Information analysis in free and confined harmonic oscillator

Neetik Mukherjee,
Amlan K. RoyE-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of chemical sciences,

Indian institute of science education and research Kolkata,

Mohanpur-741246, Nadia, India

[L]In: Harmonic Oscillators: Types,

Functions and Applications

Editor:Yilun Shang, pp. -lastpage-01
[R]ISBN 978-1-53615-810-6

2019 Nova Science Publishers, Inc.

PACS:  03.65-w, 03.65Ca, 03.65Ta, 03.65.Ge, 03.67-a.

Keywords: Rényi entropy, Shannon entropy, Fisher information, Onicescu energy, Complexities,
Confined isotropic harmonic oscillator, Particle in a symmetric box.

[EC]Neetik Mukherjee and Amlan K. Roy
[OC]Free and confined harmonic oscillator $\cdots$

In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined
harmonic oscillator (CHO). Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology,
etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and
chemical changes. The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly,
in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained
situations. The most prominent amongst these are: Fisher information ($I$), Shannon entropy ($S$), Rényi entropy ($R$), Tsallis
entropy ($T$), Onicescu energy ($E$) and several complexities. Arguably, these are the most effective measures of uncertainty, as
they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several
physico-chemical properties of a system under investigation. Kullback$–$Leibler divergence or relative entropy describes
how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination
between two states. In other words, it extracts the change of information in going from one state to another.

The one-dimensional confined harmonic oscillator (1DCHO), defined by $v(x)=\frac{1}{2}(x-d_{m})^{2}+v_{c}$ ($k$ is the force
constant, $v_c=0,~x<x_c$ and $v_c=\infty,~x \ge x_c$), can be classified into two forms, (a) symmetrically confined harmonic
oscillator (SCHO) (when $d_m=0$), (b) asymmetrically confined harmonic oscillator (ACHO) (corresponding to $d_m \ne 0$). Further,
in latter case, confinement can be accomplished two different ways: (i) by changing the box boundary, keeping box length and
$d_m$ fixed at zero; (ii) another route is to adjust $d_m$ by keeping box length and boundary fixed. SCHO can be treated as an
intermediate between a particle-in-a-box (PIB) and a 1DQHO. Though the Schrödinger equation for SCHO
can be solved exactly, for ACHO it cannot be. We have employed two different methods for the latter, viz., (i) an
imaginary time propagation (ITP), leading to minimization of an expectation value (ii) a variation-induced exact diagonalization
procedure that utilizes SCHO eigenfunctions as basis. It is found that, at very low $x_c$ region, $I,~S,~R,~T,~E$ remain
invariant with change confinement length, $x_c$. At moderate $x_c$ region $S_{x},~R_{x},~T_{x}$ progress and $I_{x},~E_{x}$
decrease with rise in $x_c$. Additionally, special attention has been paid to study relative information in SCHO and ACHO.

Analogously, a 3DCHO (radically confined within an impenetrable spherical well) can act as a bridge between particle in a spherical
box (PISB) and isotropic 3DQHO. The time-independent Schrödinger equation for D-dimensional harmonic oscillator in both free and
confined condition can be solved exactly, within a Dirichlet boundary condition. Here a detailed exploration of information
measures has been carried out in $r$ and $p$
spaces, for a 3DCHO. Some exact analytical expressions for these quantities have also been presented, wherever possible. Besides
we also consider some recent works on relative information and several complexity measures in composite $r$ and $p$ spaces. Also
discussion is made of a recently proposed virial theorem in the context of confinement. In essence, this chapter provides a
detailed in-depth investigation about the information theoretical analysis in 1DCHO and 3DCHO, as done in our laboratory.

§ INTRODUCTION

“A quantum particle inside an infinite impenetrable box" usually constitutes the first model problem taught in quantum
mechanics curriculum. Later, the attention shifts towards the study of “free" systems in some characteristic potential,
such as a harmonic oscillator or a H atom in a Coulombic potential, which typify a quantum mechanical problem in a whole
region of space. This is itself a recognition of the importance of sub-region $\Omega$ of the space, which in this
particular scenario is achieved by modeling the boundary condition. Such a situation arises when one tries to explore systems
in highly inhomogeneous media or in intense external field. The limit, when the box becomes very small, approaching quantum
size is generally termed as “quantum confinement". In this microscopic domain, confinement occurs on a scale comparable to
atomic size (object and cavity sizes are commensurate), which is in sharp contrast to the ordinary confinement, where the
cavity size is usually much larger than the atom. Classic examples of such macroscopic-size boxes are found in celebrated
physics problems like kinetic theory of gas, black-body radiation, atom in a microwave enclosure, etc.

The first seminal work in this direction is due to a pioneering article [1] published in the fourth decade of
twentieth century, where the effect of pressure on static polarizability of a H atom enclosed inside a spherical impenetrable
hard enclosure was studied. It was realized that certain properties (such as energy/eigenvalue, orbital/eigenfunction,
electron localization, shell-filling, polarizability, photonic ionization/absorption, etc., to name a few) of an atom could be
significantly altered by “squeezing" or ”shrinking" it by placing inside a hard sphere, which acts as an idealized
classical piston. Depending on the geometrical form and dimension of cavity, such characteristic features of a caged-in atom
exhibits numerous deviations from corresponding free atom. Apparently there was some paucity in their immediate
applications, but in the last few decades, these models have found wide-spread applications in various branches in physics and
chemistry. Lately there has been a plethora of activities in harnessing their potential in various physico-chemical situations as
evidenced by an almost exponential growth in the number of articles published in journals, and this continues to grow day by day.
Excellent elegant reviews have been made available on the subject; we mention here a very selected set
[2, 3, 4, 5, 6].

With the advancement of modern technology, confined quantum systems has emerged as an extremely important contemporary research
area, from both theoretical and experimental perspectives. The discovery and development of modern experimental techniques have
provided the desired impetus about these boxed-in systems. Constrained atoms, molecules inside various cavities has shed light on
their electronic structure, chemical reactivity under high pressure. Apart from that, their relevance has been advocated in
numerous other instances, such as impurity in semiconductor/nanostructure, atoms trapped in zeolite sieves or fullerene cage,
solutes under solvent environment, artificial atom like quantum dot, quantum wire, quantum well, etc. Recently endohedral
metallofullerene clusters where atoms/molecules are typically encapsulated in carbon (or fullerene-like inorganic and hybrid)
cages of varying size, shape has found promising applications [7, 8, 9] in engineering. The subtle
interaction between host cage and inserted guest molecule is manifested in their marked structural, energetic and spectroscopic
properties of endocluster, in comparison to the isolated guest molecule or empty cavity.

The confined harmonic oscillator (CHO) remains one of the oldest, heavily studied model of quantum confinement. In this case the
quadratic potential acts through only a finite distance corresponding to the length of box; beyond this an infinite repulsive
potential exists. This has been used in the context of various complicated physical situations ever since its inception.
Some noteworthy applications include energy generation in dense stars from proton-deuteron transformation [10],
mass-radius relation in white dwarf theory [11], rate of escape of a star from a galaxy or globular cumulus
[12], electric and magnetic properties of metals including small system of electrons within a cylinder
[13, 14], vibronic spectra of point defects, impurities and luminescence in solids [15], specific
heats of metal under high pressure, effect of high pressure on properties of materials, etc.

Since the forty's decade, a vast amount of theoretical methods have been reported for accurate solution of a 1DCHO, placed inside
a hard impenetrable symmetric box. The usual boundary condition of wave function vanishing at infinity of unenclosed or free
harmonic oscillator (QHO) is replaced by the requirement that it vanishes at the walls of enclosure. Eigenfunctions in such systems
could then be written in terms of Kummer's function [10, 11], whereas the zeros of confluent hypergeometric
function provided energy eigenvalues. Further, the authors employed various approaches to expand the latter in order to find an
analytical expression in terms of box size. They found the correct qualitative behavior, i.e., energy levels of a 1DCHO increases
rapidly as box length diminishes. An asymptotic expression [16] for energy valid in the region of small length of box
was derived. An attempt [17] was made to locate the zeros of hypergeometric function numerically, albeit with
limited success. It was further established that while in a QHO (wave function vanishing at $\infty$), transitions can occur
only between two adjacent levels, in a bounded oscillator, there occurs a non-zero transition probability between any two
states of different parity. Simple analytical expressions [18] of energy in certain special cases were put forth.
The effect of finite boundary on energy levels has also been pursued graphically and from some series expansion [19].
Later, within a semi-classical Wentzel-Kramers-Brillouin (WKB)-type approximation [20], it was pointed out that
eigenvalues of the constrained
problem reduce to respective unenclosed oscillators, provided the classical turning points remain inside the potential
enclosure and not near the walls. Moreover, when the separation of turning points remains large in comparison to the box size,
they become plane-wave box eigenvalues. Subsequently, a series analytical solution [21] was proposed for
eigenvalues by keeping the center of oscillation at the center of potential enclosure. Eigenvalues were also computed by a
numerical procedure as the roots of a polynomial [22]. It was observed that when the box length remains below
a certain critical value (corresponding to the effective oscillation length for an unconfined oscillator of lowest energy),
the individual energies dramatically increase from uncaged values. Closed-form energy expressions valid for boxes of
any size were derived by constructing Padé approximants [23] as interpolation between the perturbative and
asymptotic solutions. Methods such as diagonal hyper-virial [24, 25] as well as hyper-virial
perturbation theory [26] were also employed for such enclosed systems for a variety of boundary conditions.
A Rayleigh-Ritz variation method utilizing the trigonometric basis functions was suggested in [27].
A numerical scheme [28], based on a set of theorems that guarantee that the solution of SE corresponding to a
bounded system strongly converges in the norm of Hilbert space $L_2 (-\infty, \infty)$ to exact solution of respective
unbounded problem. Spatially confined 1DQHO was treated by (WKB) along with modified airy function
method [29], as well as super-symmetric WKB approach [30]. Highly accurate eigenvalues were published in
[31] by numerically finding zeros of hypergeometric functions. Accurate eigenvalues were computed by power-series
method [32], perturbation theory [33], ITP method [34] in conjunction
with the minimization of an energy expectation value; in the former also the Einstein coefficients were considered. Very recently,
this problem has been revisited by diagonalizing the Hamiltonian using a PIB basis [35]. However, all the above
referenced works relate to the so-called symmetric confinement; asymmetric confinement studies are relatively less.
For example, in an asymmetric CHO (ACHO), the energy spectrum have been reported by means of perturbation theory [36],
power-series method [32] and ITP [34] method. Other properties such dipole moment [36] as well as
Einstein coefficients [36, 32] were also considered.

In parallel to the 1DCHO problem, much attention was also paid on 2D, 3D and $N$-dimensional counterparts, albeit with
less vigour. A number of interesting unique phenomena occur in higher dimensions, especially in connection to simultaneous,
incidental and inter-dimensional degeneracy, which are elaborated later in due course of time. In one of the earliest attempts,
bounded multi-dimensional isotropic harmonic oscillators, including a 3DCHO were investigated using a hyper-virial treatment
[37]. Energy eigenvalues of 2DCHO and 3DCHO in circular and spherical boxes respectively were reported through
Rayleigh-Schrödinger perturbation expansion, considering the free PIB as corresponding unperturbed systems, as well as
Padé approximant solution [38]. A 3DCHO within an impenetrable spherical cavity was approached by a direct
variational method [39], where the trial wave function was assumed as product of the “free" solution of corresponding
SE and a simple function obeying the respective boundary condition. For the 3DCHO, super-symmetric semi-classical approach
[40] has
been advocated. The radial SE corresponding to the 2DCHO and 3DCHO was solved quite efficiently by means of a variational procedure,
whereby the wave function was expanded into a Fourier-Bessel series [41, 42], with matrix elements involving
Bessel functions, that could be evaluated analytically. First detailed and systematic calculation of energies, eigenfunctions and
spatial expectation values in isotropic 3DCHO was undertaken in [43], employing a variational strategy. Like the 1DCHO
case, a WKB approach was also put forth in 3DCHO by the same author [44], where the centrifugal term is expanded
perturbatively (in powers of $\hbar$), partitioning in two parts, viz., the usual centrifugal potential governed by
classical laws and a quantum correction. A recipe was also provided for 3DCHO within the super-symmetric quantum mechanics
[45] in association with variational principle, as well as a generalized pseudo-spectral (GPS) method
[46, 47]. Eigenspectra of 2DCHO [48] and 3DCHO [49] were investigated analytically
in terms of annihilation and creation operators. A combination of semi-classical WKB method and a proper quantization rule
[50]
has also been suggested for the spherically confined harmonic oscillator.
Some reports have also been made on higher dimensional CHO, although in much lesser intensity. Reasonably [51] and
very accurate [52] eigenvalues have been reported for $D$-dimensional CHO from numerical calculations that find
the roots of confluent hypergeometric functions satisfying the necessary degeneracy conditions. High-precision energies were
presented [31] by exploiting certain special properties of hypergeometric functions within MAPLE computer
algebra system.

