1+\tilde{u}^{2})^{2}}+\dfrac{1-4l^{2}}{4(1+\tilde{u}^{2})}\right)\psi=\epsilon\psi,$
(41)

where $\tilde{u}=u/a$ is the radial variable and $\epsilon=2ma^{2}E/\hbar^{2}$
is the energy, both in dimensionless units [22]. Thus, (41) is a one-
dimensional Schrödinger equation with an effective potential (also
dimensionless) given by

$V_{l}(\tilde{u})=-\dfrac{1}{4(1+\tilde{u}^{2})^{2}}-\dfrac{1-4l^{2}}{4(1+\tilde{u}^{2})},$
(42)

which is always attractive only if $|l|\leq 1/2$. Effective potentials which
are equivalent to (42) were previously found in Refs. [12, 22, 25, 11].

We see that Eq. (38) becomes exactly (31) when considering $l=1/2$ and the arc
length $s=u$. Since $l$ is associated with the angular momentum, it does not
appear in the CPF when considering a particle in the catenary, but $l=1/2$ can
be used as a mere mathematical artifice such that the solutions of the
Schrödinger equation for the catenary can be obtained through Eq. (38).

On the other hand, for the finite helicoid we have that
$l=\dfrac{j\pi}{\phi_{0}}$, such that the effective potential in (42) is
always attractive for $j\leq\dfrac{\phi_{0}}{2\pi}$. Thus, if $\phi_{0}$ is a
multiple of $2\pi$, the maximum value of $j$ for which $V_{l}(\tilde{u})$ is
always attractive is precisely the number of complete loops of the helicoid.
Due to the local isometry between the catenoid and the helicoid, we will
discuss only the case where $\phi_{0}=2\pi$, which implies that
$V_{l}(\tilde{u})$ is attractive only for $j=1$, i.e., for $l=1/2$. Finally,
for the infinite helicoid and the catenoid we have $l=0,\pm 1,\pm 2,\dots$,
thus in these cases the effective potential is always attractive only if
$l=0$, i.e., when the particle has zero angular momentum. Concisely, to find
the possible bound states due to the effective potential we will consider in
(42) that

$l=\begin{cases}1/2\text{ for the catenary,}\\\ 1/2\text{ for the finite
helicoid }(\phi_{0}=2\pi),\\\ 0\text{ for the catenoid,}\\\ 0\text{ for the
infinite helicoid }(\phi\in\mathbb{R}).\end{cases}$ (43)

In Fig. 6 we present the graphs of the effective potential $V_{l}(\tilde{u})$
for some values of $l$.

Figure 6: Effective potential $V_{l}(\tilde{u})$ for $l=0,1/2,\pm 1,\pm 2$.
When $|l|$ increases, $V_{l}(\tilde{u})$ becomes increasingly repulsive.

The second term in the effective potential (42) can be separated in both
repulsive and attractive parts, given by $\dfrac{l^{2}}{1+\tilde{u}^{2}}$ and
$-\dfrac{1}{4(1+\tilde{u}^{2})}$, respectively. The repulsive part, which
naturally disappears for $l=0$, corresponds to the usual centrifugal term in
quantum mechanics. On the other hand, the attractive one arises due to the
confinement of the particle, and it is related to the change made in Eq. (35),
being associated with the so-called “quantum anticentrifugal force”, as
discussed in Refs. [28, 29].

###  V.1 Variational method

Considering the Hamiltonian in Eq. (41),

$\hat{\mathcal{H}}=-\dfrac{d^{2}}{d\tilde{u}^{2}}-\left(\dfrac{1}{4(1+\tilde{u}^{2})^{2}}+\dfrac{1-4l^{2}}{4(1+\tilde{u}^{2})}\right),$
(44)

we can use the variational method (also known as the Rayleigh-Ritz method) to
obtain approximations for the energy. For a given tentative wave function
$\psi_{n,l}=\psi_{n,l}(\tilde{u})$, which is normalized, the expectation value
of $\hat{\mathcal{H}}$ is

$\begin{split}\epsilon_{n,l}=&-\int_{-\infty}^{+\infty}\psi_{n,l}^{*}\dfrac{d^{2}\psi_{n,l}}{d\tilde{u}^{2}}d\tilde{u}\\\
&-\dfrac{1}{4}\int_{-\infty}^{+\infty}\dfrac{\psi_{n,l}^{*}\psi_{n,l}}{(1+\tilde{u}^{2})^{2}}d\tilde{u}\\\
&-\dfrac{1-4l^{2}}{4}\int_{-\infty}^{+\infty}\dfrac{\psi_{n,l}^{*}\psi_{n,l}}{1+\tilde{u}^{2}}d\tilde{u},\end{split}$
(45)

where $n=1,2,3,\dots$ is the quantum number associated with the energy levels.
Such method consists of intuitively proposing a wave function $\psi_{n,l}$
which depends on a parameter $\beta_{l}$ adjusted in order to minimize the
energy $\epsilon_{n,l}(\beta_{l})$ in Eq. (45). Consequently, this method give
us an upper limit for the true energy [30, 31].

