# Effective description of a quantum particle constrained to a catenoid

G. Chacón-Acosta<EMAIL_ADDRESS>Departamento de Matemáticas Aplicadas y
Sistemas, Universidad Autónoma Metropolitana-Cuajimalpa, Av. Vasco de Quiroga
4871, Ciudad de México, 05348, Mexico.  H. Hernández-Hernández
<EMAIL_ADDRESS>Universidad Autónoma de Chihuahua, Facultad de Ingeniería,
Nuevo Campus Universitario, Chihuahua 31125, Mexico.  J. Ruvalcaba-Rascón
<EMAIL_ADDRESS>Abbe School of Photonics, Friedrich-
Schiller-Universität Jena, Albert-Einstein-Straße 6, 07745, Jena, Germany

###### Abstract

We describe a quantum particle constrained on a catenoid, employing an
effective description of quantum mechanics based on expected values of
observables and quantum dispersions. We obtain semiclassical trajectories for
particles, displaying general features of the quantum behaviour; most
interestingly, particles present tunnelling through the throat of the
catenoid, a characteristic having important physical applications.

Keywords: effective quantum mechanics, constrained systems, particles on
surfaces

##  I Introduction

The exploration of quantum mechanics on curved surfaces has attracted the
attention of the scientific community, mainly due to its potential
applications across diverse fields such as condensed matter physics, quantum
information processing, nanotechnology, and materials science. Recent focus
has turned towards describing the intricacies of quantum mechanics on curved
surfaces DASILVA201713  since, for instance, experimental techniques have
synthesized various types of nanostructures with shapes resembling spheres,
cylinders, and other non-trivial geometries DASILVA201713 , castro . Among
these geometries, there is one of particular interest, the catenoid, a surface
of revolution obtained by rotating a catenary curve about its axis. Our work
delves into the behavior of a quantum particle constrained to move on such a
surface, discussing both the internal metric of the constraint manifold and
the external metric of the embedding space. Motivated by the catenoid’s status
as a minimal surface with unique geometric properties, our study becomes a
conduit for investigating the interplay between quantum mechanics and curved
spaces. We narrow our focus on the impact of constraining potentials on a
particle confined to a catenoid, starting with the Schrödinger equation and
discussing quantum evolution through an effective method, suited for the
analysis of the system as a semiclassical dynamical one. The semiclassical
evolution exhibits interesting quantum mechanical properties, displaying the
profound effects of curvature on the particle’s behavior.

The curvature of the catenoid emerges as a pivotal factor influencing the
behavior of confined particles, offering a tunable parameter for manipulating
electronic properties de_Lima_2021 ,onoe . Additionally, the length of the
catenoid becomes a controlling factor, influencing particle behavior and
allowing strategic control over confinement in distinct regions of the surface
szameit . This study not only enriches our understanding of the electronic
structure of curved nanotubes and graphene but also hints at novel
possibilities for designing innovative nanoscale devices silva_2021 ; willmore
; alsham .

Starting from the standard description for a quantum particle constrained to a
curved surface, given in the seminal work of da Costa daCosta81 ; daCosta82 ,
we separate the Schrödinger equation into normal and tangential components and
constrain the particle to the surface through a confining potential acting in
the normal direction daCosta81 ; daCosta82 ; teixeira_schrodinger_2019 . The
analysis of the quantum evolution of the wave function has been discussed in
the literature ikegami_quantum_1992 ; 2014anphy.341..132x , where the
complexity of the dynamical equations often requires approximation methods to
provide physical insight. We apply an effective description of quantum
mechanics based on expectation values of observables and quantum dispersions
to describe the evolution of the quantum particle on the surface bojowald-
aureliano ; bojowald-sandhofer . This semiclassical implementation for the
quantum evolution allows a description for individual quantum particles, whose
behavior can be understood in terms of the physical parameters in the system
baytas_2020 ; bojowald_2022 ; tunneling ; javi . In the domain of
semiclassical quantum mechanics on a catenoid, our effective study presents a
compelling exploration into the behavior of quantum particles traversing the
catenoid throat. The semiclassical particle exhibits transmission and
reflection phenomena, aligning seamlessly with quantum mechanical analyses
dandoloff_2009 ; dandoloff_2010 . Beyond theoretical insights, our findings
hold potential applications in quantum information processing, enriching our
understanding of quantum particle behavior in non-trivial geometries and
pointing towards exciting future developments in quantum physics applications
savage ; mannini . Moreover, we explore the practical implications of our
findings for materials science, nanotechnology, and emerging interdisciplinary
domains silva_2021 ; alsham ; 2014anphy.341..132x ; danani .

The structure of the article is the following. In section II we present the
derivation of the classical and quantum description of a particle constrained
to a curved surface. In section III we introduce the effective description of
quantum systems, momentous quantum mechanics. In section IV the semiclassical
discussion of the quantum particle on the catenoid is presented, with the
analysis of the evolution being presented in section V. Finally, in section VI
we discuss our findings, and provide possible future extensions and
applications of our work.

##  II Quantum Particles on Curved Surfaces

In recent years it has been widely accepted that in dealing with the quantum
mechanical problem for particles on surfaces, one has to adopt the confinement
potential formalism, which predicts a geometrical potential whose physical
effects can be experimentally explored de_Lima_2021 ; onoe ; szameit ; oli .
In this work, we will deal with the confinement of a particle to a two-
dimensional surface embedded in a three-dimensional Euclidean space.