In the last few decades, quantum information theory has emerged as a subject of topical interest. This concept has been exploited
to understand various phenomena in physics and chemistry, with potential applications in broad topics like thermodynamics
[53, 54, 55], quantum mechanics [53, 54, 55], spectroscopy etc. Lately, it has been
extensively used to understand quantum entanglement and quantum steering problems [53, 54, 55]. The
information theoretic measures like Shannon entropy ($S$) [56, 57], Rényi entropy ($R^{\alpha}$) [58],
Fisher information ($I$), Onicescu energy ($E$) and complexities ($C$) are invoked to get knowledge about diffusion of atomic
orbitals, spread of electron density, periodic properties, correlation energy and so forth. Perhaps, these are the most effective
measures of uncertainty [56, 58, 59, 60], as
they do not make any reference to some specific points of the respective Hilbert space. In principle, they can have any real
value. However, $(-)$ve value in $R$ and $S$ indicates extreme localization, whereas, $E$ is always ($+$)ve. Likewise, changing
the numerical values of $R^{\alpha},~S$ from $(-)$ve to ($+$)ve only interprets enhancement of de-localization. Another related
concept is complexity. A system has finite complexity when it is either in a state with less than some maximal order or not
at a state of equilibrium. In a nutshell, it becomes zero at two limiting cases, viz., when a system is (i) completely
ordered (maximum distance from equilibrium) or (ii) at equilibrium (maximum disorder). Complexity has its contemporary interest
in chaotic systems, spatial patterns, language, multi-electronic systems, molecular or DNA analysis, social science, astrophysics
and cosmology etc.

In this chapter, we summarize the recent progress that has taken place in the information theoretical analysis in case of a
quantum harmonic oscillator (QHO) contained in inside a hard, impenetrable cage. To this end, a thorough investigation is made for
energy spectra and a host of information quantities, like $S, R, I, E, C$, in 1DCHO (symmetric and asymmetric) as well as a
radially confined QHO. In the 1D case, we introduce two accurate methods for solution of relevant eigenvalue equations in
presence of the Dirichlet boundary conditions, namely (i) ITP and (ii) variation-induced exact diagonalization–both developed in
our laboratory. They provide accurate eigenvalues and eigenfunctions, which are used throughout. In the 3D counterpart, we employ
the very accurate GPS method. One important conclusion is that, a 1DCHO behaves as an intermediate between a PIB and
an 1DQHO. Detailed results are compiled in tabular and graphical form for the afore-mentioned information-related quantities,
always indicating the transition from bounded to free system. Additionally we also offer a brief account of the relative
information studies that has been studied by us. Finally, we examine the validity and feasibility of a recently proposed virial
theorem in the context of such confined systems, which will consolidate the former's success further. Available literature results
have been consulted as and when possible. A few concluding remarks are made at the end.

§ THEORETICAL ASPECTS

This section is devoted to the theoretical methods that have been employed for calculation of relevant eigenvalues and
eigenfunctions for the spatially confined CHO; both in 1D and 3D. Then we proceed for a discussion on the evaluation of
momentum ($p$)-space wave functions from that in position space, as well as for the computation of information-related quantities.

§.§ Position-space wave function

In 1DCHO problem can be categorized into two distinct forms, viz., (i) in the SCHO case, potential minimum is at the
origin ($x=0$), leading to a symmetric confinement ($R$ is length of box),
($v_c(x)$ is the confinement potential),
and (ii) ACHO, where the potential minimum is beyond the origin, as exemplified here through the following relations,

\[
-\frac{1}{2} \frac{d^2 \psi}{dx^2} + \frac{1}{2}(x-d)^2 \psi +V(x) \psi = E \psi,
\]

where $V(x)= +\infty$, for $|x| \geq R$ and $V(x)=0$, when $|x| < R$. It represents an infinite square well of width $2R$ with
$d$ signifying position of minimum in the potential. The SCHO potential can be solved exactly, whereas in latter case
one needs an appropriate numerical method to get the best possible wave function and energy. Like the SCHO case, 3DCHO is also
exactly solvable and both wave functions are obtained in the form of Kummer confluent hypergeometric function. It is
crucial to point out that, in order to construct the exact wave functions of SCHO and 3DCHO for a specific state, however, one
requires to provide energy eigenvalue of that state. In our calculation, these energies of SCHO and 3DCHO have been generated
from ITP and GPS methods.

§.§.§ Exact Solution

In this part we are going to discuss the exact solutions of SCHO and 3DCHO respectively.

The time-independent non-relativistic SE is given by ($\alpha$ is force constant),

\begin{equation} \label{eq:1}
-\frac{1}{2}\frac{d^{2}\psi_{n}(x)}{dx^{2}}+4\alpha^{2}x^{2}\psi_{n}(x)+v_{c} (x) \psi_{n} (x) = \mathcal{E}_{n}\psi_{n} (x),
\end{equation}

where the confining potential is defined as, $v_{c}(x)=0$ for $x<|x_{c}|$ and $v_{c}(x)=\infty$ for $x \ge |x_{c}|$, with $x_c$
signifying length of the impenetrable box. In case of an 1DQHO, $v_{c} (x) =0$. The exact analytical solution of Eq. (<ref>)
for even and odd states are as follows,

\begin{equation} \label{eq:2}
\begin{aligned}
\psi_{e}(x)= N_{e}\ _{1}F_{1}\left[\left(\frac{1}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right),
\frac{1}{2},2\sqrt{2}\alpha x^{2}\right]e^{-\sqrt{2}\alpha x^{2}}, \\
\psi_{o}(x)=N_{o} \ x \ _{1}F_{1}\left[\left(\frac{3}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right),
\frac{3}{2},2\sqrt{2}\alpha x^{2}\right]e^{-\sqrt{2}\alpha x^{2}}.
\end{aligned}
\end{equation}

In this equation, $N_{e},N_{o}$ represent normalization constants for even, odd states respectively. It is imperative to mention
that, in order to obtain the specific solution for a particular state at a definite $x_c$, one can adopt either of the following two
procedures, viz., (i) use the Dirichlet boundary condition, $\psi_{n} (-x_{c})=\psi_{n}(x_{c})=0$ at a fixed $x_c$
to obtain $\mathcal{E}_{n}$ by solving the SE for certain $n$, and obtain wave function therefrom (ii) provide $\mathcal{E}_{n}$ as
input to get the allowed $x_c$, which further leads to the wave function. Here, we have solved SE by using ITP procedure at a
particular $x_c$ and calculated the respective energy spectrum. Then using this $\mathcal{E}_{n}$, the desired wave function is
constructed to proceed for further calculation. In Eq. (<ref>), $_{1}F_{1}(a;b;x)$ symbolizes the Kummer confluent
hypergeometric or confluent hypergeometric function of 1st kind assuming following form,

\begin{equation} \label{eq:3}
_1F_1(a;b;y)=1+\frac{a}{b}y+\frac{a(a+1)}{b(b+1)}\frac{y^{2}}{2!}+ \cdots =\sum_{k=0}^{\infty}\frac{a_{k}}{b_{k}}\frac{y^{k}}{k!}.
\end{equation}

Here $a_{k},b_{k}$ denote the  Pochhammer symbols. It is noteworthy to indicate that, if one replaces $\mathcal{E}_{n}$ by the 1DQHO
energy, $(n+\frac{1}{2})2\sqrt{2}\omega$ in Eq. (<ref>), then $\psi_{e}(x), \psi_{o}(x)$ modify as,

\begin{equation} \label{eq:4}
\begin{aligned}
\psi_{e}(x)= N_{e}\ _{1}F_{1}\left[-\left(\frac{j}{2}\right),
\frac{1}{2},2\sqrt{2}\alpha x^{2}\right]e^{-\sqrt{2}\alpha x^{2}}=\\ N_{e}H_{j}(2\sqrt{2}\alpha x^{2})e^{-\sqrt{2}\alpha x^{2}}, \\
\psi_{o}(x)=N_{o} x \ _{1}F_{1}\left[-\left(\frac{m}{2}\right),
\frac{3}{2},2\sqrt{2}\alpha x^{2}\right]e^{-\sqrt{2}\alpha x^{2}}=\\ N_{o} x \ H_{m}(2\sqrt{2}\alpha x^{2})e^{-\sqrt{2}\alpha x^{2}}.
\end{aligned}
\end{equation}

Here $H_{i}(2\sqrt{2}\alpha x^{2})$ signifies Hermite polynomials and  $j, m$ correspond to even, odd positive
integers respectively. Equation (<ref>) clearly suggests that, in an 1DQHO the hypergeometric function reduces to
Hermite polynomial.

Incidental degeneracy in SCHO: In this occasion, quantization is outcome of the boundary condition, which is
imposed by making the wave function vanish at $x_c$. Actually allowed energies are obtained by satisfying $_{1}F_{1}
[a;b;x]=0$ at $x_{c}$. The hypergeometric function takes the form given below, at $x=x_{c}$ (considering even-parity states),

\begin{equation} \label{eq:5}
\begin{aligned}
_{1}F_{1}\left[\left(\frac{1}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right),\frac{1}{2},2\sqrt{2}\alpha x_{c}^{2}\right]= 1+
\left(\frac{1}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)4\sqrt{2}\alpha x_{c}^{2} + \\ \left(\frac{1}{4}-
\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)
\left(\frac{5}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)\frac{32}{3}\alpha^{2} x_{c}^{4}+ \\ \left(\frac{1}{4}-
\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)
\left(\frac{5}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)\left(\frac{9}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)
\frac{128}{15}\alpha^{3} x_{c}^{6}+ \cdots
\end{aligned}
\end{equation}

Let us consider only two terms in the right-hand series of Eq. (<ref>). Then we get,

\begin{equation} {\label{eq:6}}
\left(\frac{5}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right)=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{E}_{n}=5\sqrt{2}\alpha.
\end{equation}

Now, putting the value of $\mathcal{E}_{n}=5\sqrt{2}\alpha$ and $x=x_{c}$ in the boundary condition we get,

\begin{equation} {\label{eq:7}}
\begin{aligned}
1+\left(\frac{1}{4}-\frac{E_{n}}{4\sqrt{2}\alpha}\right)4\sqrt{2}\alpha x_{c}^{2}=0,  \\
1+\left(\frac{1}{4}-\frac{5\sqrt{2}\alpha}{4\sqrt{2}\alpha}\right)4\sqrt{2}\alpha x_{c}^{2}=0, \\
(1-4\sqrt{2}\alpha x_{c}^{2})=0, \\
4\sqrt{2}\alpha x_{c}^{2}=1, \\
x_{c}=\pm \frac{1}{2\sqrt{\sqrt{2}\alpha}}.
\end{aligned}
\end{equation}

Since box length is positive, we choose $x_{c}=\frac{1}{2\sqrt{\sqrt{2}\alpha}}$. This shows that, at this $x_c$,
$_{1}F_{1}\left[\left(\frac{1}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}\right),\frac{1}{2},2\sqrt{2}\alpha x^{2}\right]$ with
two terms always represents a non-degenerate ground state having $\mathcal{E}_{0}=5\sqrt{2}\alpha$.
Next, let us proceed to the function having three terms. Therefore,

\begin{equation}
\frac{9}{4}-\frac{\mathcal{E}_{n}}{4\sqrt{2}\alpha}=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \  \mathcal{E}_{n}=9\sqrt{2}\alpha.
\end{equation}

Using the boundary condition and putting $\mathcal{E}_{n}=9\sqrt{2}\alpha$ and $x=x_{c}$ we obtain,

\begin{equation} {\label{eq:8}}
\begin{aligned}
1+2\left(\frac{1}{4}-\frac{E_{n}}{4\sqrt{2}\alpha}\right)2\sqrt{2} \alpha x_{c}^{2}+\\ \left(\frac{1}{4}-\frac{E_{n}}{4\sqrt{2}
\alpha}\right)
\left(\frac{1}{4}-\frac{E_{n}}{4\sqrt{2}\alpha}+1\right)\frac{32}{3}\alpha^{2} x_{c}^{4}=0, \\
64\alpha^{2} x_{c}^{4}-24\sqrt{2} \alpha x_{c}^{2}+3=0,\\
x_{c}^{2}=\frac{3+\sqrt{3}}{8\sqrt{2}\alpha} \ \ \  \rightarrow \ \ \  x_{c}=\pm \sqrt{\frac{3+\sqrt{3}}{8\sqrt{2}\alpha}}, \\
\ \ \ \ \ \ \ \
x_{c}^{2}=\frac{3-\sqrt{3}}{8\sqrt{2}\alpha} \ \ \  \rightarrow \ \ \  x_{c}=\pm \sqrt{\frac{3-\sqrt{3}}{8\sqrt{2}\alpha}}.
\end{aligned}
\end{equation}

Equation (<ref>) suggests that, in SCHO, $\mathcal{E}_{n}=9\sqrt{2}\alpha$ represents a pair of states; one at
$x_c=\sqrt{\frac{3+\sqrt{3}}{8\sqrt{2}\alpha}}$ and other at $x_{c}= \sqrt{\frac{3-\sqrt{3}}{8\sqrt{2}\alpha}}$. The former
represents a second excited state and latter symbolizes ground state. Similar analysis will reveal that, a function
$_{1}F_{1}[a,b,y]$ with $(n+1)$ terms has $n$ number of such degenerate states at $n$ different $x_c$ values. Additionally,
in such degenerate set, the smallest box length contains the ground state and longest box length holds the $n$th state. This
procedure helps us to identify incidental degeneracy in SCHO at different box lengths.