####  V.1.1 Gaussian function

For the ground state ($n=1$) we are looking for bell-shaped functions. We
firstly propose the Gaussian wave function

$\psi_{1,l}(\tilde{u})=\left(\dfrac{\beta_{l}}{\pi}\right)^{1/4}e^{-\beta_{l}\tilde{u}^{2}/2},$
(46)

where $\beta_{l}>0$ is a parameter to be adjusted in order to obtain a minimum
value for the energy of the particle. By replacing (46) in (45), we obtain

$\displaystyle\epsilon_{1,l}=-\sqrt{\dfrac{\beta_{l}}{\pi}}\Bigg{[}$
$\displaystyle\beta_{l}\int_{-\infty}^{+\infty}\left(\beta_{l}\tilde{u}^{2}-1\right)e^{-\beta_{l}\tilde{u}^{2}}d\tilde{u}$
(47)
$\displaystyle+\dfrac{1}{4}\int_{-\infty}^{+\infty}\dfrac{e^{-\beta_{l}\tilde{u}^{2}}}{(1+\tilde{u}^{2})^{2}}d\tilde{u}$
$\displaystyle+\dfrac{1-4l^{2}}{4}\int_{-\infty}^{+\infty}\dfrac{e^{-\beta_{l}\tilde{u}^{2}}}{1+\tilde{u}^{2}}d\tilde{u}\Bigg{]},$

which leads us to

$\begin{split}\epsilon_{1,l}=\
&\dfrac{\sqrt{\pi\beta_{l}}e^{\beta_{l}}}{4}\left(\beta_{l}+4l^{2}-\dfrac{3}{2}\right)\left[1-\operatorname{erf}\left(\sqrt{\beta_{l}}\right)\right]\\\
&+\dfrac{\beta_{l}}{4},\end{split}$ (48)

where $\operatorname{erf}$ is the error function [32, 33]. By minimizing the
energy with respect to $\beta_{l}$, we find

$\begin{split}\dfrac{\sqrt{\pi}e^{\beta_{l}}}{4\sqrt{\beta_{l}}}\left[\beta_{l}^{2}+2l^{2}\left(2\beta_{l}+1\right)-\dfrac{3}{4}\right]\left[1-\operatorname{erf}\left(\sqrt{\beta_{l}}\right)\right]&\\\
+\dfrac{5}{8}-l^{2}-\dfrac{\beta_{l}}{4}=\ &0.\end{split}$ (49)

Now, we need to solve Eq. (49) numerically for the corresponding values of
$l$. For the infinite helicoid and for the catenoid we have $l=0$, such that
(49) becomes

$\dfrac{\sqrt{\pi}e^{\beta_{0}}}{4\sqrt{\beta_{0}}}\left(\beta_{0}^{2}-\dfrac{3}{4}\right)\left[1-\operatorname{erf}\left(\sqrt{\beta_{0}}\right)\right]+\dfrac{5}{8}-\dfrac{\beta_{0}}{4}=0,$
(50)

whose approximate solution is $\beta_{0}\approx 0.14133$, therefore, we get
the energy $\epsilon_{1,0}\approx-0.11977$. On the other hand, we have $l=1/2$
in the cases of the finite helicoid ($\phi_{0}=2\pi$) and the catenary, such
that we find

$\begin{split}\dfrac{\sqrt{\pi}e^{\beta_{1/2}}}{4\sqrt{\beta_{1/2}}}\left(\beta_{1/2}^{2}+\beta_{1/2}-\dfrac{1}{4}\right)\left[1-\operatorname{erf}\left(\sqrt{\beta_{1/2}}\
\right)\right]&\\\ +\dfrac{3}{8}-\dfrac{\beta_{1/2}}{4}=\ &0,\end{split}$ (51)

thus $\beta_{1/2}\approx 0.04120$ and $\epsilon_{1,1/2}\approx-0.02299$. Fig.
7 shows the graphs of the corresponding tentative wave functions from Eq.
(46).

Figure 7: Gaussian wave functions obtained with the variational method for the
ground state considering $l=0$ for the infinite helicoid and the catenoid, as
well as $l=1/2$ in for the finite helicoid ($\phi_{0}=2\pi$) and the catenary.

####  V.1.2 Lorentzian function

Also for the ground state, we propose (for comparison purposes) the Lorentzian
wave function

$\psi_{1,l}(\tilde{u})=\left(\dfrac{2\beta_{l}^{3}}{\pi}\right)^{1/2}\dfrac{1}{\beta_{l}^{2}+\tilde{u}^{2}},$
(52)

where $\beta_{l}>0$. By replacing Eq. (52) in (45), we obtain

$\begin{split}\epsilon_{1.l}=-\dfrac{2\beta_{l}^{3}}{\pi}&\Bigg{[}2\int_{-\infty}^{+\infty}\dfrac{3\tilde{u}^{2}-\beta_{l}^{2}}{(\beta_{l}^{2}+\tilde{u}^{2})^{4}}d\tilde{u}\\\
&+\dfrac{1}{4}\int_{-\infty}^{+\infty}\dfrac{d\tilde{u}}{(1+\tilde{u}^{2})^{2}(\beta_{l}^{2}+\tilde{u}^{2})^{2}}\\\
&+\dfrac{1-4l^{2}}{4}\int_{-\infty}^{+\infty}\dfrac{d\tilde{u}}{(1+\tilde{u}^{2})(\beta_{l}^{2}+\tilde{u}^{2})^{2}}\Bigg{]}.\end{split}$
(53)

From equation (53) we get

$\epsilon_{1,l}=\dfrac{1}{2\beta_{l}^{2}}-\dfrac{\beta_{l}^{2}+3\beta_{l}+1}{4(\beta_{l}+1)^{3}}+\dfrac{4l^{2}-1}{4}\dfrac{2\beta_{l}+1}{(\beta_{l}+1)^{2}},$
(54)

whose minimization gives us

$\begin{split}(3-8l^{2})\beta_{l}^{5}+\left(2-8l^{2}\right)\beta_{l}^{4}-16\beta_{l}^{3}&\\\
-24\beta_{l}^{2}-16\beta_{l}-4&=0.\end{split}$ (55)