In this formalism daCosta81 ; daCosta82 ; teixeira_schrodinger_2019 , we take
the Schrödinger equation,

$\text{i}\hbar\frac{\partial\xi}{\partial
t}=-\frac{\hbar^{2}}{2m}\Delta\xi+V\xi,$ (1)

and expressing the Laplacian in terms of the coordinates of the curved surface
as in the Laplace-Beltrami operator

$\Delta
f=\frac{1}{\sqrt{\absolutevalue{g}}}\sum\limits_{i,j=1}^{n}\partial_{i}\left(\sqrt{\absolutevalue{g}}g^{ij}\partial_{j}f\right).$
(2)

One decouples the resulting differential equation by splitting the wave
function, as the product of normal ($\xi_{\text{N}}$) and tangential
components ($\xi_{\text{T}}$). The most adequate parametrization of the
surface is obtained by introducing a Gauss map, which yields the corresponding
coordinate on the surface ($q_{1},q_{2}$) and a normal coordinate ($q_{3}$).
Expressing the wave function as $\xi=\xi_{\text{T}}(q_{1},q_{2})\
\xi_{\text{N}}(q_{3})$, separates the Schrödinger equation into two decoupled
equations. In the case of an interaction-free particle, we get

$\text{i}\hbar\frac{\partial\xi_{\text{N}}}{\partial
t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\xi_{\text{N}}}{\partial
q_{3}^{2}}+V_{\lambda}(q_{3})\xi_{\text{N}},$ (3)

and

$\text{i}\hbar\frac{\partial\xi_{\text{T}}}{\partial
t}=-\frac{\hbar^{2}}{2m}\Delta_{\text{g}}\xi_{\text{T}}-\frac{\hbar^{2}}{2m}\left(H^{2}-K\right)\xi_{\text{T}},$
(4)

where $\Delta_{\text{g}}$ corresponds to the operator (2) for the two-
dimensional induced metric of the surface, and $H$ and $K$ correspond to its
mean and Gaussian curvatures, respectively teixeira_schrodinger_2019 . In the
first equation a confining potential is added to constraint the particle to
the surface

$\lim\limits_{\lambda\rightarrow\infty}V_{\lambda}(q_{3})=\begin{cases}0&q_{3}=0,\\\
\infty&q_{3}\neq 0,\end{cases}$ (5)

where $\lambda$ denotes the magnitude of the binding potential. Additionally,
the last term in (4) is known as the quantum geometric potential, or da Costa
potential daCosta81 ; daCosta82 .

###  II.1 Schrödinger Equation for a Particle Constrained to a Catenoid

Particularly, the Schrödinger equation for a particle constrained to a
catenoid with an axis along $z$ and waist radius $R$ can be derived by
considering the line element dandoloff_2010 ,

$\text{d}s^{2}=\cosh^{2}(z/R)\text{d}z^{2}+R^{2}\cosh^{2}(z/R)\text{d}\phi^{2},$
(6)

from which the mean and Gaussian curvatures can be calculated, and are given
respectively as

$H=0,$ (7)

and

$K=-\frac{1}{R^{2}}\sech^{4}\left(\frac{z}{R}\right).$ (8)

The geometric potential in this case is

$V_{G}(z)=-\frac{\hbar^{2}}{2mR^{2}}\sech^{4}\left(\frac{z}{R}\right),$ (9)

which allows us to write the time-independent Schrödinger equation
dandoloff_2009  in the following form

$-\frac{\hbar^{2}\sech^{2}\left(\frac{z}{R}\right)}{2mR}\left[R\frac{\partial^{2}\xi_{\text{T}}}{\partial
z^{2}}+\frac{1}{R}\frac{\partial^{2}\xi_{\text{T}}}{\partial\phi^{2}}+\sech^{2}\left(\frac{z}{R}\right)\xi_{\text{T}}\right]=E,$
(10)

with $E$ the energy eigenvalue. It is important to note that the confining
potential formalism predicts a non-trivial geometric potential, in this case,
given by the expression (9). The geometrical nature of this potential needs to
be distinguished from the characteristics of the so-called effective
potential, which arises from the semiclassical approximation.

###  II.2 Quantization of the Classical System

The quantization of constrained classical systems is a long-standing problem
that was earlier noticed by Dirac dirac1930 ; dirac2001 , and has been treated
for the particular case of a particle whose motion is constrained to a curved
surface. It has been shown that quantizing a particle constrained to a surface
yields the Schrödinger equation (4), ikegami_quantum_1992 .

To achieve this, we need to consider a conservative, constrained classical
Hamiltonian written in Cartesian coordinates, obeying usual canonical
relations $\left\\{x_{a},P_{b}\right\\}=\delta_{ab}$, ikegami_quantum_1992 ,

$H=\frac{1}{2m}\sum\limits_{a,b=1}^{N}P_{a}\left(\delta_{ab}-n_{a}n_{b}\right)P_{b}+V(x).$
(11)

where $n_{a}$ are the components of the unit vector normal to the surface and
$a,b=1,2,3$. Or, in terms of the surface coordinates, which are also canonical
$\left\\{q_{\mu},p_{\nu}\right\\}=\delta_{\mu\nu}$, and the induced metric on
the surface $g_{\mu\nu}$

$H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}.$ (12)

where $\mu,\nu=1,2$.

When quantizing this Hamiltonian one has to define the corresponding position
operators in the usual way $\hat{q}_{\mu}$, and the momentum operators as the
geometric momentum operators defined in liu2007 ; pauli ,

$\hat{p}_{\mu}=-\text{i}\hbar\left(\partial_{\mu}+\frac{1}{2}\Gamma_{\mu}\right).$
(13)

where $\Gamma_{\mu}$ corresponds to the contracted connection symbols. One
obtains the right Schrödinger equation on the surface (4). The classical
phase-space is coordinatized as before by
$q_{\mu}=\langle\hat{q}_{\mu}\rangle$ and,
$p_{\mu}=\langle\hat{p}_{\mu}\rangle$ which remain as canonical pairs.

We emphasize that the constraint has been applied to the classical
Hamiltonian, which will have no further information about the normal motion of
the particle, classically there is no motion in this direction. This means
that one cannot obtain (3) by quantizing this classical constrained
Hamiltonian. For consistency, we will only consider the extreme case for the
confining potential given in (5).