The time-independent, non-relativistic wave function for a 3DCHO system, in $r$ space may be written as ($n_r$ identifies
radial quantum number),

\begin{equation} \label{eq:9}
\Psi_{n_r,\ell,m} (\rvec) = \psi_{n_r, \ell}(r)  \ Y_{\ell,m} (\Omega),
\end{equation}

with $r$ and $\Omega$ illustrating radial distance and solid angle successively. Here $\psi_{n_{r},\ell}(r)$ represents the
radial part and $Y_{\ell,m}(\Omega)$ identifies spherical harmonics of the wave function. The pertinent radial Schrödinger
equation under the influence of confinement, with $v(r)=\frac{1}{2}\omega^{2}r^{2}$, is (atomic unit employed unless
otherwise mentioned),

\begin{equation} \label{eq:10}
\left[-\frac{1}{2} \ \frac{d^2}{dr^2} + \frac{\ell (\ell+1)} {2r^2} + v(r) +v_c (r) \right] \psi_{n_r,\ell}(r)=
\mathcal{E}_{n_r,\ell}\ \psi_{n_r,\ell}(r).
\end{equation}

Our required confinement effect is introduced by invoking the following potential: $v_c(r) = +\infty$ for $r > r_c$, and $0$
for $r \leq r_c$, where $r_c$ signifies radius of confinement.

The exact generalized radial wave function is mathematically expressed [52] as,

\begin{equation} \label{eq:11}
\psi_{n_{r}, \ell}(r)= N_{n_{r}, \ell} \ r^{\ell} \ _{1}F_{1}\left[\frac{1}{2}\left(\ell+\frac{3}{2}-
\frac{\mathcal{E}_{n_{r},\ell}}{\omega}\right), (\ell+\frac{3}{2}),\omega r^{2}\right] e^{-\frac{\omega}{2}r^{2}}.
\end{equation}

Here $N_{n_r, \ell}$ represents normalization constant, $\mathcal{E}_{n_r,\ell}$ corresponds to energy of a given state
characterized by quantum numbers $n_r,\ell$, whereas $_1F_1\left[a,b,r\right]$ signifies confluent hypergeometric
function. Allowed energies are computed by applying the boundary condition $\psi_{n_r,\ell} (0)= \psi_{n_r,\ell} \ (r_c)=0$.
In this work, the GPS method has been used to calculate eigenvalues and eigenfunctions of these states. It has
provided highly accurate results for various model and real systems including atoms, molecules, some of which could be found
in the references [61, 62, 63, 64, 65, 66, 46, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 47, 77, 78].

Incidental degeneracy in 3DCHO:
Just like the 1DCHO, quantization in a D-dimensional CHO is also an manifestation of the effect of making radial wave function vanishing
at $r_{c}$. In this occasion, allowed energies are obtained when,

\begin{equation} \label{eq:12}
\frac{D}{2}\right); 2\sqrt{2}\alpha r_{c}^{2}\right]=0,
\end{equation}

where, at a fixed $\ell$, the successive roots are numbered $n_{r}=0,1,2,...$. Note that, the levels are designated by $n_{r} +1$
and $\ell$ values, such that $n_{r} =\ell = 0$, $n_r = \ell=2$ correspond to $1s$ and $3d$ states respectively. The radial quantum
number $n_r$ relates to $n$ as $n = 2n_r + \ell$. Invoking the series expansion [79] in left-hand side, we get,

\begin{equation} \label{eq:13}
\begin{aligned}
& _{1}F_{1}\left[\frac{1}{2}\left(\ell+\frac{D}{2}-\frac{\mathcal{E}_{n_{r},\ell}}{2\sqrt{2}\alpha}\right);\left(\ell+
\frac{D}{2}\right); 2\sqrt{2}\alpha r_{c}^{2}\right]  =\\
{4\sqrt{2}\alpha}\right)}{\left(\ell+ \frac{D}{2}\right)}2\sqrt{2}\alpha r_{c}^{2}+ \\
&  \frac{\left(\frac{\ell}{2}+\frac{D}{4}-\frac{\mathcal{E}_{n_{r},\ell}}{4\sqrt{2}\alpha}\right) \left(\frac{\ell}{2}+\frac{D}{4}-
\frac{\mathcal{E}_{n_{r},\ell}}{4\sqrt{2}\alpha}+1\right)}{\left(\ell+\frac{D}{2}\right) \left(\ell+\frac{D}{2}+1\right)}\frac{8}
{2!}\alpha^{2} r_{c}^{4}+ \cdots
\end{aligned}
\end{equation}

In order to terminate the series after 2nd term we need,

\begin{equation} \label{eq:14}
\begin{aligned}
\left(\frac{\ell}{2}+\frac{D}{4}-\frac{\mathcal{E}_{n_{r},\ell}}{4\sqrt{2}\alpha}+1\right)=0 \\
\frac{\mathcal{E}_{n_{r},\ell}}{2\sqrt{2}\alpha} = 1+\frac{\ell}{2}+\frac{D}{4} \\
\mathcal{E}_{n_{r},\ell}=\sqrt{2} \alpha \left(D+2\ell+4\right).
\end{aligned}
\end{equation}

Now, it is important to determine the corresponding $r_{c}$ values at which Eq. (<ref>) is valid. According to Eq. (<ref>),
this corresponds to the requirement that,

\begin{equation}
\begin{aligned}
2\sqrt{2}\alpha r_{c}^{2}=0 \\
2\sqrt{2}\alpha r_{c}^{2}=0 \\ r_{c}=\sqrt{\frac{1}{2\sqrt{2}\alpha}\left(\ell+\frac{D}{2}\right)}.
\end{aligned}
\end{equation}

This shows that, degeneracy depends on $\ell$ and $D$ values. Let us consider some examples,

* $l=1,~D=2$, $\mathcal{E}_{n_{r},1}=8\sqrt{2}\alpha$,   $l=0,~D=4$, $\mathcal{E}_{n_{r},0}=8\sqrt{2}\alpha$   at  $r_{c}=\sqrt{\frac{1}{\sqrt{2}\alpha}}$,

* $l=1,~D=3$, $\mathcal{E}_{n_{r},1}=9\sqrt{2}\alpha$,   $l=0,~D=5$, $\mathcal{E}_{n_{r},0}=9\sqrt{2}\alpha$   at  $r_{c}=\sqrt{\frac{5}{4\sqrt{2}\alpha}}$,

* $l=1,~D=2$, $\mathcal{E}_{n_{r},2}=10\sqrt{2}\alpha$,   $l=2,~D=4$, $\mathcal{E}_{n_{r},1}=10\sqrt{2}\alpha$   at  $r_{c}=\sqrt{\frac{3}{2\sqrt{2}\alpha}}$,

* $l=1,~D=3$, $\mathcal{E}_{n_{r},2}=11\sqrt{2}\alpha$,   $l=2,~D=5$, $\mathcal{E}_{n_{r},1}=11\sqrt{2}\alpha$   at  $r_{c}=\sqrt{\frac{7}{4\sqrt{2}\alpha}}$,

* $l=2,~D=4$, $\mathcal{E}_{n_{r},2}=12\sqrt{2}\alpha$,   $l=3,~D=2$, $\mathcal{E}_{n_{r},1}=12\sqrt{2}\alpha$   at  $r_{c}=\sqrt{\frac{2}{\sqrt{2}\alpha}}$.

From above instances it is clear that, degeneracy appears at a certain $r_{c}$. However, the selection rule for this
degeneracy can be achieved from the following relation. Suppose, at a fixed $r_{c}$ we have a pair of states, characterized by
quantum numbers $l_{1},l_2$, in dimensions $D_{1}, D_2$ respectively, having same energy. As a consequence of the above, we have,

\begin{equation} \label{eq:15}
\sqrt{2} \alpha (D_{1}+2l_{1}+4)= \sqrt{2} \alpha (D_{2}+2l_{2}+4).
\end{equation}

\begin{equation} \label{eq:16}
\sqrt{\frac{1}{2\sqrt{2}\alpha}\left(l_{1}+\frac{D_{1}}{2}\right)}= \sqrt{\frac{1}{2\sqrt{2}\alpha}\left(l_{2}+\frac{D_{2}}{2}
\right)}
\end{equation}

Equations (<ref>) and (<ref>) lead to the common relation,

\begin{equation} \label{eq:17}
\end{equation}

It is noticeable from Eq. (<ref>) that, if $D_{2}>D_{1}$ then $l_{1}>l_{2}$ and vice-versa. Thus the incidental degeneracy
selection rule is,

\begin{equation} \label{eq:18}
(n_{r},l,D,r_{c}) \ \ \rightarrow \ \ (n_{r},l \pm j,D \mp 2j, r_{c}),
\end{equation}

where $j$ is an integer. It is noticed that, Eq. (<ref>) may further be written in following form,

\begin{equation}
\begin{aligned}
\mathcal{E}_{n_{r},(l+j)}^{D}(r_{c})= \mathcal{E}_{n_{r},l}^{(D+2j)}(r_{c}), \\
\mathcal{E}_{n_{r},l}^{(D-2j)}(r_{c})= \mathcal{E}_{n_{r},(l-j)}^{D}(r_{c}).
\end{aligned}
\end{equation}

Thus, in a 3DCHO, incidental and inter-dimensional degeneracy occur simultaneously.

Simultaneous degeneracy in 3DCHO:
It can be proved that, at $r_{c}=\sqrt{\frac{1}{2\sqrt{2}\alpha}\left(l+\frac{D}{2}\right)}$, the pair of energies
$\mathcal{E}_{n_{r},(l+2)}^{D}$ and $\mathcal{E}_{(n_{r}+1),l}^{D}$ are related by the following relation,

\begin{equation}
\mathcal{E}_{(n_{r}+1),l}^{D}\left(\sqrt{\frac{1}{2\sqrt{2}\alpha}\left(l+\frac{D}{2}\right)}\right) =
\mathcal{E}_{n_{r},(l+2)}^{D}\left(\sqrt{\frac{1}{2\sqrt{2}\alpha}\left(l+\frac{D}{2}\right)}\right).
\end{equation}

Inter-dimensional degeneracy in 3DCHO:
Such a degeneracy occurs when,

\begin{equation}
(n_{r},l,D) \rightarrow (n_{r},l \pm j, D \mp 2j).
\end{equation}

This can be demonstrated by the example as given below,

\begin{equation}
\mathcal{E}_{0,4}^{2}(r_{c})= \mathcal{E}_{0,3}^{4}(r_{c}) = \mathcal{E}_{0,2}^{6}(r_{c}) =\mathcal{E}_{0,1}^{8}(r_{c})
\end{equation}

In this scenario, the terms inter-dimensional and incidental degeneracy are synonymous.

§.§.§ Imaginary-time propagation (ITP) method

In this subsection, we discuss the general concepts of the imaginary-time evolution technique, as employed here for a particle under
confinement. The scheme has been found to provide accurate bound-state solutions through a transformation of the appropriate
time-dependent (TD) SE into a diffusion equation in imaginary time.  This resulting equation is numerically solved through a
finite-difference procedure, in amalgamation with a minimization of expectation value to hit the ground state. After the original
proposal that came several decades ago, a number of successful implementations [80, 81, 82, 83, 84, 85, 86] have been reported in the literature since then. In this work we have adopted an implementation,
which has been successfully applied to a number of physical systems, such as atoms, diatomic molecules within a quantum fluid
dynamical density functional theory (DFT) [87, 88], as well as some model (harmonic, anharmonic, self-interacting,
spiked oscillators) potentials [89, 90, 91, 92, 93], in both 1D, 2D and 3D. Recently this was also
extended to confinement (SCHO and ACHO) problems [34] with very good success.