Thus, for the infinite helicoid and for the catenoid we set $l=0$ in Eq. (55),
which give us

$3\beta_{0}^{5}+2\beta_{0}^{4}-16\beta_{0}^{3}-24\beta_{0}^{2}-16\beta_{0}-4=0,$
(56)

whose positive solution is approximately $\beta_{0}\approx 2.70072$, and from
that, we get the energy $\epsilon_{1,0}\approx-0.12918$. Similarly, we have
$l=1/2$ in the cases of the finite helicoid ($\phi_{0}=2\pi$) and of the
catenary, thus

$\beta_{1/2}^{5}-16\beta_{1/2}^{3}-24\beta_{1/2}^{2}-16\beta_{1/2}-4=0,$ (57)

therefore $\beta_{1/2}\approx 4.67957$ and $\epsilon_{1,1/2}\approx-0.02757$.
Figure 8 shows the graphs of the tentative wave functions obtained with the
respective $\beta_{l}$.

Figure 8: Lorentzian wave functions obtained with the variational method for
the ground state considering $l=0$ for the infinite helicoid and the catenoid,
as well as $l=1/2$ for the finite helicoid ($\phi_{0}=2\pi$) and the catenary.

As we can see with the Gaussian and Lorentzian functions, the wave function is
more concentrated at the origin for $l=0$ than for $l=1/2$. This was expected,
as the second term in Eq. (42) cancels out for $l=1/2$, but is negative for
$l=0$, making the particle in the infinite helicoid or in the catenoid feel a
more attractive effective potential (see Fig. 6). Furthermore, in both cases,
the respective energies found with the Gaussian function are greater than
those found with the Lorentzian function, which indicates that Eq. (52) is a
better approximation for the wave function, since the variational method
always provides an upper limit for the exact energy of the particle [31].

###  V.2 Confluent Heun function

The variational method provides reasonable approximations for the ground state
of the problems in question, but it does not necessarily works well in the
search for possible excited states. Consequently, we need to turn our
attention again to the radial equation,

$\dfrac{d^{2}\psi}{d\tilde{u}^{2}}+\left(\dfrac{1}{4(1+\tilde{u}^{2})^{2}}+\dfrac{1-4l^{2}}{4(1+\tilde{u}^{2})}+\epsilon\right)\psi=0,$
(58)

in order to find exact solutions or better approximations.

Introducing a new variable $\xi=-\tilde{u}^{2}$, such that
$\xi\in(-\infty,0],$ we find from Eq. (58) that

$\begin{split}&\dfrac{d^{2}\psi}{d\xi^{2}}+\dfrac{1}{2\xi}\dfrac{d\psi}{d\xi}\\\
&-\dfrac{4\epsilon\xi^{2}+(4l^{2}-8\epsilon-1)\xi+4\epsilon-4l^{2}+2}{16\xi(1-\xi)^{2}}\psi=0.\end{split}$
(59)

Considering similar changes to those used in Refs. [34, 35, 36], we use the
ansatz $\psi(\xi)=(1-\xi)^{(\tilde{\gamma}+1)/2}H(\xi)$, where
$\tilde{\gamma}>0$ is a parameter to be conveniently adjusted. From (59) we
get

$\dfrac{d^{2}H}{d\xi^{2}}+\left[\dfrac{1}{2\xi}-\dfrac{\tilde{\gamma}+1}{1-\xi}\right]\dfrac{dH}{d\xi}+\dfrac{\mathcal{A}(\xi)}{16\xi(1-\xi)^{2}}H=0,$
(60)

where we have the function

$\mathcal{A}(\xi)=(1-\xi)\left[4l^{2}-4\tilde{\gamma}(\tilde{\gamma}+1)-1-4\epsilon(1-\xi)\right]+4{\tilde{\gamma}}^{2}-5.$

Since $\tilde{\gamma}$ is “arbitrary”, we can consider it in a way that Eq.
(60) becomes as simple as possible. Thus, choosing $\tilde{\gamma}$ such that
$4{\tilde{\gamma}}^{2}-5=0$ is the most convenient here, because these are
precisely the terms that are not being multiplied by $(1-\xi)$ in
$\mathcal{A}(\xi)$. Therefore, we have the parameter
$\tilde{\gamma}=\sqrt{5}/2$, thus it follows from Eq. (60) that

$\begin{split}\dfrac{d^{2}H}{d\xi^{2}}+\left[\tilde{\alpha}+\dfrac{\tilde{\beta}+1}{\xi}+\dfrac{\tilde{\gamma}+1}{\xi-1}\right]\dfrac{dH}{d\xi}&\\\
+\left[\dfrac{\tilde{\mu}}{\xi}+\dfrac{\tilde{\nu}}{\xi-1}\right]H&=0,\end{split}$
(61)

for $\tilde{\alpha}=0$, $\tilde{\beta}=-1/2$,
$\tilde{\mu}=\left(2l^{2}-\sqrt{5}-3-2\epsilon\right)/8$, and
$\tilde{\nu}=\left(\sqrt{5}+3-2l^{2}\right)/8$. Equation (61) is known as the
confluent Heun equation [37, 38, 36, 39, 40].