##  III Semiclassical description of quantum mechanics

Momentous quantum mechanics describes effectively the quantum dynamics via the
Taylor expansion of the expectation value of the quantum Hamiltonian operator.
This expansion leads to a quantum-corrected semiclassical Hamiltonian that
yields enhanced classical equations of motion and introduces dynamical
dispersions called quantum variables bojowald-aureliano .

The quantum variables are defined as

$\displaystyle G^{a_{1},b_{1},\cdots,a_{k},b_{k}}$ $\displaystyle\equiv$
$\displaystyle\langle\left(\hat{q}_{1}-q_{1}\right)^{a_{1}}(\hat{p}_{1}-p_{1})^{b_{1}}$
$\displaystyle\cdots$
$\displaystyle\left(\hat{q}_{k}-q_{k}\right)^{a_{k}}(\hat{p}_{k}-p_{k})^{b_{k}}\rangle,$
(14)

where $k$ corresponds to the number of degrees of freedom of the system, and
for any of the operator products, from now on, we will consider the Weyl
ordering. Expectation values of canonical operators
$q_{k}=\langle\hat{q}_{k}\rangle$ and $p_{k}=\langle\hat{p}_{k}\rangle$,
satisfying $\left[\hat{q},\hat{p}\right]=\text{i}\hbar\mathds{1}$, now
corresponding to the classical variables $x_{k}$, and $p_{k}$, coordinatize
the augmented phase space with the $G^{a_{1},b_{1},\cdots,a_{k},b_{k}}$
variables bojowald-sandhofer .

The symplectic structure of phase space determines a Poisson bracket bojowald-
aureliano  algebra for the expectation values,

$\left\\{\langle\hat{A}\rangle,\langle\hat{B}\rangle\right\\}=\frac{1}{\text{i}\hbar}\Big{\langle}\left[\hat{A},\hat{B}\right]\Big{\rangle}.$
(15)

We can see that classical variables are canonical. The Poisson brackets of the
quantum variables are given in Appendix A.

Momentous quantum mechanics is based on a quantum-corrected Hamiltonian, which
is defined as the expectation value of the quantum Hamiltonian operator in the
following manner,

$\langle\hat{H}\rangle\equiv
H_{\text{Q}}=H\left(\\{x_{i}\\},\\{P_{i}\\}\right)+\sum\limits_{a_{1},b_{1}}^{\infty}\cdots\sum\limits_{a_{k},b_{k}}^{\infty}\frac{\partial^{a_{1}+b_{1}+\cdots+a_{k}+b_{k}}H}{\partial
x_{1}^{a_{1}}\partial P_{1}^{b_{1}}\cdots\partial x_{k}^{a_{k}}\partial
P_{k}^{b_{k}}}\frac{G^{a_{1},b_{1};\cdots;a_{k},b_{k}}}{a_{1}!b_{1}!\cdots
a_{k}!b_{k}!},$ (16)

where $H$ is the classical Hamiltonian of the system and $\hat{H}$ is the
quantum Hamiltonian operator. In this paper, we refer to the quantum-corrected
Hamiltonian $H_{\text{Q}}$ simply as the quantum Hamiltonian, and should not
be confused with the quantum Hamiltonian operator $\hat{H}$, which will not be
used unless explicitly specified.

The symplectic structure allows us to calculate the equations of motion of the
variables as

$\dot{f}=\left\\{f,H_{\text{Q}}\right\\},$ (17)

where $f$ can be either a classical or a quantum variable. Observe that,
starting from the Poisson brackets between classical and quantum variables
(34) and (35), quantum variables only see other quantum variables, here the
interaction between classical and quantum variables is given by the potential
of the quantum Hamiltonian.

The effective description inherits Heisenberg’s uncertainty relations, which
are now written in terms of the dispersions, as

$\displaystyle G^{2,0,0,0}G^{0,2,0,0}-(G^{1,1,0,0})^{2}$ $\displaystyle\geq$
$\displaystyle\frac{\hbar^{2}}{4},$ (18) $\displaystyle
G^{0,0,2,0}G^{0,0,0,2}-(G^{0,0,1,1})^{2}$ $\displaystyle\geq$
$\displaystyle\frac{\hbar^{2}}{4},$ (19)

for a system with two degrees of freedom.

In momentous quantum mechanics, it is required that the extended classical
Hamiltonian (16) must be a conservatively constrained one. Additionally, the
momentum operators in the definition of the quantum variables must be the
geometric momenta as defined in (13). These requirements are necessary for our
description to agree with the full quantization scheme for the constrained
classical particle.

It is worth mentioning that a new complete and canonical formulation of
momentous quantum mechanics has been recently introduced baytas_2020 ;
bojowald_2022 . However, the characteristics of the restricted classical
Hamiltonian that we will consider restrict the use of this formulation, as can
be followed in Appendix B. Given this fact, we use the formulation of
momentous quantum mechanics presented above for the following sections.

##  IV Semiclassical Description for the Particle on a Catenoid Problem

We consider the conservative classical Hamiltonian describing the motion of a
particle constrained on a catenoid with fixed $R$ in cylindrical coordinates,
whose Hamiltonian is 2014anphy.341..132x ,

$H=\frac{\sech^{2}\left(z/R\right)}{2mR^{2}}\left(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2}\right).$
(20)

We use the definition for the geometric momentum operators (13) for the
moments, which in this particular case are

$\displaystyle\hat{P}_{\theta}$ $\displaystyle=$
$\displaystyle-\text{i}\hbar\partial_{\theta},$ (21)
$\displaystyle\hat{P}_{\text{z}}$ $\displaystyle=$
$\displaystyle-\text{i}\hbar\left[\partial_{\text{z}}+\frac{1}{R}\tanh\left(\frac{z}{R}\right)\right].$
(22)

The quantum moments using (14), with canonical coordinates $\theta$ and $z$
are defined as

$G^{a,b,c,d}\equiv\langle(\hat{\theta}-\theta_{0})^{a}(\hat{P}_{\theta}-P_{\theta_{0}})^{b}\left(\hat{z}-z_{0}\right)^{c}(\hat{P}_{\text{z}}-P_{z_{0}})^{d}\rangle,$
(23)

where the product of operators is symmetrical Weyl ordered. The quantum
effective Hamiltonian obtained from (20), using (16) is