In the following, we provide a short account of the essentials of the methodology; more details could be found in the references
quoted above. Let us begin with the TDSE, which for a particle under the influence of a potential $v(x)$ in 1D, in atomic unit, is
given by ($v(x)=4 \omega^2 x^2$ in an SCHO),

\begin{equation} \label{eq:itp1}
i \frac{\partial}{\partial t} \psi (x, t) = H \psi(x, t) =
\left[ -\frac{1}{2} \frac{d^2}{dx^2} + v(x) + v_c(x) \right] \psi(x, t).
\end{equation}

The Hamiltonian operator consists of usual kinetic and potential energy operators. This method is, in principle, exact.
Here we have provided the equations for 1D SCHO; however this has been easily extended to higher dimensions (see, e.g.,
[92, 93]. The confinement condition can be achieved by reducing the boundary from infinity to finite region,
as expressed in the following equation (symmetric box of length $2R$),

\begin{equation} v_c(x) = \begin{cases}
0,  \ \ \ \ \ \ \ -R < x < +R   \\
+\infty, \ \ \ \ |x| \geq R.  \\
\end{cases}
\end{equation}

Equation (<ref>) can be expressed in imaginary time, $\tau= i t$ ($t$ is real time) to yield a non-linear diffusion-type
equation similar to a diffusion quantum Monte Carlo equation [94],

\begin{equation} \label{eq:itp2}
- \ \frac{d \psi(x, \tau) }{d \tau} =  H \psi(x, \tau).
\end{equation}

Its formal solution can be written as,

\begin{equation}
\psi(x, \tau) = \sum_{k=0}^{\infty} c_k \psi_k(x) \exp{(-\epsilon_k \tau)}.
\end{equation}

The initial guessed wave function has the form, $\psi(x, \tau)$ at $\tau=0$ and if propagated for a sufficiently long time, it will
finally converge towards the desired stationary ground-state wave function;
\lim_{\tau \rightarrow \infty} \psi(x, \tau) \approx c_0 \psi_0 (x) e^{-\epsilon_0 \tau}.
Thus, provided $c_0 \neq 0$, apart from a normalization constant, this leads to the global minimum corresponding to an
expectation value $\langle \psi(x, \tau) |H| \psi(x,\tau) \rangle$.

The numerical solution of Eq. (<ref>) can be obtained by using a Taylor series expansion of
$\psi(x,\tau+\Delta \tau)$ around time $\tau$ as follows,

\begin{equation} \label{eq:itp3}
\psi(x, \tau+\Delta t)= e^{-\Delta \tau H} \psi(x, \tau).
\end{equation}

Here, the exponential in right side represents a time-evolution operator, which propagates diffusion function $\psi(x, \tau)$
at an initial time $\tau$ to an advanced time step to $\psi(x, \tau+\Delta \tau)$. Since this is a non-unitary operator,
normalization of the function at a given time $\tau$ does not necessarily preserve the same at a future time $\tau+\Delta \tau$.
Transformation of Eq. (<ref>) into an equivalent, symmetrical form leads to ($j,n$ signify space, time indices

\begin{equation}
e^{(\Delta \tau/2) H_j} \ \psi_j^{'(n+1)}= e^{-(\Delta \tau/2)H_j} \ \psi_j^n.
\end{equation}

A prime is introduced in above equation to indicate the unnormalized diffusion function. Taking the full form of
Hamiltonian from Eq.(<ref>), one can further write,

\begin{equation}
e^{(\Delta \tau/2) [-\frac{1}{2}D_x^2+v(x_j)]} \ \psi_j^{'(n+1)} = e^{-(\Delta \tau/2) [-\frac{1}{2}D_x^2+v(x_j)]}
\ \psi_j^n,
\end{equation}

where the spatial second derivative has been defined as $D_x^2= \frac{d^2}{dx^2}$. Now, expanding the exponentials on both sides,
followed by truncation after second term and approximation of second derivative by a five-point difference formula
[79] ($\Delta x= h$),

\begin{equation}
D_x^2 \ \psi_j^n  \approx  \frac{-\psi_{j-2}^n +16\psi_{j-1}^n -30\psi_j^n+ 16\psi_{j+1}^n-\psi_{j+2}^n}{12 h^2},
\end{equation}

yields a set of $N$ simultaneous equations, as follows,

\begin{equation}
\alpha_j \psi_{j-2}^{'(n+1)} + \beta_j \psi_{j-1}^{'(n+1)} + \gamma_j \psi_{j}^{'(n+1)}
\delta_j \psi_{j+1}^{'(n+1)} + \zeta_j \psi_{j+2}^{'(n+1)} = \xi_j^n.
\end{equation}

After some straightforward algebra, the quantities $\alpha_j, \beta_j, \gamma_j, \delta_j, \zeta_j, \xi_j^n$ are identified as,

\begin{eqnarray}
\alpha_j & = & \zeta_j = \frac{\Delta \tau}{48 h^2}, \ \
\beta_j = \delta_j = -\frac{\Delta \tau}{3 h^2}, \ \ \
\gamma_j =  1+\frac{ 5\Delta \tau}{8 h^2} + \frac{\Delta \tau }{2} \ v(x_j), \\ \nonumber
\xi_j^n & = & \left[ -\frac{\Delta \tau}{48 h^2} \right] \psi_{j-2}^n + \left[ \frac{\Delta \tau}{3 h^2} \right] \psi_{j-1}^n +
\left[ 1- \frac{5 \Delta \tau}{8 h^2} - \frac{\Delta \tau}{2} \ v(x) \right] \psi_j^n     \\
& & + \left[ \frac{\Delta \tau}{3 h^2} \right] \psi_{j+1}^n + \left[ -\frac{\Delta \tau}{48 h^2} \right] \psi_{j+2}^n. \nonumber
\end{eqnarray}

Since discretization and truncation occur on both sides, there may be some cancellation of errors. Here, $\psi^{'(n+1)}$ signifies
the unnormalized diffusion function at some future time $\tau_{n+1}$. The quantities like $\alpha_j, \beta_j, \gamma_j, \delta_j,
\zeta_j$, after some algebraic manipulation, can be easily expressed in terms of space and time spacings as well as the potential
(appears only in $\gamma_j$ and $\xi_j^n$).  The latter also requires knowledge of normalized diffusion functions $\psi_{j-2}^n,
\psi_{j-1}^n, \psi_j^n, \psi_{j+1}^n, \psi_{j+2}^n$ at spatial grid points $x_{j-2}, x_{j-1}, x_j, x_{j+1}, x_{j+2}$ at an earlier time
$\tau_n$. It is convenient to recast this in a pentadiagonal matrix form, as below,

\begin{equation}
\left[ \begin{array}{ccccccc}
\gamma_1  &  \delta_1  & \zeta_1    &               &                &                 &      (0)       \\
\beta_2   &  \gamma_2  & \delta_2   &  \zeta_2      &                &                 &                \\
\alpha_3  &  \beta_3   & \gamma_3   & \delta_3      &  \zeta_3       &                 &                \\
&  \ddots    & \ddots     & \ddots        &  \ddots        &  \ddots         &                \\
&            & \ddots     & \ddots        & \ddots         &  \ddots         & \zeta_{N-2}    \\
&            &            & \alpha_{N-1}  & \beta_{N-1}    &  \gamma_{N-1}   & \delta_{N-1}   \\
(0)   &            &            &               & \alpha_N       & \beta_N         & \gamma_N       \\
\end{array} \right]
\left[ \begin{array}{c}
\psi_1^{'(n+1)} \\
\psi_2^{'(n+1)} \\
\psi_3^{'(n+1)} \\
\vdots \\
\psi_{N-2}^{'(n+1)} \\
\psi_{N-1}^{'(n+1)} \\
\psi_N^{'(n+1)}
\end{array} \right]
\left[ \begin{array}{c}
\xi_1^n \\
\xi_2^n \\
\xi_3^n \\
\vdots \\
\xi_{N-2}^n \\
\xi_{N-1}^n \\
\xi_N^n
\end{array} \right].
\end{equation}

The above matrix equation can be readily solved for $\{\psi^{'(n+1)}\}$ using standard routine, satisfying the required boundary condition
$\psi_1^n=\psi_N^n=0$, at all time. Hence, from an initial trial function $\psi_j^0$ at $n=0$ time step, the diffusion function can
be propagated according to Eq. (29) following the sequence of steps as delineated above. Then at a given time level $(n+1)$,
the following series of instructions are carried out, viz., (a) $\psi_j^{'(n+1)}$ is normalized to $\psi_j^{(n+1)}$ (b) if one
is interested in an excited-state calculation, then $\psi_j^{'(n+1)}$ is required to be made orthogonalized with respect to all lower
states (we use the standard Gram-Schmidt method) (c) desired expectation values are computed as $\epsilon_0 = \langle \psi^{(n+1)} | H |
\psi^{(n+1)} \rangle$ (d) difference in the observable expectation values between two successive time steps, $\Delta \epsilon = \langle H
\rangle^{(n+1)} - \langle H \rangle^n$, is monitored, and (e) until the above discrepancy $\Delta \epsilon$, goes below a certain
threshold tolerance limit, one proceeds with the calculation of $\psi_j^{(n+2)}$ at the next time level iteratively. The guessed functions
for even and odd states were chosen as simple Gaussian-type functions such as $e^{-x^2}$ and $xe^{-x^2}$ respectively. All the integrals
were evaluated with the help of standard Newton-Cotes quadratures [79].

§.§.§ Generalized pseudospectral (GPS) method

All the 3DCHO calculations in this work, were done by using the the GPS formalism. It provides accurate eigenvalues and eigenfunctions
easily various $r_c$. By means of an optimal, non-uniform spatial grid, it leads to a symmetric eigenvalue problem, which can be
easily solved by standard diagonalization routines available. It has been applied to a series of model and real systems, in both
free and confined cases, viz., spiked harmonic oscillators [61, 70], power-law and logarithmic
[62], Húlthen and Yukawa [64], Hellmann [68], rational [69], exponential-screened
Coulomb [71], Morse [72], hyperbolic [73], Deng-Fan [74], Manning-Rosen
[75], Tietz-Hua [76], other singular [65] potentials, as well as many-electron systems
within the broad domain of DFT [95, 96, 88, 63, 66, 67]. Of late
this has also produced excellent quality results in various radial confinement [46, 47, 77, 78] studies
in several Coulombic systems, as well as in 3DCHO.

The key characteristic of this method is to approximate an exact wave function $f(x)$ defined in the period $[-1,1]$
by the $N$th-order polynomial $f_{N}(x)$,

\begin{equation} \label{eq:23}
f(x) \cong f_{N}(x)=\sum_{j=0}^{N} f(x_{j})g_{j}(x),
\end{equation}

and confirm the estimation to be exact at the collocation points $x_{j}$,

\begin{equation}
\end{equation}

In this work, the authors have employed the Legendre pseudo-spectral method where $x_{0}=-1$,$x_{N}=1$, and $x_{j}(j=1,....,N-1)$ are
defined by the roots of first derivative of Legendre polynomial $P_{N}(x)$, with respect to $x$, namely,

\begin{equation}
\end{equation}

In Eq. (<ref>), $g_{j}(x)$ are termed cardinal functions, and as such, are given by,

\begin{equation}
\end{equation}

fulfilling the unique property that, $g_{j}(x_{j^{'}})=\delta_{j^{'},j}$.

The co-efficients $a_k, b_j$ for even-$l$ $p$-space wave functions in central potential [97].