Around the regular singular point $\xi=0$, the power series solution of Eq.
(61) converges on the unit disc, $|\xi|<1$, and it is called the confluent
Heun function,

$\operatorname{HeunC}\left(\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta},\xi\right)=\sum_{n=0}^{\infty}v_{n}\left(\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta}\right)\xi^{n},$
(62)

with the parameters

$\displaystyle\tilde{\delta}$
$\displaystyle=\tilde{\mu}+\tilde{\nu}-\dfrac{\tilde{\alpha}}{2}\left(\tilde{\beta}+\tilde{\gamma}+2\right)=-\dfrac{\epsilon}{4},$
(63) $\displaystyle\tilde{\eta}$
$\displaystyle=\dfrac{\tilde{\alpha}}{2}\left(\tilde{\beta}+1\right)-\tilde{\mu}-\dfrac{\tilde{\beta}+\tilde{\gamma}+\tilde{\beta}\tilde{\gamma}}{2}=\dfrac{2\epsilon-2l^{2}+5}{8},$
(64)

and where
$v_{n}=v_{n}\left(\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta}\right)$
are coefficients [37, 41]. Even though the convergence of (62) is assured for
$|\xi|<1$, in many cases it is possible to use numerical methods of
extrapolation (like those implemented in Maple) to obtain approximations of
$H(\xi)$ when $|\xi|\geq 1$, depending on the parameters $\tilde{\alpha}$,
$\tilde{\beta}$, $\tilde{\gamma}$, $\tilde{\delta}$, and $\tilde{\eta}$.

Since $\tilde{\beta}$ is non-integer, a second solution for (61) can be
written as
$\xi^{-\tilde{\beta}}\operatorname{HeunC}\left(\tilde{\alpha},-\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta},\xi\right)$
[42, 33]. Thus, from (62), the solutions of Eq. (58) are separable into an
even function,

$\psi_{\text{e}}(\tilde{u})=C_{0}\left(1+\tilde{u}^{2}\right)^{\frac{\sqrt{5}}{4}+\frac{1}{2}}\operatorname{HeunC}\left(\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta},-\tilde{u}^{2}\right),$
(65)

and an odd function,

$\psi_{\text{o}}(\tilde{u})=C_{1}\left(1+\tilde{u}^{2}\right)^{\frac{\sqrt{5}}{4}+\frac{1}{2}}\tilde{u}\operatorname{HeunC}\left(\tilde{\alpha},-\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta},-\tilde{u}^{2}\right),$
(66)

where $C_{0}$ and $C_{1}$ are normalization constants.

Once the analytical solutions of (58) have been found, the next step is to
apply the boundary conditions
$\displaystyle{\lim_{\tilde{u}\rightarrow\pm\infty}\psi=0}$. We cannot use
here the approach of reducing $\operatorname{HeunC}$ to a polynomial (see [38,
41, 43]), because we would get solutions that do not obey the boundary
conditions. Similar problems have been reported in some recent works [44, 45,
46, 47]. An alternative would be writing a solution to (61) around the regular
singular point $\xi=1$ and use the so-called Wronskian method to obtain the
corresponding eigenvalues [34, 42, 48]. However, here we have the variable
$\xi=-\tilde{u}^{2}$, which implies that $\xi=1$ is not part of the solutions
domain of Eq. (61) in the problem of interest, therefore, the Wronskian method
cannot be applied here.

To circumvent the above mentioned adversities, we propose that the particle is
initially in a system limited in $\tilde{u}$, i.e., subject to the potential

$V(\tilde{u})=\begin{cases}+\infty\text{ for
}\tilde{u}\leq-\tilde{u}_{0}\text{ or }\tilde{u}\geq\tilde{u}_{0},\\\
V_{l}(\tilde{u})\text{ for
}-\tilde{u}_{0}<\tilde{u}<\tilde{u}_{0},\end{cases}$ (67)

where $V_{l}(\tilde{u})$ is the effective potential in (42) and
$\pm\tilde{u}_{0}$ corresponds to the edges of the one-dimensional box, whose
width is $2\tilde{u}_{0}$. Thus, when we have finite catenaries, helicoids, or
catenoids in $\tilde{u}$, both equations

$\lim_{\tilde{u}\rightarrow\pm\tilde{u}_{0}}\operatorname{HeunC}\left(\tilde{\alpha},\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta},-\tilde{u}^{2}\right)=0$
(68)

and

$\lim_{\tilde{u}\rightarrow\pm\tilde{u}_{0}}\operatorname{HeunC}\left(\tilde{\alpha},-\tilde{\beta},\tilde{\gamma},\tilde{\delta},\tilde{\eta},-\tilde{u}^{2}\right)=0$
(69)

can be used to obtain the energies $\epsilon_{n,l}$ (for $n=1,2,3,\dots$) that
cause wave functions to tend to zero at $\pm\tilde{u}_{0}$ [49]. When
assigning values to $\tilde{u}_{0}$, solutions to (68) and (69) can be found
using Maple’s fsolve command and its numeric methods for $|\xi|\geq 1$ [50].
Next, we will conduct a study of how the energies and the wave functions
behave as the box size increases. We call this “the box method”.