$\displaystyle H_{\text{Q}}$ $\displaystyle=$ $\displaystyle
H+\frac{1}{2mR^{2}}\sum\limits_{b+c+d\geq
2}^{\infty}\sech^{2}\left(\frac{z}{R}\right)\text{P}_{c}\left(\tanh\left(\frac{z}{R}\right)\right)$
$\displaystyle\times$
$\displaystyle\frac{G^{0,b,c,d}}{b!c!d!}\left[\prod\limits_{j=0}^{b-1}(2-j)P_{\theta}^{2-b}+R^{2}\prod\limits_{k=0}^{d-1}(2-k)P_{\text{z}}^{2-d}\right],$
(24)

where $0\leq b\leq 2$, $0\leq d\leq 2$ and $\text{P}_{n}(\tanh x)$ is the
polynomial boyadzhiev2010derivative ; Feng12017tan ,

$\text{P}_{n}\left(\tanh
x\right)=-2^{n+1}\sum\limits_{k=1}^{n+1}\frac{k!}{2^{k}}\mathcal{S}_{n+1}^{(k)}\left(\tanh
x-1\right)^{k-1},$ (25)

where $\mathcal{S}_{n}^{(k)}$ are the Stirling numbers of second kind.

###  IV.1 Second Order Truncation

The dynamical system obtained from (IV) comprises the equations of motion of
the classical variables and the infinite many quantum variables. Evidently,
solving such a system analytically is impossible, so it is necessary to
perform consistent truncations.

Therefore, up to the second order, we write this Hamiltonian simply as,

$\displaystyle H_{\text{Q}}$ $\displaystyle=$
$\displaystyle\frac{\sech^{2}\left(z/R\right)}{2mR^{2}}\bigg{\\{}p_{\theta}^{2}+R^{2}p_{\text{z}}^{2}-\frac{4}{R}\tanh\left(\frac{z}{R}\right)p_{\theta}^{2}G^{0,1,1,0}$
(26) $\displaystyle-$ $\displaystyle
4R\tanh\left(\frac{z}{R}\right)p_{\text{z}}G^{0,0,1,1}+2G^{0,2,0,0}+2R^{2}G^{0,0,0,2}$
$\displaystyle+$
$\displaystyle\frac{2}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right]\left(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2}\right)G^{0,0,2,0}\bigg{\\}}.$

From the Poisson brackets with the effective Hamiltonian (26), and employing
the relation (34), the system of equations of motion for the classical
variables is

$\displaystyle\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{\sech^{2}(z/R)}{mR^{2}}\bigg{\\{}p_{\theta}-\frac{4}{R}\tanh\left(\frac{z}{R}\right)p_{\theta}G^{0,1,1,0}$
$\displaystyle+$
$\displaystyle\frac{2p_{\theta}}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right]\right)G^{0,0,2,0}\bigg{\\}},$
$\displaystyle\dot{p}_{\theta}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\dot{z}$ $\displaystyle=$
$\displaystyle\frac{\sech^{2}(z/R)}{mR^{2}}\bigg{\\{}Rp_{\text{z}}-2\tanh\left(\frac{z}{R}\right)G^{0,0,1,1}$
$\displaystyle+$
$\displaystyle\frac{2p_{\text{z}}}{R}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right]G^{0,0,2,0}\bigg{\\}},$
$\displaystyle\dot{p}_{\text{z}}$ $\displaystyle=$
$\displaystyle\frac{2}{R}\tanh(\frac{z}{R})H_{\text{Q}}-\frac{2}{mR^{2}}\sech^{4}\left(\frac{z}{R}\right)$
(27) $\displaystyle\times$
$\displaystyle\bigg{[}\frac{3}{R^{3}}\left(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2}\right)\tanh(\frac{z}{R})G^{0,0,2,0}$
$\displaystyle-$ $\displaystyle
p_{\text{z}}G^{0,0,1,1}-\frac{p_{\theta}^{2}}{R^{2}}G^{0,1,1,0}\bigg{]}.$

Given the symplectic structure in (35), and defining the common factor
$\eta(z)=\sech^{2}\left(z/R\right)/2mR^{2}$ we have the system of equations of
motion for the quantum variables as