$l$     $b_{0}$     $b_{2}$     $b_{4}$     $b_{6}$     $b_{8}$     $a_{1}$     $a_{3}$     $a_{5}$     $a_{7}$

$0$     $\frac{1}{\sqrt{\pi}}$        –             –            –         –         –         –          –           –

$2$     $\frac{1}{\sqrt{\pi}}$        $-$$\frac{3}{\sqrt{\pi}}$          –            –             –            $\frac{3}{\sqrt{\pi}}$
–             –           –

$4$     $\frac{1}{\sqrt{\pi}}$      $-$$\frac{105}{\sqrt{\pi}}$        $\frac{315}{\sqrt{\pi}}$          –             –
$\frac{30}{\sqrt{\pi}}$        $-$$\frac{315}{\sqrt{\pi}}$        –           –

$6$     $\frac{1}{\sqrt{\pi}}$       $-$$\frac{210}{\sqrt{\pi}}$        $\frac{4725}{\sqrt{\pi}}$        $-$$\frac{10395}{\sqrt{\pi}}$
–      $\frac{21}{\sqrt{\pi}}$        $\frac{-1260}{\sqrt{\pi}}$       $\frac{10395}{\sqrt{\pi}}$       –

$8$     $\frac{1}{\sqrt{\pi}}$      $-$$\frac{630}{\sqrt{\pi}}$      $\frac{51975}{\sqrt{\pi}}$      $-$$\frac{945945}{\sqrt{\pi}}$
$\frac{2027025}{\sqrt{\pi}}$     $\frac{36}{\sqrt{\pi}}$       $-$$\frac{6930}{\sqrt{\pi}}$     $\frac{270270}{\sqrt{\pi}}$

In order to solve the radial SE for a central potential using finite-difference methods, it requires a significantly large number
of points in an equal-spacing grid arrangement. However, in GPS method, this is alleviated by (i) mapping semi-infinite domain
$r \in [0,\infty]$ onto a finite domain $x \in [-1,1]$ via a transformation $r=r(x)$, and then (ii) employing a Legendre
pseudo-spectral discretization technique. Utilizing an algebraic non-linear mapping,

\begin{equation} \label{eq:25}
\end{equation}

where $L, \alpha=\frac{2L}{r_{max}}$ denote two mapping parameters, plus a symmetrization procedure,

\begin{equation}
\psi(r(x))=\sqrt{r^{'}(x)}f(x),
\end{equation}

leads to the transformed Hamiltonian, as given below,

\begin{equation}
\hat{H}(x)=-\frac{1}{2}\frac{1}{r^{'}(x)}\frac{d^{2}}{dx^{2}}\frac{1}{r^{'}(x)}+V(r(x))+V_{m}(x),
\end{equation}

\begin{equation}
\end{equation}

This turns out to be a symmetric matrix eigenvalue problem. The mapping used in Eq. (<ref>) is such that $V_{m}(x)=0$.
Eventually, we obtain following discrete set of coupled equations,

\begin{equation} \label{eq:26}
\sum_{j=0}^{N}\left[-\frac{1}{2}D^{(2)}_{j^{'}j}+\delta_{j^{'}j}V(r(x_{j}))+\delta_{j^{'}j}V_{m}(r(x_{j}))\right]A_{j}=
\mathrm{E}A_{j'}, \  J=1,...,N-1,
\end{equation}

\begin{equation}
A_{j}=R^{'}(x_{j})f(x_{j})[P_{N}(x_{j})]^{-1}= [r^{'}(x_{j})]^{\frac{1}{2}}\psi(r(x_{j}))[P_{N}(x_{j})]^{-1}.
\end{equation}

Here, $D^{(2)}_{j^{'}j}$ signifies symmetrized second derivative of cardinal function with respect to $r$,

\begin{equation}
\end{equation}

\begin{equation} d^{(2)}_{j^{'}j} = \begin{cases}
\frac{(N+1)(N+2)}{6(1-x^{2}_{j})},  \ \ \ \  j=j^{'},   \\
\frac{1}{(x_{j}-x_{j^{'}})^{2}}, \ \ \ \ j \neq j^{'}.  \\
\end{cases}
\end{equation}

The symmetric eigenvalue problem can be easily and efficiently be solved by standard library routines, to obtain accurate
eigenvalues and eigenfunctions.

The co-efficients $a_k,b_j$ for odd-$l$ $p$-space wave functions in central potential [97].

$l$     $a_{0}$     $a_{2}$     $a_{4}$     $a_{6}$     $a_{8}$

$1$     $\frac{1}{\sqrt{\pi}}$       –         –            –                –

$3$     $\frac{1}{\sqrt{\pi}}$      $-$$\frac{15}{\sqrt{\pi}}$        –            –                 –

$5$     $\frac{1}{\sqrt{\pi}}$      $-$$\frac{105}{\sqrt{\pi}}$        $\frac{945}{\sqrt{\pi}}$       –                –

$7$     $\frac{1}{\sqrt{\pi}}$     $-$$\frac{378}{\sqrt{\pi}}$       $\frac{17325}{\sqrt{\pi}}$     $-$$\frac{135135}{\sqrt{\pi}}$

$9$     $\frac{1}{\sqrt{\pi}}$       $-$$\frac{990}{\sqrt{\pi}}$     $\frac{135135}{\sqrt{\pi}}$      $-$$\frac{4729725}{\sqrt{\pi}}$

$l$       $b_{1}$     $b_{3}$     $b_{5}$     $b_{7}$     $b_{9}$

$1$     $-$$\frac{1}{\sqrt{\pi}}$         –          –               –             –

$3$     $-$$\frac{3!}{\sqrt{\pi}}$        $\frac{15}{\sqrt{\pi}}$        –               –             –

$5$     $-$$\frac{15}{\sqrt{\pi}}$        $\frac{420}{\sqrt{\pi}}$      $-$$\frac{945}{\sqrt{\pi}}$          –         –

$7$     $-$$\frac{28}{\sqrt{\pi}}$      $\frac{3150}{\sqrt{\pi}}$     $-$$\frac{2370}{\sqrt{\pi}}$
$\frac{135135}{\sqrt{\pi}}$     –

$9$     $-$$\frac{45}{\sqrt{\pi}}$      $\frac{13860}{\sqrt{\pi}}$      $-$$\frac{945945}{\sqrt{\pi}}$
$\frac{16216200}{\sqrt{\pi}}$      $-$$\frac{34459425}{\sqrt{\pi}}$

§.§ Momentum-space wave function

The $p$-space wave function ($\pvec = \{ p, \Omega \}$) for a particle in a central potential is obtained from respective
Fourier transform of its $r$-space counterpart [97], and as such, is given below,

\begin{equation} \label{eq:psi_p}
\begin{aligned}
\psi_{n,l}(p) & = & \frac{1}{(2\pi)^{\frac{3}{2}}} \  \int_0^\infty \int_0^\pi \int_0^{2\pi} \psi_{n,l}(r) \ \Theta(\theta)
\Phi(\phi) \ e^{ipr \cos \theta}  r^2 \sin \theta \ \mathrm{d}r \mathrm{d} \theta \mathrm{d} \phi,  \\
& = & \frac{1}{2\pi} \sqrt{\frac{2l+1}{2}} \int_0^\infty \int_0^\pi \psi_{n,l} (r) \  P_{l}^{0}(\cos \theta) \
e^{ipr \cos \theta} \ r^2 \sin \theta  \ \mathrm{d}r \mathrm{d} \theta.
\end{aligned}
\end{equation}

Note that $\psi(p)$ is not normalized; thus needs to be normalized. Integrating over $\theta$ and $\phi$ variables,
Eq. (<ref>) can be further reduced to,

\begin{equation}
\psi_{n,l}(p)=(-i)^{l} \int_0^\infty \  \frac{\psi_{n,l}(r)}{p} \ f(r,p)\mathrm{d}r.
\end{equation}

Depending on $l$, this can be rewritten in following simplified form ($m'$ starts with 0),

\begin{equation}
\begin{aligned}
f(r,p) & = & \sum_{k=2m^{\prime}+1}^{m^{\prime}<\frac{l}{2}} a_{k} \ \frac{\cos pr}{p^{k}r^{k-1}} +
\sum_{j=2m^{\prime}}^{m^{\prime}=\frac{l}{2}} b_{j} \ \frac{\sin pr}{p^{j}r^{j-1}}, \ \ \ \ \mathrm{for} \
\mathrm{even} \ l,   \\
f(r,p) & = & \sum_{k=2m^{\prime}}^{m^{\prime}=\frac{l-1}{2}} a_{k} \ \frac{\cos pr}{p^{k}r^{k-1}} +
\sum_{j=2m^{\prime}+1}^{m^{\prime}=\frac{l-1}{2}} b_{j} \ \frac{\sin pr}{p^{j}r^{j-1}}, \ \ \ \ \mathrm{for} \ \mathrm{odd} \ l.
\end{aligned}
\end{equation}

The co-efficients $a_{k}$, $b_{j}$ of even- and odd-$l$ states are collected in Tables 1 and 2 respectively.

§ FORMULATION OF INFORMATION-THEORETICAL QUANTITIES

In this section we shall briefly discuss the various information-theoretic quantities along with their mathematical form.s
This will provide the context where these quantities are defined and the relations between them.

§.§ Shannon entropy ($S$)

Information is carried over from one place to another. We seek information when there are more than one alternatives, and we are
not certain about the outcome of the event. If an event occurs in just one way, there is no uncertainty and no information is called
for. In summary, we get some information due to the occurrence of an event if there was some uncertainty prevailing before the
occurrence [56, 98, 99].

\begin{equation}
\begin{aligned}
\mathrm{Information~received~by~occurrence~of~an~event~} = \mathrm{Amount~of~uncertainty~} \\
\mathrm{prevailed~before~its~occurrence}
\end{aligned}
\end{equation}

Let us consider a discrete probability distribution ($p_{1},p_{2},p_{3},...,p_{n}$) consisting of $n$ different events. To
quantify the uncertainty in results, Shannon in 1948 defined the information entropy as [57],

\begin{equation}
\end{equation}

where $k$ is a positive constant depending on the choice of unit. This definition can be explained by choosing two limiting cases:
(i) at first, when any of the $p_{i}=1$ and others are zero, then for this certain event $S=0$, which is minimum (ii)
in case of equiprobability, where all $p_{i}=\frac{1}{n}$, and the uncertainty of the outcome is maximum, then $S$ is also maximum
($S=k\ln n$) [98]. In essence, it can be said that, for a given distribution, lesser the probability of occurring an
event, higher will be the uncertainty associated with it. Hence, after occurrence of that event, more information will come out.
Arguably, $S$ is the best measure of information [56].

In wave mechanics, the concept of $S$ has been used to explain and interpret various phenomena. Illustrative examples involve in
illuminating Colin conjecture [100, 101], atomic avoided crossing [102, 103], orbital-free
DFT [104, 105], electron correlation [106, 107, 108, 109, 110, 111], configuration-interaction, entanglement in artificial atoms [112], aromaticity [113] in
many-electron systems, etc. Moreover, it can be proved that a stronger version (compared to the traditional position-momentum
uncertainty, due to Heisenberg), has been derived by adopting the idea of $S$ [56],

\begin{equation} \label{eq:20}
S_{\rvec}+S_{\pvec} \ge D(1+\mathrm{ln}~\pi),
\end{equation}

where $D$ refers to dimensionality of the system. Here, $S_{\rvec}$, $S_{\pvec}$ have the form,

\begin{equation}
\begin{aligned}
S_{\rvec}  & =  -\int_{{\mathcal{R}}^3} \rho(\rvec) \ \ln [\rho(\rvec)] \ \mathrm{d} \rvec   =
2\pi \left(S_{r}+S_{(\theta,\phi)}\right), \\
S_{\pvec}  & =  -\int_{{\mathcal{R}}^3} \Pi(\pvec) \ \ln [\Pi(\pvec)] \ \mathrm{d} \pvec   =
2\pi \left(S_{p}+S_{(\theta, \phi)}\right), \\
S_{r} & =  -\int_0^\infty \rho(r) \ \ln [\rho(r)] r^2 \mathrm{d}r, \ \ \ \ \ \ \ \ \rho(r) = |\psi_{n,l}(r)|^{2}, \\
S_{p} & =  -\int_{0}^\infty \Pi(p) \ln [\Pi(p)]  \ p^2 \mathrm{d}p, \ \ \ \ \ \ \ \ \Pi(p) = |\psi_{n,l}(p)|^{2}, \\
S_{(\theta, \phi)}  & =   -\int_0^\pi \chi(\theta) \ \ln [\chi(\theta)] \sin \theta \mathrm{d} \theta, \ \ \ \ \ \
\chi(\theta)   =  |\Theta(\theta)|^2.  \\
\end{aligned}
\end{equation}

§.§ Rényi entropy ($R$)

Rényi entropy is a one-parameter extension of $S$. Rényi in 1961 defined this quantity as, “the measure of information of order
$\alpha$ associated with the probability distribution $P=(p_{1},p_{2},....,p_{n})$" [114]. The mathematical form of $R$ is,

\begin{equation}
R_{\alpha}=\frac{1}{1-\alpha} \ln \left(\sum_{k} p_{k}^{\alpha}\right).
\end{equation}

It is important to note that, $R_{\alpha}$ is considered as another measure of information because (i) $R_{\alpha}$ is the
exponential mean of information entropy whereas $S$ provides the arithmetic mean of it, and (ii) at $\alpha \rightarrow \infty$,
$R_{\alpha}$ reduces to $S$ [56].