####  V.2.1 Attractive effective potential

Using the box method, let us discuss here the cases in which the effective
potential in Eq. (42) is attractive, i.e., when we set $l=0$ in the cases of
the infinite helicoid and of the catenoid, and $l=1/2$ in the case of the
finite helicoid ($\phi_{0}=2\pi$) and the catenary, according to Eq. (43).
Table LABEL:tab:catenoid-box shows the energies for some values of
$\tilde{u}_{0}$. Thus, we can see that when $\tilde{u}_{0}$ has values around
$1$, the effects of the effective potential are not so relevant. For example,
when considering $\tilde{u}_{0}=1$, the energies of the first four states
($n=1,2,3,4$) when we set $l=0$ ($l=1/$ 2) are given approximately by
$2.03806$ ($2.26282$), $9.50983$ ($9.70867$), $21.84675$ ($22.04447$) and
$39.11990$ ($39.31700$). In this case, the radial wave functions
$\psi(\tilde{u})$ for $l=1/2$ behave similarly to the corresponding wave
functions for $l=0$ (see graph (a) in Fig. 9).

Table 1: Energies $\epsilon_{n,l}$ of the first four states ($n=1,2,3,4$) for $l=0$ and for $l=1/2$, considering increasing values for the box size, $2\tilde{u}_{0}$. $\tilde{u}_{0}$ | $\epsilon_{1,0}$ | $\epsilon_{2,0}$ | $\epsilon_{3,0}$ | $\epsilon_{4,0}$
---|---|---|---|---
$1$ | $2.03806$ | $9.50983$ | $21.84675$ | $39.11990$
$3.00420$ | $0.00000$ | $0.96307$ | $2.27982$ | $4.21282$
$10$ | $-0.12999$ | $0.07853$ | $0.15781$ | $0.36069$
$50$ | $-0.13051$ | $0.00289$ | $0.00388$ | $0.01370$
$100$ | $-0.13051$ | $0.00071$ | $0.00087$ | $0.00338$
$500$ | $-0.13051$ | $0.00003$ | $0.00003$ | $0.00013$
$1000$ | $-0.13051$ | $0.00001$ | $0.00001$ | $0.00001$
|  |  |  |
$\tilde{u}_{0}$ | $\epsilon_{1,1/2}$ | $\epsilon_{2,1/2}$ | $\epsilon_{3,1/2}$ | $\epsilon_{4,1/2}$
$1$ | $2.26282$ | $9.70867$ | $22.04447$ | $39.31700$
$5.97477$ | $0.00000$ | $0.26695$ | $0.57495$ | $1.08526$
$10$ | $-0.02401$ | $0.09601$ | $0.19112$ | $0.38755$
$50$ | $-0.02892$ | $0.00392$ | $0.00502$ | $0.01568$
$100$ | $-0.02892$ | $0.00098$ | $0.00111$ | $0.00393$
$500$ | $-0.02892$ | $0.00004$ | $0.00004$ | $0.00016$
$1000$ | $-0.02892$ | $0.00001$ | $0.00001$ | $0.00004$

As the box size increases, the energy associated with each level decreases.
Thus, the ground state energy of the particle, $\epsilon_{1,l}$, becomes zero
for $l=0$ ($l=1/2$) when $\tilde{u}_{0}\approx 3.00420$ ($\tilde{u}_{0}\approx
5.97477$), such that if the respective $\tilde{u}_{0}$ is greater than that,
$\epsilon_{1,l}$ becomes negative. Therefore, if we have $\tilde{u}_{0}=100$,
the approximate energies of the first four states for $l=0$ ($l=1/2$) are
$-0.13051$ ($-0.02892$), $0.00071$ ($0.00098$), $0.00087$ ($0.00111$), and
$0.00338$ ($0.00393$), thus, more significant differences between the
respective wave functions for $l=0$ and $l=1/2$ appear, as shown in graphs (b)
and (c) in Fig. 9. Furthermore, in these graphs it is clear that the square
module of the respective wave functions is practically the same for $n=2$ and
$n=3$; this happens for $n=4$ and $n=5$ for larger values of $\tilde{u}_{0}$,
and so on.

Figure 9: In graph (a) are the radial wave functions (numerically normalized)
corresponding to the first four states when considering $\tilde{u}_{0}=1$ and
$l=0$ (if $\tilde{u}_{0}=1$ and $l=1/2$, the solutions behave similarly). In
graph (b) are the radial wave functions (numerically normalized) for the first
four states when $l=0$ for $\tilde{u}_{0}=100$ and, in graph (c), the same
when $l=1/2$.

Thus, when the box size increases, the effects of the effective potential on
the wave functions becomes increasingly apparent, while the effects due to the
box starts to become less relevant. For $l=0$ ($l=1/2$), the energy of the
ground state $\epsilon_{1.0}$ ($\epsilon_{1,1/2}$) converges to approximately
$-0.13051$ ($-0.02892$) as the box size is increased. As we can see in both
cases, the respective energies of the following three states decreases when
$\tilde{u}_{0}$ increases, in such a way that they become practically
indistinguishable from each other. These results show that, for both values of
$l$, there will be a state with negative energy starting from a certain value
of $\tilde{u}_{0}$, therefore, this is only due to $V_{l}(\tilde{u})$.

To better illustrate this, Fig. 10 shows how the energy of the particle varies
according to the box size, $2\tilde{u}_{0}$, where the blue continuous (red
dashed) lines represent the states with even (odd) wave functions. Thus, the
energy of the ground state with $l=0$ ($l=1/2$) tends to approximately
$-0.13051$ ($-0.02892$) as the box size increases, while the energies of the
excited states tend to form a continuous spectrum. This leads to the inference
that there is only a single bound state for the effective potential presented
in equation (42) for the catenary, helicoid, or catenoid with infinite size in
$\tilde{u}$.