$\displaystyle\dot{G}^{1,1,0,0}$ $\displaystyle=$ $\displaystyle
4\eta(z)\left[-\frac{\tanh\left(z/R\right)}{R}p_{\theta}^{2}G^{0,1,1,0}+G^{0,2,0,0}\right],$
$\displaystyle\dot{G}^{1,0,1,0}$ $\displaystyle=$ $\displaystyle
4\eta(z)\bigg{[}-\frac{\tanh\left(z/R\right)}{R}p_{\theta}^{2}G^{0,0,2,0}+R^{2}G^{1,0,0,1}$
$\displaystyle+$ $\displaystyle
G^{0,1,1,0}-R\tanh\left(\frac{z}{R}\right)p_{\text{z}}G^{1,0,1,0}\bigg{]},$
$\displaystyle\dot{G}^{1,0,0,1}$ $\displaystyle=$ $\displaystyle
4\eta(z)\bigg{\\{}-\frac{\tanh\left(z/R\right)}{R}p_{\theta}^{2}\left(G^{0,0,1,1}-G^{1,1,0,0}\right)$
$\displaystyle+$ $\displaystyle
R\tanh\left(\frac{z}{R}\right)p_{\text{z}}G^{1,0,0,1}+G^{0,1,0,1}$
$\displaystyle-$
$\displaystyle\frac{(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2})}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right]G^{1,0,1,0}\bigg{\\}},$
$\displaystyle\dot{G}^{0,1,1,0}$ $\displaystyle=$ $\displaystyle
4R\eta(z)\left[-\tanh\left(\frac{z}{R}\right)p_{\text{z}}^{2}G^{0,1,1,0}+RG^{0,1,0,1}\right],$
$\displaystyle\dot{G}^{0,1,0,1}$ $\displaystyle=$
$\displaystyle\frac{4}{R}\eta(z)\bigg{\\{}\tanh\left(\frac{z}{R}\right)\left[p_{\theta}^{2}G^{0,2,0,0}+R^{2}p_{\text{z}}G^{0,1,0,1}\right]$
$\displaystyle-$
$\displaystyle\frac{(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2})}{R}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right]G^{0,1,1,0}\bigg{\\}},$
$\displaystyle\dot{G}^{0,0,1,1}$ $\displaystyle=$
$\displaystyle\eta(z)\bigg{\\{}\frac{4\tanh\left(z/R\right)}{R}p_{\theta}^{2}G^{0,1,1,0}+4R^{2}G^{0,0,0,2}$
$\displaystyle-$
$\displaystyle\frac{4(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2})}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right]G^{0,0,2,0}\bigg{\\}},$
$\displaystyle\dot{G}^{2,0,0,0}$ $\displaystyle=$
$\displaystyle\eta(z)\left[-\frac{4\tanh\left(z/R\right)}{R}p_{\theta}^{2}G^{1,0,1,0}+8G^{1,1,0,0}\right],$
$\displaystyle\dot{G}^{0,2,0,0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\dot{G}^{0,0,2,0}$ $\displaystyle=$
$\displaystyle\eta(z)\left[-8R\tanh\left(\frac{z}{R}\right)p_{\text{z}}G^{0,0,2,0}+8R^{2}G^{0,0,1,1}\right],$
$\displaystyle\dot{G}^{0,0,0,2}$ $\displaystyle=$
$\displaystyle\eta(z)\bigg{\\{}\frac{8\tanh\left(z/R\right)}{R}p_{\theta}^{2}G^{0,1,0,1}$
(28) $\displaystyle-$
$\displaystyle\frac{8(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2})}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right]G^{0,0,1,1}$
$\displaystyle+$ $\displaystyle
8R\tanh\left(\frac{z}{R}\right)p_{\text{z}}G^{0,0,0,2}\bigg{\\}}.$

To find the initial conditions to solve the equations of motion, we propose a
normalized Gaussian wave packet as our initial state,

$\displaystyle\psi_{0}(\theta,z)$ $\displaystyle=$
$\displaystyle\left[\frac{\lambda^{1/2}}{R\pi\sigma_{\text{z}}\erf(\pi\lambda^{1/2})}\right]^{1/2}$
$\displaystyle\times$
$\displaystyle\exp{\text{i}(lz+m\theta)-\frac{(z-z_{0})^{2}}{2\sigma_{\text{z}}^{2}}-\frac{\lambda(\theta-\theta_{0})^{2}}{2}},$
(29)

where $-\infty<z<\infty$ and $-\pi\leq\theta<\pi$ and $\lambda$ and
$\sigma_{z}$ are related to the initial width of dispersion in $\theta$ and
$z$, respectively. Using this state, we find the initial values of the quantum
variables by evaluating the respective expected values in (14) and using the
generalized momentum operators in (21-22). We get

$\displaystyle~{}G_{0}^{1,1,0,0}$ $\displaystyle=$
$\displaystyle~{}G_{0}^{1,0,1,0}=G_{0}^{1,0,0,1}=G_{0}^{0,1,1,0}$
$\displaystyle=$ $\displaystyle G_{0}^{0,1,0,1}=G_{0}^{0,0,1,1}=0,$
$\displaystyle~{}G_{0}^{2,0,0,0}$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda\erf(\pi\lambda^{1/2})}\bigg{(}\erf\left(\pi\lambda^{1/2}\right)$
$\displaystyle-$
$\displaystyle(\pi\lambda)^{1/2}\exp{-\pi^{2}\lambda}\bigg{)},$
$\displaystyle~{}G_{0}^{0,2,0,0}$ $\displaystyle=$
$\displaystyle\hbar^{2}\lambda\left(1-\lambda G_{0}^{2,0,0,0}\right),$
$\displaystyle~{}G_{0}^{0,0,2,0}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sigma^{2}\left(2+\frac{\sigma^{2}}{R^{2}}\right).$
(30)

The initial value for $G^{0,0,0,2}$ does not have an analytical expression,
thus its value was calculated by numerical integration. We point out that
although the initial values for the moments were calculated for a specific
initial wave function, the dynamical evolution for these quantum variables
does not depend on a specific state. The system of equations of motion for the
moments comes from the effective Hamiltonian.

This effective setting allows exploring general quantum effects, such as
quantum tunneling tunneling , that for certain states for the analysis of the
catenoid may not appear. We will see that this is the case here as we report
the reflection and transmission of particles through semiclassical
trajectories. In previous studies, as in dandoloff_2010 , no reflection has
been found due to the choice of states. We will discuss this later on.

##  V Numerical evolution

Numerically, we will use $\hbar=m=1$, $R=1/2$, and the $z$-momentum will be
taken as a varying parameter $-a$. We employ the following initial numerical
values of the classical and quantum variables

$\begin{split}\theta_{0}&=0,\\\ P_{\theta_{0}}&=1,\\\ z_{0}&=1\\\
P_{z_{0}}&=-a,\\\ G_{0}^{1,1,0,0}&=0,\\\ G_{0}^{1,0,1,0}&=0,\\\
G_{0}^{1,0,0,1}&=0,\end{split}\hskip
28.45274pt\begin{split}G_{0}^{0,1,1,0}&=0,\\\ G_{0}^{0,1,0,1}&=0,\\\
G_{0}^{0,0,1,1}&=0,\\\ G_{0}^{2,0,0,0}&=0.05,\\\ G_{0}^{0,2,0,0}&=5,\\\
G_{0}^{0,0,2,0}&=0.06,\\\ G_{0}^{0,0,0,2}&=4.169094014469417.\end{split}$ (31)

With these initial conditions, one can solve the system of equations
numerically since no analytical solution exists. We now explore the behavior
of the resulting quantum-corrected trajectories.