It is well known that, $R^{\alpha}$, the so-called information generating functional, is closely related to $\alpha$ order
entropic moments, and completely characterize density $\rho(\rvec)$. In case of continuous density distribution, it is expressed in
terms of expectation values of density, in the following conventional form [114, 115],

\begin{equation}
R^{\alpha}[\rho (\rvec)]  =  \frac{1}{(1-\alpha)} \ \mathrm{ln} \ \langle \rho(\rvec)^{(\alpha-1)}\rangle,  \ \ \ \ 0 < \alpha
< \infty, \ \ \alpha \neq 1.
\end{equation}

Quantum mechanical uncertainty principle such as, $\Delta x \Delta p \ge \hbar$, does not comment about the accuracy of our
measuring instruments. On the contrary, entropic uncertainty relations depend on the accuracy of a given measurement. Because
they manifest the area of phase-space ($\delta \rvec \delta \pvec$) obtained by the resolution of measuring instruments. It
suggests that, with an increase of localization of a particle in phase space, the sum of uncertainties in position and momentum
space enhances. The quantum mechanical uncertainty relation containing the phase space is as follows,

\begin{equation} \label{eq:21}
R_{\rvec}^{\alpha}+R_{\pvec}^{\alpha} \ge -\frac{1}{2}\left(\frac{\ln~\alpha}{1-\alpha}+\frac{\ln~\beta}{1-\beta}\right) -
\ln~\frac{\delta \rvec \delta \pvec}{\pi \hbar};  \ \ \ \ \ \left(\frac{1}{\alpha}+\frac{1}{\beta}\right)=2.
\end{equation}

In the limit, when $\alpha \rightarrow 1$ and $\beta \rightarrow 1$, this uncertainty relation reduces to the relation given in
Eq. (<ref>). The relation given in Eq. (<ref>) provides better idea about uncertainty as it contains all-order entropic
moments [58]. But its improvement is still necessary, which remains an open challenging problem. Interestingly,
Eq. (<ref>) becomes sharper and sharper when the relative size of phase-space area $\left(\frac{\delta \rvec \delta
\pvec}{\pi \hbar}\right)$ defined by the experimental resolution decreases; as it is when we enter deeper and deeper into the
quantum regime.

In quantum mechanics, $R^{\alpha}$ has been successfully employed to investigate and predict various quantum properties and
phenomena like entanglement, communication protocol, correlation de-coherence, measurement, localization properties of Rydberg
states, molecular reactivity, multi-fractal thermodynamics, production of multi-particle in high-energy collision, disordered
systems, spin system, quantum-classical correspondence, localization in phase space [116, 117, 118, 119, 120, 121, 122, 123, 124, 125], etc.

Rényi entropies of order $\lambda (\neq 1)$ ($\lambda$ is either $\alpha$ or $\beta$) are obtained by taking logarithm of
$\lambda$-order entropic moment [115, 58]. In spherical polar coordinate these can be written in following simplified form by some
straightforward mathematical manipulation [97],

\begin{equation}
\begin{aligned}
R_{\rvec}^{\lambda}  = & \frac{1}{1-\lambda} \ln \left(\int_{{\mathcal{R}}^3} \rho^{\lambda}(\rvec)\mathrm{d} \rvec \right) \\ = &
\frac{1}{(1-\lambda)} \ln \left(2\pi\int_0^\infty [\rho(r)]^{\lambda} r^2 \mathrm{d}r \int_0^\pi [\chi(\theta)]^{\lambda} \sin
\theta \mathrm{d}\theta \right) \\
= & \frac{1}{(1-\lambda)}\left( \ln 2\pi + \ln [\omega^{\lambda}_r] + \ln [\omega^{\lambda}_{(\theta, \phi)}] \right),  \\
R_{\pvec}^{\lambda}  = & \frac{1}{1-\lambda} \ln \left[\int_{{\mathcal{R}}^3} \Pi^{\lambda}(\pvec)\mathrm{d} \pvec \right] \\ = &
\frac{1}{(1-\lambda)} \ln \left(2\pi \int_{0}^\infty [\Pi(p)]^{\lambda} p^2 \mathrm{d}p \int_0^\pi [\chi(\theta)]^{\lambda} \sin
\theta \mathrm{d}\theta \right) \\
= & \frac{1}{(1-\lambda)}\left( \ln 2\pi + \ln [\omega^{\lambda}_p] + \ln [\omega^{\lambda}_{(\theta, \phi)}] \right).
\end{aligned}
\end{equation}

Here $\omega^{\lambda}_{\tau}$s are entropic moments in $\tau$ ($r$ or $p$ or $\theta$) space with order $\lambda$, having forms,

\begin{equation}
\omega^{\lambda}_r= \int_0^\infty [\rho(r)]^{\lambda} r^2 \mathrm{d}r, \ \ \
\omega^{\lambda}_p= \int_{0}^\infty [\Pi(p)]^{\lambda} p^2 \mathrm{d}p, \ \ \
\omega^{\lambda}_{(\theta, \phi)}= \int_0^\pi [\chi(\theta)]^{\lambda} \sin \theta \mathrm{d}\theta.
\end{equation}

Three different categories of Complexities, as functions of disorder.

§.§ Fisher information ($I$)

The idea of entropy can adequately explain the degree of disorder of a given phenomenon. Apart from that, however, it is necessary
to find out a suitable measure of disorder whose variation derives the event. The concept of entropy is not able to do this.
However, $I$ can serve as a good candidate in this context, having the ability to estimate a parameter. Hence it becomes a
cornerstone of the statistical field of study, called parameter estimation [126]. $I$ measures the expected error
in a smart measurement. Let $e^{2}$ be the mean-square error in an estimation of $\hat{\theta}$ then it obeys the relation,

\begin{equation} \label{eq:22}
\begin{aligned}
e^{2}I \ge 1; \ \ \ \
I \ge \frac{1}{e^{2}}.
\end{aligned}
\end{equation}

Equation (<ref>) suggests that, $I$ is always greater than the reciprocal of $e^{2}$; only in case of Gaussian
distribution it becomes equal to inverse of $e^{2}$. The general form of $I$ is,

\begin{equation}
I=\int \frac{|\nabla \rho(\tau)|^{2}}{\rho(\tau)} \mathrm{d}\tau,
\end{equation}

which is a gradient functional of density, measuring the local density fluctuation in a given space. In case of a sharp
distribution $I$ is higher; whereas, for a flat distribution it is lower. Thus, it can be concluded
that, with a rise in uncertainty, $I$ decreases. It resembles the Weizsäcker kinetic energy functional, $T_{\omega}[\rho]$ often
used in DFT [127, 104].

In case of a central potential $I_{\rvec}$ and $I_{\pvec}$, the net Fisher information, in $\rvec$ and $\pvec$ spaces respectively,
are expressed as [128],

\begin{equation}
\begin{aligned}
I_{\rvec}  =  \int_{{\mathcal{R}}^3} \left[\frac{|\nabla\rho(\rvec)|^2}{\rho(\rvec)}\right] \mathrm{d}\rvec  =
4\langle p^2\rangle - 2(2l+1)|m|\langle r^{-2}\rangle \\
I_{\pvec} =  \int_{{\mathcal{R}}^3} \left[\frac{|\nabla\Pi(\pvec)|^2}{\Pi(\pvec)}\right] \mathrm{d} \pvec  =
4\langle r^2\rangle - 2(2l+1)|m|\langle p^{-2}\rangle, \\
\langle r^2\rangle=\int_{0}^{r_c} \psi_{n_{r}, l}^{*}(r) r^{4} \ \psi_{n_{r}, l}(r) \mathrm{d}r, \  \  \ \ \
\langle p^2\rangle=\int_{0}^{r_c} \psi_{n_{r}, l}^{*}(r) [-\nabla^{2}\psi_{n_{r}, l}(r)] r^2 \mathrm{d}r  \\
\left\langle \frac{1}{r^2}\right\rangle=\int_{0}^{r_c} \psi_{n_{r}, l}^{*}(r) \psi_{n_{r}, l}(r) \mathrm{d}r, \  \  \ \ \
\left\langle \frac{1}{p^2}\right\rangle=\int_{0}^{r_c} \psi_{n_{r}, l}^{*}(p) \psi_{n_{r}, l}(p) \mathrm{d}p.
\end{aligned}
\end{equation}

The above equations can be further recast in following equivalent forms [129, 130],

\begin{equation}
\begin{aligned}
I_{\rvec} & = 8\mathcal{E}_{n,l}-8\langle v(r)\rangle-2(2l+1)|m|\langle r^{-2}\rangle \\
I_{\pvec} & = 8\mathcal{E}_{n,l}-8\langle T\rangle-2(2l+1)|m|\langle p^{-2}\rangle \\
& = \frac{8}{\omega^{2}}\langle v(r)\rangle-2(2l+1)|m|\langle p^{-2}\rangle,
\end{aligned}
\end{equation}

where $v(p)$ is the $p$-space counterpart of $v(r)$.

In case of a 3DCHO, $I$'s in $r$ and $p$ space can be expressed analytically as [131],

\begin{equation}
\begin{aligned}
I_{\rvec}(\omega)  =\frac{\omega}{\sqrt{2}} I_{\rvec}(\omega=1), \hspace{3mm} \ \ \ \  I_{\pvec}(\omega) =
\frac{\sqrt{2}}{\omega}I_{\pvec}(\omega=1).
\end{aligned}
\end{equation}

Thus, an increase in $\omega$ leads to rise in $I_{\rvec}(\omega)$ and fall in $I_{\pvec}(\omega)$. However, it is obvious that
$I_{t} \ (=I_{\rvec} I_{\pvec})$ remains invariant with $\omega$. Throughout the article, for brevity, $I_{\rvec}(\omega=1)$ and
$I_{\pvec}(\omega=1)$ will be symbolized as $I_{\rvec}$, $I_{\pvec}$ respectively.

When $m=0$, $I_{\rvec}$ and $I_{\pvec}$ in Eq. (62) reduce to further simplified forms as below,

\begin{equation}
\begin{aligned}
I_{\rvec}  =  4\langle p^2\rangle, \hspace{3mm} \ \ \ \  I_{\pvec} =4\langle r^2\rangle.
\end{aligned}
\end{equation}

It is seen that, at fixed $n_{r}, l$, both $I_{\rvec}, I_{\pvec}$ provide maximum values when $m=0$, and both of them
decrease with rise in $m$. Hence one obtains the following upper bound for $I_t$,

\begin{equation}
I_{\rvec} I_{\pvec} \ (=I_t) \leq 16 \langle r^2\rangle \langle p^2\rangle .
\end{equation}

Further adjustment using Eq. (65) leads to following uncertainty relations [128],

\begin{equation}
\frac{81}{\langle r^2\rangle \langle p^2\rangle} \leq I_{\rvec} I_{\pvec} \leq 16 \langle r^2\rangle \langle p^2\rangle.
\end{equation}

Therefore, in a central potential, $I$-based uncertainty product is bounded by both upper as well as lower limits. They are
state-dependent, varying with $n_{r}$ and $l$ quantum numbers.

Schematic representation of a confined 1DQHO potential: (a) SCHO at two separate box lengths
(b) ACHO at six chosen values of $d_m$ [132].

§.§ Onicescu energy ($E$)

Information energy is the counterpart of information entropy. In 1966, Onicescu defined this quantity as [133],

\begin{equation}
E=\sum_{i} p_{i}^{2}.
\end{equation}

It is interesting to note that, in case of a certain event $E$ is maximum. On the contrary, for an equiprobable distribution it is
minimum [134, 127, 135]. Qualitatively it is inverse of $S$. When $S$ increases, $E$ decreases and vise-versa. In case
of continuous probability distribution, $E$ is an expectation value of probability density. Its mathematical form is,

\begin{equation}
E=\int \rho^{2}(\tau) \ \mathrm{d}\tau.
\end{equation}

In addition, Disequilibrium ($D_e$) for a continuous probability distribution is defined as,

\begin{equation}
De=\int_{-\infty}^{+\infty} \rho^{2}(\tau) \mathrm{d}\tau.
\end{equation}

Therefore, in this occasion, disequilibrium and information energy have same form. Like the previous measures, it is also
utilized in orbital-free DFT [105], testing normality [136], electron correlation
[110], Colin conjecture [100, 101], configuration interaction [137] etc.

By definition, $E$ refers to the 2nd-order entropic moment [127]; for central potential it assumes the form ($E_{t}$ is
the Onicescu energy product),

\begin{equation}
\begin{aligned}
E_{r} & = \int_0^\infty [\rho(r)]^{2} r^2 \mathrm{d}r, \ \
E_{p}  = \int_{0}^\infty [\Pi(p)]^{2} p^2 \mathrm{d}p, \\
E_{\theta, \phi} & = \int_0^\pi [\chi(\theta)]^{2} \sin \theta \mathrm{d}\theta.  \ \
E_{t} = E_{r}E_{p}E_{\theta, \phi}^{2}.
\end{aligned}
\end{equation}

Uncertainty product for such measures are studied in [138].