Figure 10: Graph (a) shows the energies as a function of the box size,
$2\tilde{u}_{0}$, for $l=0$ and, in graph (b), the same for $l=1/2$. Each line
corresponds to an energy level, with the blue continuous lines associated with
the even solutions, and the red dashed lines with the odd ones.

The ground state energy $\epsilon_{1,l}$ was obtained in an approximate way
with the box method for $\tilde{u}_{0}\gg 1$, but the wave functions in (65)
and (66) are the exact eigenfunctions of Eq. (41), thus the solution obtained
in this work is actually “semi-exact” [46]. Fig. 11 compares these solutions
with the wave functions found using the variational method for both $l=0$ and
$l=1/2$. As one can see, the confluent Heun function goes to zero faster than
the Lorentzian, but slower than the Gaussian. For both values of $l$, the
respective energy found with the box method is considerably near those
obtained with the variational method, especially for the Lorentzian function.

Figure 11: Comparison between the radial wave functions obtained by the box
method using $\operatorname{HeunC}$, and for the Gaussian and Lorentzian
functions used in the variational method (a) for $l=0$ and (b) for $l=1/2$.

####  V.2.2 Repulsive effective potential

As we can see, the situations in which the effective potential in (42) is
always repulsive correspond to $|l|>1/2$. Thus, for a helicoid with
$\phi_{0}=2\pi$ this occurs if the quantum number $j>1$, i.e., when
$l=1,3/2,2,5/2,\dots$, while it happens for $l=\pm 1,\pm 2,\pm 3,\dots$ in the
case of the infinite helicoid and of the catenoid. In these cases all bound
states have positive energy and are due only to the box, such that at the
limit where $\tilde{u}_{0}\rightarrow\infty$ there will be no bound state for
both surfaces. As examples, the cases where $l=\pm 1$ and $l=\pm 2$ for
$\tilde{u}_{0}=1$ and $\tilde{u_{0}}=100$ are discussed below. For the semi-
integer values $l=3/2,5/2,\dots$, the results are completely analogous.

When we have $\tilde{u}_{0}=1$, the energies for the first four states where
$l=\pm 1$ ($l=\pm 2$) are given approximately by $2.93669$ ($5.62598$),
$10.30488$ ($12.68535$), $22.63779$ ($25.01384$), and $39.90841$ ($42.27562$),
whose respective wave functions are also very similar to those shown in graph
(a) in Fig. 9. In such cases, this also suggests that, for values of
$\tilde{u}_{0}$ near $1$, the effects of the box have more influence on the
eigenfunctions than the presence of the effective potential. However, as the
values of $|l|$ increase, $V_{l}(\tilde{u})$ becomes more and more repulsive,
tending to “push” the probability density $|\psi|^{2}$ further away from the
origin, even for relatively small $\tilde{u}_{0}$ values, i.e., near $1$.

When we set $\tilde{u}_{0}=100$, the energies for the first four states where
$l=\pm 1$ ($l=\pm 2$) are given approximately by $0.00146$ ($0.00264$),
$0.00147$ ($0.00264$), $0.00490$ ($0.00708$), and $0.00491$ ($0.00708$). From
these results, we show in Fig. 12 the corresponding wave functions. There, it
is possible to observe in both $l=\pm 1$ and $l=\pm 2$ cases that the
probability densities $|\psi|^{2}$ are very similar for states where $n=1$ and
$n=2$, as well as for $n=3$ and $n=4$, and so on. Furthermore, the effects of
the repulsive potential become evident here, since the probability density
near the origin is practically null in all the studied states. Again, all of
this indicates that the possible energies of the particle start to form a
continuous spectrum for the repulsive effective potential when the values of
$\tilde{u}_{0}$ are large enough.

Figure 12: Numerically normalized radial wave functions corresponding to the
first four states (a) for $l=\pm 1$ and (b) for $l=\pm 2$, i.e., for the
repulsive effective potential, and considering $\tilde{u}_{0}=100$.

##  VI On the ground state

In subsection IV.1 we showed that, in the CPF, a quantum particle constrained
to an infinite helix has its angular momentum quantized, such that the
probability density is always constant, since
$|\psi_{l}(\phi)|^{2}=\dfrac{1}{2\pi}$, for any integer $l$. On the other
hand, if the helix is finite for $0<\phi<\phi_{0}$, thus the wave functions
$\psi_{j}(\phi)$ and the energies $E_{j}$ are analogous to the one-dimensional
infinite potential well problem. Figure 13 shows the probability density of
the ground state, $|\psi_{1}|^{2}$, plotted on a finite helix with
$\phi_{0}=2\pi$.

Figure 13: Color map representation of the probability density
$|\psi_{1}|^{2}$ corresponding to the ground state of a particle constrained
to a finite helix with $\phi_{0}=2\pi$. The brighter (darker) the region, the
greater (smaller) the probability of finding the particle.

Furthermore, we showed in section V that a particle constrained to a catenary
can be found in a bound state due to the GIP. When we make $l=1/2$ and
$\tilde{u}=u/a$ in Eq. (66), we get the wave function

$\begin{split}\psi(u)&=\dfrac{C_{0}}{\sqrt{a}}\left(1+\dfrac{u^{2}}{a^{2}}\right)^{\frac{\sqrt{5}}{4}+\frac{1}{2}}\times\\\
&\
\times\operatorname{HeunC}\left(0,-\dfrac{1}{2},\dfrac{\sqrt{5}}{2},-\dfrac{\epsilon}{4},\dfrac{9}{16}+\dfrac{\epsilon}{4},-\dfrac{u^{2}}{a^{2}}\right),\end{split}$
(70)

where $C_{0}\approx 0.37842$ e $\epsilon\approx-0.02892$, according to the box
method. Thus, the energy of the particle in the bound state is given by
$E\approx-0.02892\dfrac{\hbar^{2}}{2ma^{2}}$. Using (70), the probability
density plotted on the catenary is shown in Fig. 14.