###  V.1 Single Particle Effective Trajectories

Let us discuss the case of a single particle by fixing $a=1$ for the classical
and effective solutions. Figure 1 compares both trajectories on the surface of
the catenoid for a range of $0\leq t\leq 0.25$. Notice that the particle
starts on the upper half of the surface and, while the classical trajectory
continues to the lower half, the semiclassical presents a rebound on the
throat, confining its motion to $z>0$. This effect is due entirely to the
quantum nature of the system.

Referring to Figure 2(a), we can see that the trajectories do not vary much in
the angular variable since the quantum correction for $\theta$ is small. In
Figure 2(b), we observe that for the $z$-variable, the trajectories acquire
considerable quantum corrections, the classical particle goes through the
catenoid throat, while the quantum-corrected one does not reach $z=0$. We also
show the semiclassical trajectory’s lower and upper uncertainty bounds, making
its probabilistic nature explicit. As we mentioned above, the semiclassical
trajectory has a bounce.

Figure 1: Semiclassical and classical trajectories for a particle on a
catenoid with $a=1$.

(a) $\theta$ evolution.

(b) $z$ evolution.

Figure 2: Variable evolution for the semiclassical and classical cases with
$a=1$, where the uncertainty belt (in grey) is shown.

As a validity condition, we verify that the uncertainty relations for the
moments in (18) and (19) are satisfied. In Figures 3(a) and 3(b) we show that
this is the case.

(a) $G^{2,0,0,0}G^{0,2,0,0}-(G^{1,1,0,0})^{2}\geq\hbar^{2}/4$.

(b) $G^{0,0,2,0}G^{0,0,0,2}-(G^{0,0,1,1})^{2}\geq\hbar^{2}/4$.

Figure 3: Uncertainty relations for the variables with $a=1$.

###  V.2 Multiple Quantum-Corrected Trajectories

We now analyze multiple trajectories by varying $a$ from $(0-9)$, with an
incremental step of 1. We obtain classical trajectories in Figure 4(a) and the
corresponding effective trajectories in Figure 4(b). In this case, an even
more interesting effect appears for the quantum case: particles with higher
energy tunnel through the barrier at the waist; in contrast, the lower
energetic ones still get reflected off the barrier.

(a) Classical trajectories.

(b) Semiclassical trajectories.

Figure 4:  Classical (blue) and effective (red) trajectories with $0\leq a\leq
9$. Semiclassical trajectories get either reflected or transmitted at the
throat.

In Figures 5(a) and 5(b) we compare the $\theta$ and $z$ coordinates for these
trajectories, for both the classical and semiclassical solutions. As can be
noted, quantum corrections make each variable get more oscillations around its
initial value. We also verify, as for the one-particle case, that the
uncertainty relation is always valid through the dynamical evolution, as can
be appreciated in Figures 6(a) and 6(b) for $\theta$ and $z$ variables,
respectively. The trajectories follow from the upper branch to the lower
branch of the catenoid.

(a) $\theta$.

(b) $z$ .

Figure 5: Coordinates evolution for the semiclassical and classical
trajectories for free particles with $0\leq a\leq 9$.

(a) $G^{2,0,0,0}G^{0,2,0,0}-(G^{1,1,0,0})^{2}\geq\hbar^{2}/4$.

(b) $G^{0,0,2,0}G^{0,0,0,2}-(G^{0,0,1,1})^{2}\geq\hbar^{2}/4$.

Figure 6: Uncertainty relations for each variable with $0\leq a\leq 9$.

###  V.3 Effective Potential

We now turn our attention to the effective quantum potential. As our treatment
is semiclassical, we obtain the evolution of the system through an extended
classical dynamical system, whose qualitative behavior can be determined by
analyzing the potential, hence its importance.

The potential can be read off the effective Hamiltonian (IV), or from its
second-order expression (26)

$\displaystyle V_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{\sech^{2}\left(z/R\right)}{mR^{2}}\bigg{\\{}-\frac{2\tanh\left(z/R\right)}{R}p_{\theta}^{2}G^{0,1,1,0}+G^{0,2,0,0}$
(32) $\displaystyle-$ $\displaystyle
2R\tanh\left(\frac{z}{R}\right)p_{\text{z}}G^{0,0,1,1}+R^{2}G^{0,0,0,2}$
$\displaystyle+$
$\displaystyle\frac{1}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right](p_{\theta}^{2}+R^{2}p_{\text{z}}^{2})G^{0,0,2,0}\bigg{\\}}.$

The classical potential, which in our context, can be interpreted as the
initial value of $V_{\text{eff}}$,

$\displaystyle V_{\text{class}}$ $\displaystyle=$
$\displaystyle\frac{\sech^{2}\left(z/R\right)}{mR^{2}}\bigg{\\{}G^{0,2,0,0}_{0}+R^{2}G^{0,0,0,2}_{0}$
$\displaystyle+$
$\displaystyle\frac{1}{R^{2}}\left[2-3\sech^{2}\left(\frac{z}{R}\right)\right](p_{\theta}^{2}+R^{2}p_{\text{z}}^{2})G^{0,0,2,0}_{0}\bigg{\\}}.$
(33)

In Figure 7 we plot this classical potential for three momentum values,
$p_{\text{z}}=-7.0,-8.4,-9.6$. In contrast, the dynamical effective potential
for these same initial values is shown in Figure 8.

(a) $p_{\text{z}}=-7.0$

(b) $p_{\text{z}}=-8.4$

(c) $p_{\text{z}}=-9.6$

Figure 7: Classical potential, $V_{\text{class}}$, for different initial
values of $p_{\text{z}}$.

(a) $p_{\text{z}}=-7.0$

(b) $p_{\text{z}}=-8.4$

(c) $p_{\text{z}}=-9.6$

Figure 8: Effective potential $V_{\text{eff}}$ for different initial values of
$p_{\text{z}}$.