§.§ Complexities

In a system, statistical complexity arises due to breakdown of symmetry. It illustrates a competing effect of two complementary
quantities, offering a qualitative idea of the organization, structure and correlation. According to [139]
it can be broadly characterized into three categories: (i) advances monotonically with disorder (ii) reaches its minimal
value for both completely ordered and disordered systems, and a maximum at some intermediate level (iii) increases with order. It
has finite value in a state lying between two limiting cases of complete order (maximum distance from equilibrium) and maximum
disorder (at equilibrium). Figure 1 pictorially demonstrates above three different kind of complexities as a function
of disorder.

The statistical measure of complexity ($C_{LMC}$) is nothing but the product of the information content ($H$) (such as $S$, $R$
etc.) and concentration of spatial distribution ($D$), and can be written as $C_{LMC} = H. D$. This was later criticized
[140] and modified [141] to the form of $C_{LMC} = D. e^{S}$, in order to satisfy few conditions such as
reaching minimal values for both extremely ordered and disordered limits, invariance under scaling, translation and replication.
Various definitions were put forth in literature. Some notable ones include Shiner, Davidson and Landsberg (SDL) [142, 139], Fisher-Shannon $(C_{IS})$ [143, 144, 145], Cramér-Rao [145, 146], generalized
Rényi-like [147, 148, 149] complexity, etc. Amongst these, $C_{IS}$ corresponds to a measure which probes a
system in terms of complementary global and local factors, and also satisfies certain desirable properties in complexity [127],
like invariance under translations and re-scaling transformations, invariance under replication, near-continuity, etc. This has
remarkable applications in the study of atomic shell structure, ionization processes [144, 145, 150], as well as
in molecular properties like energy, ionization potential, hardness, dipole moment in the localization-delocalization plane showing
chemically significant pattern [151], molecular reactivity studies [152]. Some elementary chemical
reactions such as hydrogenic-abstraction reaction [153], identity $SN^{2}$ exchange reaction [154], and also
concurrent phenomena occurring at the transition region [155] of these reactions have been investigated through
composite information-theoretic measures in conjugate spaces.

Without any loss of generality, let us define complexity in following general form $C = Ae^{b.B}$. The order ($A$) and disorder
parameters ($B$) may include ($E, I$) and ($R, S$) respectively. With this in mind, we are interested in the following four

\begin{equation}
\begin{aligned}
C_{ER} & = E e^{bR}, \ \ \ \ \ \ \ \ C_{IR} = Ie^{bR}, \ \ \ \ \ \ \ C_{ES} & = E e^{bS}, \ \ \ \ \ \ \ \ C_{IS} = Ie^{bS}.
\end{aligned}
\end{equation}

Plot of (a) $S_x$ (b) $S_p$, of SCHO potential, as function of $x_c$, for first five states [132].

§.§ Relative information

Kullback-Leibler divergence or relative entropy is a descriptor or quantifier of how a measured probability distribution function
deviates from a given reference distribution [156, 157]. In quantum mechanics, this characterizes a measure of
distinguishability between two states. It actually quantifies the change of information from one state to other [126].
Relative $R$ and $S$ was explored for various atomic systems using H atom ground state as reference [158]. A detailed
investigation reveals  that, they are directly related with atomic radii and quantum capacitance [159, 158]. Another
interesting measure is the relative Fisher information (IR) [160]. It has importance in different topics of physics
and chemistry, such as to calculate phase-space gradient of dissipated work and information [161], deriving Jensen
divergence [162], relation with score function [163], in the context of study of probability current
[164], in thermodynamics [165], etc. Of late it has been successfully used in formulating atomic densities
[166] and deriving density functionals under local-density and generalized-gradient approximations [167].
Further, IR along with Hellmann-Feynman and virial theorem has been used to develop a Legendre transform structure related to
SE [168]. It has been designed self-consistently on the basis of estimation theory [169]. In quantum chemistry
perspective, it has been successfully formulated using above two theorems and entropy maximization principle [170, 171, 172]. Very recently, IR for some exactly solvable potentials including 1D and 3D QHO in both position and
momentum spaces has been estimated analytically using ground state of definite symmetry (for example, $l=0$ for $s$ orbitals, $l=1$
for $p$ orbital, and so on) as reference [173]. Before that, a numerical estimation of IR in position space is done
for H-atom using $1s$ orbital as basis [174].

For two normalized probability densities $\rho_{n,l,m}(\tau), \rho_{n_{1},l_{1},m_{1}}(\tau)$, IR is expressed as,

\begin{equation}
\mathrm{IR} \ [\rho_{n,l,m}(\tau)|\rho_{n_{1},l_{1},m_{1}}(\tau)]=\int_{{\mathcal{R}}^3}\rho_{n,l,m}(\tau)\left|\nabla \
\mathrm{ln} \left\{ \frac{\rho_{n,l,m}(\tau)}{\rho_{n_{1},l_{1},m_{1}} (\tau)}\right \} \right|^{2}  \mathrm{d}\tau.
\end{equation}

Here $n,l,m$ and $n_{1},l_{1},m_{1}$ are the descriptors of target and reference states respectively, while $\tau$ is a
generalized variable. In case of central potential, these probability densities $\rho_{n,l,m}(\tau)$, $\rho_{n_{1},l_{1},m_{1}}
(\tau)$ can be expressed in following forms, without any loss of generality,

\begin{equation}
\begin{aligned}
\rho_{n,l,m}(\tau) & = R_{n,l}^{2}(s) \Theta_{l,m}^{2}(\theta);  \      \hspace{3mm}
\rho_{n_{1},l_{1},m_{1}}(\tau) = R_{n_{1},l_{1}}^{2}(s) \Theta_{l_{1},m_{1}}^{2}(\theta).
\end{aligned}
\end{equation}

In the above equation, $R_{n,l}(s), R_{n_{1},l_{1}}(s)$ signify radial parts, $\Theta_{n,l}(\theta), \Theta_{n_{1},l_{1}}
(\theta)$ represent angular contributions of two wave functions, whereas $``s"$ implies either $r$ or $p$ variable in respective
radial functions. Thus Eq. (73) may be rewritten as,

\begin{multline}
\mathrm{IR} \ [\rho_{n,l,m}(\tau)|\rho_{n_{1},l_{1},m_{1}}(\tau)]= \mathrm{IR} \ [\rho_{n,l}(s)|\rho_{n_{1},l_{1}}(s)]+
\left\langle \frac{1}{s^{2}} \right\rangle \mathrm{IR} \ [\Theta_{l,m}^{2}(\theta)|\Theta_{l_{1},m_{1}}^{2}(\theta)]+\\
2\int_{0}^{\infty}s~R_{n,l}^{2}(s)\left[\frac{d}{ds}\mathrm{\ln}\left\{ \frac{R^{2}_{n,l}(s)}{R^{2}_{n_{1},l_{1}}(s)}\right\}
\right]\mathrm{d}s
\int_{0}^{\pi}\Theta_{l,m}^{2}(\theta)\left[\frac{d}{d\theta}\mathrm{ln}\left\{ \frac{\Theta_{l,m}^{2}(\theta)}
{\Theta_{l_{1},m_{1}}^{2}(\theta)}\right\} \right] \sin\theta \mathrm{d}\theta
\end{multline}

where the following quantities have been defined,

\begin{equation}
\begin{aligned}
\mathrm{IR} \ [\rho_{n,l}(s)|\rho_{n_{1},l_{1}}(s)]& =\int_{0}^{\infty}s^{2}~\rho_{n,l}(s)\left|\frac{d}{ds}\mathrm{ln}
\left\{ \frac{\rho_{n,l}(s)}{\rho_{n_{1},l_{1}}(s)}\right\} \right|^{2}\mathrm{d}s \\
\mathrm{IR} \ [\Theta_{l,m}^{2}(\theta)|\Theta_{l_{1},m_{1}}^{2}(\theta)]&=\int_{0}^{\pi}\Theta_{l,m}^{2}(\theta)\left|
\frac{d}{d\theta}\mathrm{ln}\left\{ \frac{\Theta_{l,m}^{2}(\theta)}{\Theta_{l_{1},m_{1}}^{2}(\theta)}\right\} \right|^{2}
\sin\theta~\mathrm{d}\theta
\end{aligned}
\end{equation}

Plot of $I_x$, $I_p$, $E_x$, $E_p$ for first five states of SCHO potential, as functions of $x_c$, in panels
(a), (b), (c) and (d). See text for details [132].

§ RESULT AND DISCUSSION

§.§  1D Confined harmonic oscillator (1DCHO)

Panel (a) of Fig. 2 represents an SCHO scenario for two box lengths $1$ and $2$, respectively. Next, the asymmetric confinement
in a 1DQHO can be invoked in two different ways: (i) by varying the box boundary, keeping box length and $d_m$ fixed at zero (ii)
the other way is to change $d_m$ by keeping box length and boundary constant. We have chosen the second option; where a rise in
$d_m$ transfers the minimum towards the right of origin, keeping box length, $b_2-b_1 = 2$ fixed, while left and right
boundaries are residing at $b_1 = -1$ and $b_2 = 1$. Moreover, since $\frac{1}{2} k(x \pm d_{m})^{2}$ illustrates a mirror-image pair
potential trapped within $b_1$ and $b_2$, their eigenvalues and expectation values are same for all states. As the respective
wave functions are mirror images of each other, it suffices to consider the behavior of either one of them. The other one
automatically follows from it. The right-hand panel (b) of Fig. 1 shows a schematic representation of an ACHO potential, at
five different $d_m$ values.

Comparison of energies of first six states of ACHO potential at four distinct $d_m$. Length of the box is
fixed at 2. PR implies Present Result [132]. See text for details.

$d_m$     $\epsilon_0$ (PR)      $\epsilon_0$ (Ref.)     $\epsilon_1$ (PR)      $\epsilon_1$(Ref.)

0.36     2.7177633960054      2.7177633960054$^\ddag$
10.283146010610      10.283146010610$^\ddag$

1.92     6.0383021056781      6.0383021056781$^\ddag$
13.901445986629      13.901445986629$^\ddag$

5.00     26.065225076406     26.065225076406$^\ddag$     35.462261039378     35.462261039378$^\ddag$

10.0     97.474035270680     97.474035270680$^\ddag$     110.51944554927     110.51944554927$^\ddag$

$\epsilon_2$ (PR)      $\epsilon_2$ (Ref.)     $\epsilon_3$ (PR)      $\epsilon_3$(Ref.)

0.36     22.648848755052      22.648848755052$^\ddag$
39.929984298830      39.929984298830$^\ddag$

1.92     26.249310409373      26.249310409373$^\ddag$
43.513981920357      43.513981920357$^\ddag$

5.00     47.817024422796     47.817024422796$^\ddag$      64.900200447511     64.900200447511$^\ddag$

10.0     123.593144939095    123.593144939095$^\ddag$     140.555432078323      140.555432078323$^\ddag$

$\epsilon_4$ (PR)      $\epsilon_4$ (Ref.)     $\epsilon_5$ (PR)      $\epsilon_5$(Ref.)

0.36     62.140768627508     62.140768627508$^\ddag$     89.284409553063      89.284409553063$^\ddag$

1.92     65.715672311936     65.715672311936$^\ddag$     92.854029622882      92.854029622882$^\ddag$

5.00     87.137790461503     87.137790461503$^\ddag$     114.244486402564     114.244486402564$^\ddag$

10.0     162.519960161732     162.519960161732$^\ddag$     189.515389275133     189.515389275133$^\ddag$

$^\ddag$ITP result [34].

§.§.§ Symmetrically confined harmonic oscillator (SCHO)

In this case, all calculations are performed involving the wave function given in Eq. (2). Figure 3 shows
plots of $S_{x}$, $S_{p}$ versus $x_c$, for first five states of SCHO. In panel (a), $S_{x}$ progresses with box
length and converges to a constant value ($S_{x}$ of 1DQHO) at sufficiently large box length. At small $x_c$ region, $S_x$ changes
rather insignificantly with $n$, corresponding to the behavior of a PIB problem (where $S_x$ remains unchanged with $n$).
Next, panel (b), imprints that $S_{p}$ abates with increase in box length and finally merges to corresponding 1DQHO value. It is
important to note that, there appears a minimum in $S_p$ for all excited states. Appearance of such minimum may be ascribed to
the competing effect in $p$ space. In fact, three possibilities could be contemplated ($l_{x}$, $l_{p}$ are box lengths in $x$,
$p$ space):

* When $l_{x} \rightarrow 0,$ then $l_{p} \rightarrow \infty$.