Figure 14: Color map representation of the probability density $|\psi|^{2}$
corresponding to the ground state of a particle constrained to a catenary. The
brighter (darker) the region, the greater (smaller) the probability of finding
the particle.

For a helicoid finite in the $\phi$ variable, with $\phi_{0}=2\pi$, we obtain
from Eq. (40) the angular wave function,

$\Phi_{j}(\phi)=\dfrac{1}{\sqrt{\pi}}\sin\left(\dfrac{j\phi}{2}\right),$ (71)

thus $j=1,2,3,\dots$ and $l=j/2$. Since the effective potential in (42) is
attractive only if $|l|\leq 1/2$, thus, for the helicoid in question, this
implies that it is only possible to have bound states if $j=1$. Therefore, the
radial wave function in this case is also given by Eq. (70), such that the
total wave function of the single bound state is given by

$\begin{split}\chi_{g}(\phi,&u)=\dfrac{C_{0}}{\sqrt{\pi
a}}\sin\left(\dfrac{\phi}{2}\right)\left(1+\dfrac{u^{2}}{a^{2}}\right)^{\frac{\sqrt{5}}{4}+\frac{1}{2}}\times\\\
&\
\times\operatorname{HeunC}\left(0,-\dfrac{1}{2},\dfrac{\sqrt{5}}{2},-\dfrac{\epsilon}{4},\dfrac{9}{16}+\dfrac{\epsilon}{4},-\dfrac{u^{2}}{a^{2}}\right),\end{split}$
(72)

where $C_{0}\approx 0.37842$ and $\epsilon\approx-0.02892$. Thus, the energy
of the particle in the bound state is
$E\approx-0.02892\dfrac{\hbar^{2}}{2ma^{2}}$, as in the case of the catenary.
Using Eq. (72), the probability density plotted on the helicoid is shown in
Fig. 15.

Figure 15: Color map representation of the probability density
$|\chi_{g}|^{2}$ corresponding to the ground state of a particle constrained
to a finite helicoid with $\phi_{0}=2\pi$. The brighter (darker) the region,
the greater (smaller) the probability of finding the particle.

Finally, as the angular wave function for the cases of the infinite helicoid
and the catenoid is given by
$\Phi_{l}(\phi)=\dfrac{1}{\sqrt{2\pi}}e^{il\phi}$, and the effective potential
is attractive only if $l=0$, thus, for there to be bound states, we simply
have $\Phi_{0}(\phi)=\dfrac{1}{\sqrt{2\pi}}$. From the radial wave function
(66), the total wave function for the single bound state is

$\begin{split}\chi_{g}(\phi,&u)=\dfrac{C_{0}}{\sqrt{2\pi
a}}\left(1+\dfrac{u^{2}}{a^{2}}\right)^{\frac{\sqrt{5}}{4}+\frac{1}{2}}\times\\\
&\
\times\operatorname{HeunC}\left(0,-\dfrac{1}{2},\dfrac{\sqrt{5}}{2},-\dfrac{\epsilon}{4},\dfrac{5}{8}+\dfrac{\epsilon}{4},-\dfrac{u^{2}}{a^{2}}\right),\end{split}$
(73)

where $C_{0}\approx 0.49087$ and $\epsilon\approx-0.13051$, as found with the
box method. Thus, the energy of the particle is given by
$E\approx-0.13051\dfrac{\hbar^{2}}{2ma^{2}}$. Figure 16 shows the probability
density plotted on the respective surfaces, as obtained from Eq. (73).

(a)

(b)

Figure 16: Color map representation of the probability density
$|\chi_{g}|^{2}$ corresponding to the ground state of a particle constrained
(a) to a infinite helicoid and (b) to a catenoid. The brighter (darker) the
region, the greater (smaller) the probability of finding the particle.

##  VII Topological aspects

As mentioned before, the helicoid and catenoid are locally isometric to each
other, that is, their local geometry is indistinguishable. But, as it is
obvious from Figs. 3 and 4, they are quite different from a global point of
view. In fact, these surfaces have different topologies. Any closed curve on
the helicoid can be shrunk to a point, while there is a class of loops on the
catenoid that will end up stuck at the catenoid’s neck upon reduction. Quantum
mechanics, being inherently delocalized, offers many possibilities as a probe
for topology. A well known example, the Aharonov-Bohm [51] phase stands as a
self-evident quantum observable that is directly related to topology.