###  V.4 Quantum transmission

We analyze semiclassical trajectories and their passing from the upper branch
to the lower branch of the catenoid through the throat at $z=0$, as shown in
Figures 1 and 4. In both figures we observe either reflection or transmission
off the potential barrier, depending on the initial energy of the particles;
such behavior was also obtained in dandoloff_2010  for example. We used
initial values $p_{\text{z}}=-7.0,-8.4,-9.6$.

This can be classically interpreted directly from the effective potential. We
can see in figure 9 the $z$ component of the trajectory atop the effective
potential. Initially, the potential presents a local maximum, and no tunneling
is possible; the particle gets reflected. As it evolves, the potential
changes, acquires local minima, and its maximum decreases, allowing the
particle to pass. Whether the particle can tunnel or reflect the potential
depends on initial conditions, which, in turn, determine its energy tunneling
.

(a) $p_{\text{z}}=-7.0$

(b) $p_{\text{z}}=-8.4$

(c) $p_{\text{z}}=-9.6$

Figure 9: Semiclassical trajectories on the effective potential
$V_{\text{eff}}$ for different initial values of $p_{\text{z}}$.

In figure 10 we can see the evolution of the $z(t)$ component under the
aforementioned initial conditions, with the corresponding effective potential
as a density plot. In figure 10(a) the particle gets reflected before being
able to reach the throat of the catenoid due to quantum back-reaction. By
increasing the value of the initial momentum of the particle $p_{z}$, the
semiclassical particle is able to cross the potential barrier once
$p_{\text{z}}=-9.6$, as we can appreciate in figure 10(c), this is
accomplished by the decreasing of the effective potential maxima (figure
9(c)). Under the momentous quantum mechanics formalism, this has already been
interpreted as a quantum tunnelling effect tunneling ; article1 ; article3 ;
javi .

(a) $p_{\text{z}}=-7.0$

(b) $p_{\text{z}}=-8.4$

(c) $p_{\text{z}}=-9.6$

Figure 10: Reflection and transmission effect for semiclassical trajectories
with different initial values of $p_{\text{z}}$.

The Supplementary Material presents a GIF animation of the trajectory with
initial momentum varying in the range $-9.7\leq p_{\text{z}}\leq-7.0$. We can
see how the particle crosses the catenoid throat as its initial momentum
$p_{\text{z}}$ increases.

##  VI Discussion

In this work, we have obtained the effective evolution of a quantum particle
constrained to the curved two-dimensional surface of the catenoid. The
analysis has been performed with momentous quantum mechanics, a semiclassical
description based on expectation values of observables and quantum
dispersions.

We have found that the constrained particle exhibits reflection and
transmission at the catenoid throat, described in terms of particle
trajectories. This same effect has been reported under a different approach in
dandoloff_2010 . In our case, the dynamics of quantum trajectories do not
depend on the quantum state but rather on the extended effective Hamiltonian.

In the constrained description on a catenoid, one finds that the geometric
potential is $V_{\text{G}}=-\hbar^{2}\sech^{4}\left(z/R\right)/2mR^{2}$, which
maximizes at $z=0$. The presence of this geometric potential leads us to
expect a tunneling effect at the surface throat once quantization is
considered, i.e., we can find transmission and reflection of quantum particles
on the catenoid due to the geometric potential dandoloff_2010 ; griffiths ;
sakurai . This behavior can be appreciated in the semiclassical trajectories,
a particular feature of our approach. This frame is equivalent to the full
quantum description.

A very interesting description of the quantum evolution for this constrained
system can be given due to the semiclassical nature of momentous quantum
mechanics. The classical (confining) potential gets quantum backreacted by
quantum dispersion, rendering an effective potential. It is dynamic and
evolves with the system: depending on initial conditions for the classical and
quantum variables, the potential gives way to the possibility of tunneling,
which we depict in our analysis. The occurrence of these effects, inherited
from the tunneling effect, has been studied with the effective model of
moments in other physical systems. tunneling .

We have provided a robust scheme to study several effects in constrained
quantum systems, particularly in curved spaces. By extending the conditions
discussed in this study to allow more general, or different physical systems,
one can explore several models and experimentally interesting settings. We can
mention: the transport of particles, a topic of research in vastly different
areas such as nanomaterials, radiation physics and materials science danani .
The catenoid is of particular interest because it is a minimal surface
silva_2021  and minimizes the Willmore energy willmore . This makes the
surface suitable for making connections between nanostructures of different
materials, such as graphene and boron nitride silva_2021 ; alsham , and it has
been studied using the Schrödinger formalism dandoloff_2009 ; dandoloff_2010 .
Such analysis is under further investigation.

##  Appendix A Symplectic Structure of the Quantum Variables

Classical and quantum variables are symplectically orthogonal, which means
that their Poisson brackets follow

$\bigg{\\{}x_{a},G^{a_{1},b_{1},\cdots,a_{k},b_{k}}\bigg{\\}}=\bigg{\\{}p_{a},G^{a_{1},b_{1},\cdots,a_{k},b_{k}}\bigg{\\}}=0,$
(34)

while the Poisson bracket between the quantum variables is non-zero article1
and is given by,