Topology has become an important tool both in the study of the physical
properties as well as in the design of new materials [52]. Physical systems
may have very rich hidden topologies, like in Fermi surfaces, for instance. A
change of topology of the Fermi surface signals an electronic phase
transition, as observed by Lifshitz [53] already in 1960. A large class of
modern materials, the so-called Topological Materials [54], rely their
unusual, and sometimes exotic properties, in the intrinsic topology of their
electronic band structures. A detailed account of the role of topology in
modern materials is given in Ref. [17]. It is clear then, the importance of
probing the topology of a physical system by means of specific measurements.
The concept of using the measurement of a physical observable to identify the
topology of a system, or topological metrology [52], becomes then an important
tool for the geometric/topological characterization of novel materials. How to
apply this idea to the systems studied here? How an experiment can distinguish
between two isometric surfaces with different topologies? A hint comes from
the aforementioned Aharonov-Bohm phase. The original A-B setup [51] involves a
non simply-connected (shielded) region of space which is threaded by a
magnetic flux. Electron beams traveling on either side of the “hole” that
contains the flux acquire different quantum phases whose difference can be
measured by interference. This establishes the A-B phase as a topological
metric in the sense described above. On the other hand, geometry and topology
give rise to analog A-B effects. For instance, the parallel transport of
vectors and spinors around cosmic strings (literally a cylindrical hole in
spacetime) gives rise [55] to nontrivial phases that reflects the locally flat
spacetime geometry of this cosmic object and its conical topology.
Disclinations, which are topological line defects in nematic liquid crystals,
present a similar effect for light propagation [56]. A measurement of the
phase acquired by the polarization (a spinor) of the light traveling around
the disclination can give information on its topology. Both the cosmic string
and disclination examples can be unified under the single view of a A-B-type
effect where the magnetic field is substituted by a curvature flux [57].

Wavefunctions, being scalar, are trivially parallel-transported, not yielding
any topological information in this way. Nevertheless, upon scattering, the
acquired phase shift of the wave function might provide topological
information as in the case of diffraction of light by disclinations in liquid
crystals [58]. The phase shift appears then as possible topological metric.
Another option comes from higher order tensors like vectors or spinors which,
as discussed above, may reveal topological features when subjected to parallel
transport. In modeling curved graphene with the Dirac equation, one naturally
deals with spinors, which can then be used as probes for topology in the A-B
scheme. Recent tight-binding results [59] indicated that curvature of
elastically deformed graphene can be used to split the electric current into
two beams of differently valley-polarized electrons. The curvature,
represented by the strain field, acts on the electrons as a pseudo-magnetic
field splitting the otherwise degenerate valley states. This could be a
strategy for an Aharonov-Bohm-type of experiment that would compare the phases
acquired by the valley spinors in each beam as they travel on opposing sides
of the curved region (see Fig. 1 of Ref. [59]). Then, parallel transport of
the spinors, measured by interference of valley-polarized electron beams
traveling along equivalent paths, like the ones shown on the catenoid of Fig.
17, can be used as a tool for topological metrology and thus distinguish
between two isometric surfaces with different topologies, like the ones
studied here.

Figure 17:  Representation of equivalent classical trajectories of electrons
injected at the edge of the catenoid.

##  VIII Conclusions

In this work, we studied the confinement of a quantum particle to a helix,
catenary, helicoid, or catenoid using the Confining Potential Formalism to
curves and surfaces. First, it was shown that a particle constrained to an
infinite helix has its angular momentum quantized. On the other hand, for the
confinement to a finite helix, the problem becomes similar to that of a
quantum particle in a one-dimensional potential well.

The local isometry between the finite helicoid and the catenoid was explored,
such that it was shown that the quantum dynamics on the surfaces is governed
by the same Schrödinger equation. From this, an effective Schrödinger equation
was obtained, in the radial variable $\tilde{u}$, with an effective potential
$V_{l}(\tilde{u})$ that is attractive only if $|l|\leq 1/2$. In the case of
the infinite helicoid and the catenoid, it is only possible to have bound
states when $l=0$, while in the case of a helicoid limited to $0<\phi<2\pi$
this occurs only for $l=1/2$. In addition, it was observed that, by making
$l=1/2$ in the radial equation for the helicoid and the catenoid, the same
Schrödinger equation is obtained for the case of the catenary.

The use of the variational method (with the Gaussian and the Lorentzian
functions) guarantees the existence of a bound state in the cases of the
catenary, the helicoid (finite and infinite in $\phi$) and the catenoid. Thus,
it was shown that the radial Schrödinger equation has exact solutions that are
written in terms of the confluent Heun function, however, the eigenvalues
cannot be obtained analytically, so it was necessary to use numerical methods
to estimate them. In fact, both methods were in agreement with each other, and
it was possible to infer that, in these three cases, the particle can be found
in a single bound state, which is due to the GIP that comes from the CPF.

To sum up, we recall that Refs. [12, 22, 11] provided detailed analysis of the
geometry-induced quantum potential and some of its effects due to confinement
of a particle to either a helix, catenary, helicoid, or catenoid. We extend
and complement these previous works by providing a unified treatment of the
Schrödinger equations of the corresponding cases. Furthermore, we determine
the single bound state that appears in the infinite versions of the catenary,
helicoid, and catenoid and show that the remaining states combine into a
continuous energy band.

As perspectives for future works, we intend to study the confinement of a
quantum particle with spin in the helicoid and catenoid, as well as to study
the confinement of identical particles under these conditions. Since the spin
is sensitive to torsion [13], it may be able to distinguish the surfaces in
spite of their isometry. Another interesting approach is the addition of a
geometry-induced magnetic field to the helicoid as done in Ref. [60]. We also
intend to use the methods used in the present work for the confinement problem
in other regular curves and surfaces.

###### Acknowledgements.

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001 (F.F.S.F.),
Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE),
Grants No. IBPG-0487-1.05/19 (J.D.M.L.), and BIC-1187-1.05/20 (E.G.), and
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant
No. 307687/2017-1 (F.M.). The authors are indebted to L.C.B. da Silva for
invaluable discussions and suggestions.

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