$\displaystyle\bigg{\\{}G^{a_{1},b_{1},\cdots,a_{k},b_{k}},G^{c_{1},d_{1},\cdots,c_{k},d_{k}}\bigg{\\}}=$
$\displaystyle\sum\limits_{i=1}^{k}\biggl{(}a_{i}d_{i}G^{a_{1},b_{1},\cdots,a_{i}-1,b_{i},\cdots,a_{k},b_{k}}G^{c_{1},d_{1},\cdots,c_{i},d_{i}-1,\cdots,c_{k},d_{k}}$
$\displaystyle-$ $\displaystyle
b_{i}c_{i}G^{a_{1},b_{1},\cdots,a_{i},b_{i}-1,\cdots,a_{k},b_{k}}G^{c_{1},d_{1},\cdots,c_{i}-1,d_{i},\cdots,c_{k},d_{k}}\biggl{)}$
$\displaystyle+$
$\displaystyle\sum\limits_{n}\sum\limits_{s}\sum\limits_{e_{1},\cdots,e_{k}}(-1)^{s}\left(\frac{\text{i}\hbar}{2}\right)^{n-1}\delta_{\sum_{i}e_{i},n}$
$\displaystyle\times$
$\displaystyle\mathcal{K}^{n,s\\{e\\}}_{\\{a\\},\\{b\\},\\{c\\},\\{d\\}}G^{a_{1}+c_{1}-e_{1},b_{1}+d_{1}-e_{1};\cdots;a_{k}+c_{k}-e_{k},b_{k}+d_{k}-e_{k}},$
(35)

where $n=1,\cdots,\tilde{N}$, and

$\tilde{N}=\begin{cases}1,&\eta_{i}\leq 1\\\
\eta_{i}-1,&\eta_{i}>1\end{cases}$ (36)

defining $\eta_{i}=\sum_{i}\left(\min[a_{i},d_{i}]+\min[b_{i},c_{i}]\right)$,
also, $s=0,\cdots n;0\leq e_{i}\leq\min[a_{i},d_{i},s]+\min[b_{i},c_{i},n-s]$.

On the other side, $\mathcal{K}$ is given by

$\displaystyle\mathcal{K}^{n,s\\{e\\}}_{\\{a\\},\\{b\\},\\{c\\},\\{d\\}}$
$\displaystyle=$
$\displaystyle\sum\limits_{g_{1},\cdots,g_{k}}\frac{\delta_{\sum_{i}g_{i},n-s}}{s!\left(n-s\right)!}$
$\displaystyle\prod\limits_{i}\frac{\binom{a_{i}}{e_{i}-g_{i}}\binom{b_{i}}{g_{i}}\binom{c_{i}}{g_{i}}\binom{d_{i}}{e_{i}-g_{i}}}{\binom{n-s}{g_{i}}\binom{s}{e_{i}-g_{i}}},$
(37)

with $\max[e_{i},s,e_{i}-a_{i},e_{i}-d_{i},0]\leq
g_{i}\leq\min[b_{i},c_{i},n-s,e_{i}]$.

##  Appendix B Canonical Transformation of the Classical Hamiltonian

It is interesting to write the classical Hamiltonian (20) in kinetic plus
potential form; this is because there exists an alternate formulation of
momentous quantum mechanics in terms of explicitly canonical variables, one
for each degree of freedom in the theory baytas_2020 . This formulation is
interesting because there is no need to truncate the Hamiltonian, and one
obtains an “all-order” potential.

To this end, we perform a canonical transformation to obtain the kinetic terms
explicitly in the classical Hamiltonian for the particle on a catenoid (20)

$H=\frac{\sech^{2}\left(z/R\right)}{2mR^{2}}\left(p_{\theta}^{2}+R^{2}p_{\text{z}}^{2}\right).$

We propose the following redefinition

$P_{2}=\sech\left(\frac{z}{R}\right)p_{\theta},$ (38)

and we use the second canonical generating function goldstein:mechanics ,

$\frac{\partial F_{2}}{\partial\theta}=p_{\theta}.$ (39)

We get

$F_{2}=\theta\cosh\left(\frac{z}{R}\right)P_{2}+g(z)P_{1},$ (40)

where we have introduced the second term on the r.h.s. $g(z)P_{1}$, such that
$F_{2}=\sum\limits_{i=1}^{2}Q_{i}P_{i}$, to have the canonical pairs
$\left(Q_{i},P_{i}\right)$. Taking the derivate with respect to $z$, we
identify,

$\frac{\partial F_{2}}{\partial
z}=\frac{\theta}{R}\sinh\left(\frac{z}{R}\right)P_{2}+g^{\prime}(z)P_{1}=p_{\text{z}}.$
(41)

Now we propose that $g^{\prime}(z)=\cosh\left(\frac{z}{R}\right)$, such that
the generating function becomes

$F_{2}=R\sinh\left(\frac{z}{R}\right)P_{1}+\theta\cosh\left(\frac{z}{R}\right)P_{2}.$
(42)

From the generating function, we identify the new canonical pairs of the
transformation in terms of the old variables, which are, respectively,

$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle
R\sinh\left(\frac{z}{R}\right),$ $\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\sech\left(\frac{z}{R}\right)p_{\text{z}}-\frac{\theta}{R}\sinh\left(\frac{z}{R}\right)\sech^{2}\left(\frac{z}{R}\right)p_{\theta},$
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\theta\cosh\left(\frac{z}{R}\right),$ $\displaystyle P_{2}$
$\displaystyle=$ $\displaystyle\sech\left(\frac{z}{R}\right)p_{\theta},$ (43)

and $\left\\{Q_{i},P_{j}\right\\}=\delta_{i,j}$. Upon substitution, they allow
us to rewrite the Hamiltonian $H(q,p,t)$ as a new Hamiltonian $K(Q,P,t)$,
given $\partial F_{2}/\partial t=0$. The new Hamiltonian is,

$K(Q,P,t)=\frac{P_{1}^{2}}{2m}+\frac{P_{2}^{2}}{2mR^{2}}+\frac{Q_{1}Q_{2}P_{1}P_{2}}{m\left(R^{2}+Q_{1}^{2}\right)}+\frac{Q_{1}^{2}Q_{2}^{2}P_{2}^{2}}{2m\left(R^{2}+Q_{1}^{2}\right)^{2}}.$
(44)

We can see from (44), that the Hamiltonian (20) corresponds to a constrained
Hamiltonian in the usual way, as the first two terms in $K(Q,P,t)$ are kinetic
terms, while the last two terms correspond to a constraining potential, which
depends on position and momentum. An extension of the present work in this
canonical formulation will be implemented in a future analysis.

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