* When $l_{x}$ is finite, then $l_{p}$ is also finite. But an increase in $l_{x}$ leads to a decrease in $l_{p}$.

* When $l_{x} \rightarrow \infty,$ then also $l_{p} \rightarrow \infty$.

These plots suggest that, at the beginning, with increase in $l_{x}$, particle gets localized in $p$ space ($S_{p}$ decreases),
but when potential behaves like 1DQHO, de-localization predominates. Thus, existence of minimum in $S_{p}$ is due to presence
of two conjugate forces. These variations of $S_x$, $S_p$ with $x_c$ are in complete agreement with the findings of

The behavior of $I_{x}$, $I_{p}$ with respect to $x_{c}$ of first five stationary states are exhibited in two bottom panels (a), (b)
of Fig. (4). One can see that initially $I_{x}$ decreases sharply, extent of which is maximum for the lowest state and successively
decreases with $n$. Then it attains a constant value after a certain $x_c$. On the contrary, in panel (b), $I_{p}$ strongly grows
with $x_{c}$ at first; the extent increases with $n$, lowest state producing lowest. Finally, for all states, it eventually
reaches a state-dependent constant value as in $I_x$. It is pertinent to mention that, behavior of $I_x$, $I_p$ under confinement
agrees with those of $\Delta p$, $\Delta x$ [175]. Next, panel (c) reveals that, $E_x$ for all these states remain
very close to each other at smaller $x_c$; then as $x_c$ progresses, $E_x$ for individual states branch out and decreases
indicating delocalization. In the end, it reaches some constant value for all $n$. Lastly, in panel (d), $E_p$s tend to
advance in the beginning for all states at low $x_c$ region, eventually becoming smooth after some threshold $x_c$. These
calculation have further consolidated the conclusions obtained from the study of $S_x$, $S_p$, $S$ demonstrated earlier.

Variation of $\epsilon_n$ ($n \! = \! 0-5$) of ACHO potential as function of $d_m$ [132].

§.§.§ Asymmetrically confined harmonic oscillator (ACHO)

Now, the focus is on the ACHO case. This presentation is based on results of $\epsilon_n$; $S_x, S_p$; $I_x, I_p$; $E_x, E_p$
as functions of $d_m$, for some low-lying states. Throughout the whole presentation, the box length has been kept fixed at 2 and
boundaries are placed at $-$1 and 1. At the outset it is important to note that, an increase in $d_m$ leads to localization in real
space. Thus it can be expected that, there will be delocalization in $p$ space.

Wave functions of lowest six states of ACHO potential at four specific values of $d_m$, namely 1,
10, 20, 50. Panels (a)–(f) represent $n \! = \! 0-5$ states [132]. See text for details.

Before looking into the behavior of IE in ACHO, it is pertinent to make some comments about its eigenvalues and eigenfunctions.
ACHO has been studied using power-series solution [32] and ITP method [34]. In this section, we shall, however,
present another simple method, which produces quite accurate results. In this variation induced
exact diagonalization procedure [176], an energy functional is minimized using an SCHO basis set. Recently this
method has been successfully employed in studying symmetric and asymmetric double-well potentials [177, 178], where
a 1DQHO basis was utilized.

Plot of $I_x$, $I_p$, $I$ of first five states of ACHO potential, as function of $d_m$, in left (a), middle (b),
right (c) columns; (i)–(iii) represent $n=0-2$ states [132]. See text for details.

The Hamiltonian matrix elements are evaluated by using functions given in Eq. (2). Presence of a single non-linear parameter $\alpha$
allows us to adopt
a coupled variation procedure. Further it is easy to confirm the convergence of results with respect to basis dimension $N$. Thus
it provides a secular equation at each $\alpha$. The kinetic energy part blows up to $\infty$ when $\alpha \rightarrow \infty$,
whereas at $\alpha \rightarrow 0$, potential energy part behaves in a similar fashion. This qualitative analysis through
uncertainty principle, confirms the existence of such basis. Diagonalization of $H_{mn} = \langle m | \hat{H}| n \rangle$ leads
to accurate eigenvalues and eigenfunctions, which is attained through MATHEMATICA package. In principle, to achieve exact solution,
one needs to employ the complete basis; however for practical purposes a truncated basis of finite dimension is envisaged.
Here, $N=50$ appears adequate; with further increase in basis, result improves. A cross-section of energies obtained from above scheme
is produced in Table 3, for lowest six states of ACHO at four selected $d_m$. Some literature results are provided for comparison,
wherever available. For all four $d_m$, present energies with SCHO basis, practically coincide with ITP results [34],
for all digits reported. It is expected that this SCHO basis may be useful in future for other confined and free 1D potentials.

Our calculated ACHO energies of Table 3 are presented in Fig. 5, for first six states with respect to $d_m$, in panels (a)-(f).
Ranges of $\epsilon_n$ and $d_m$ are different for each $n$. Variation in $\epsilon_0$ is unique amongst all $\epsilon_n$, where, from
an initial higher value at small $d_m$, ground-state energy continuously falls. In all excited states, however, it passes through
a maximum; with increase in $n$, this maximum shifts to higher values of $d_m$. Next Fig. 6 portrays our computed wave functions for
$n \! = \! 0-5$ states of ACHO at four selected $d_m$ values. Clearly, the maximum, minimum and nodal positions of all $\psi_n$'s
switch towards right with advancement in $d_m$. These plots suggest that particle gets localized in $x$ space as $d_m$ progresses.

Plot of $S_x$, $S_p$, $S$ of first three states of ACHO potential, as function of $d_m$, in left (a), middle (b),
right (c) columns; (i)–(iii) represent $n=0-2$ states [132]. See text for details.

Now we move on to $I_x,I_p$ variations with change of $d_m$. These are shown pictorially in Fig. 7 left (a), and right (b) panels
display these in  conjugate $x$, $p$ spaces respectively. For ground and first excited state $I_x$ tends to increase with $d_m$,
on the whole, whereas for ($n = 2$) state, the same decreases. Thus, for first two states, it indicates localization in
$x$ space and delocalization for 2nd excited state. A careful investigation of panels b(i)–b(iii) of Fig. 7 reveals that, $I_{p}$,
for all $n$ under consideration, consistently decreases with increase of $d_m$, signifying a delocalization of particle in $p$

Next, it is imperative to explore changes in behavior of $S_x, S_p$ with respect to $d_m$. These are offered in Fig. 8. It is clear
from left panels a(i)–a(iii) of Fig. 8 that, $S_x$ for all these three states generally decreases monotonically with growth in $d_m$,
signifying localization of particle in position space. Likewise, a scrutiny of right plots b(i)–b(iii) of Fig. 8 summarizes an opposite
trend in $S_p$ with increase in $d_m$. A sample of $S_x, S_p$ are reported in
Table 4 for $n \! = \! 0,1,2$ states at five selected $d_m$ values, namely, 0.12, 2.04, 5, 7 and 10. Evidently, these entries
corroborate the outcomes of Fig. 8.

Now, we move on to an analysis of $E$, using Fig. 9, for $n=0-2$. Panels a(i)–a(iii) suggest that, for all three states there
is an overall increase of $E_{x}$ with rise in $d_m$. This indicates localization in $x$ space. Panels b(i)–b(iii) portray that,
general trend of $E_{p}$ is a gradual decrease with rise in $d_m$.

On the basis of above discussion, it is clear that, only $S_{x}, S_{p}$ can adequately explain localization-delocalization
phenomena in an ACHO. Study of $E_{x}, E_{p}$ can also offer valuable knowledge about the dual nature of $E$ in composite $x$,
$p$ space. But, $I_{x}, I_{p}$ appear to be inadequate in explaining the contrasting phenomena in ACHO.

Plot of $E_x$, $E_p$, $E$ of first three states of ACHO potential, as function of $d_m$, in left (a), middle (b),
right (c) columns; (i)–(iii) represent $n=0-2$ states [132]. See text for details.

§.§ 3D Confined harmonic oscillator (3DCHO)

At the onset, it is convenient to point out that, net information measures in conjugate $r$ and
$p$ space may be segmented into radial and angular parts. In a given space, the results provided correspond to net measures
including the angular contributions. One can transform the 3DQHO into a 3DCHO by squeezing the radial boundary of former from
infinity to a finite region. This alteration in radial environment does not affect the angular boundary conditions.
Therefore, angular part of the information measures in free and confined systems remain unchanged in both spaces. Further
as we are solely focused in radial confinement, this will also not influence the characteristics of a given measure as one
changes $r_c$. Throughout this investigation, magnetic quantum number $m$ remains fixed to $0$ for $R, S, E$ calculation. However, $I$
has been studied for non-zero $m$ states. The radial wave function in $r$ and $p$ spaces depend only on $n_{r}, l$ quantum
numbers. Hence, in both space, radial wave function can be obtained by considering $m=0$. Further, a change in $m$ from zero to
non-zero value will not influence the expression of radial wave function in $p$ space. Also note that, we have followed the
spectroscopic notation, i.e., the levels are denoted by $n_r+1$ and $l$ values (see, e.g., [69]). Therefore, $n_r=0$ and
$l=4$ signifies $1g$ state. The radial quantum number $n_r$ relates to $n$ as $n=2 n_{r}+l$.

$S_{x}, S_{p}, S$ for $n \! = \! 0-2$ states of ACHO at five specific $d_m$ [132]. See text for details.

$d_m$     $S_{x}^{0}$     $S_{p}^{0}$     $S^{0}$     $S_{x}^{1}$     $S_{p}^{1}$     $S^{1}$     $S_{x}^{2}$     $S_{p}^{2}$     $S^{2}$

0.12     0.3783     1.8296     2.2079     0.3857     2.2212     2.6069     0.3862     2.3692     2.7554

2.04     0.3428    	1.8779     2.2208     0.3807     2.2512     2.6319     0.3850     2.3852     2.7703

5.0     0.2184     2.0238     2.2422     0.3636     2.3390	    2.7027     0.3782    	2.4512     2.8295

8.0     0.093     2.1576     2.2506     0.3360     2.4313     2.7674     0.3695     2.5429    	2.9124

10.0     0.0233     2.2294     2.2527     0.3108     2.4900     2.8009     0.3611     2.6057     2.9668

$I_{\rvec}, I_{\pvec}$ values are obtained from Eq. (62). In all occasions, it has been verified that, in both spaces,
as $r_c \rightarrow \infty$, $I_{\rvec}$ and $I_{\pvec}$ merge to respective 3DQHO limit. The net $I$ in $r$ and $p$ spaces are
divided into radial and angular part. But in both $I_{\rvec}$ and $I_{\pvec}$ expressions, angular contribution is normalized
to unity. Hence, evaluation of all these targeted quantities using only radial part will serve our purpose. Pilot calculations
are done for $1s$-$1g$ and $2s$-$2m$ states, with $r_c$ varying from 0.1 to 7 a.u. The former set is chosen as they represent
node-less ground states corresponding to various $l$, whereas, $2s$-$2m$ states are considered to perceive the effect of nodes on
$I$ at non-zero $m$.

It is worthwhile noting that, $I$'s can be accurately calculated from a knowledge of $\langle p^2\rangle, \left\langle
\frac{1}{r^2}\right\rangle, \langle r^2\rangle, \left\langle \frac{1}{p^2}\right\rangle$. Two possibilities may be invoked: (i)
first three quantities evaluated in $r$ space, while $\langle p^{-2}\rangle$ in $p$ space (ii) $\langle r^2\rangle,
\langle r^{-2}\rangle$ in $r$ space, while $\langle p^2\rangle, \langle p^{-2}\rangle$ in $p$ space. Here we have opted for
first route removing the necessity to do numerical differentiation in either spaces. This has been discussed in
detail in a recent article [97].

Exact analytical form of $I_{\rvec}$ and $I_{\pvec}$ in isotropic 3DQHO was given in [128],

\begin{equation}
I_{\rvec}(\omega)= 4\omega \left(2n_{r}+l-|m|+\frac{3}{2}\right), \ \ \
I_{\pvec}(\omega)=\frac{4}{\omega}\left(2n_{r}+l-|m|+\frac{3}{2} \right).
\end{equation}

Thus, at a certain $m$, both $I_{\rvec}(\omega), I_{\pvec}(\omega)$ increase as $n_{r}$ and $l$ approach higher values.
Similarly, for specific $n_{r}$ and $l$, both $I_{\rvec}(\omega), I_{\pvec}(\omega)$ regress with growth in $|m|$. Effect of
$\omega$ on $I_{\rvec}(\omega), I_{\pvec}(\omega)$ is quite straightforward. $I_{\rvec}(\omega)$ advances and $I_{\pvec}
(\omega)$ decreases with rise of $\omega$. By putting $\omega=1$ in Eq. (77) one easily recovers expressions for $I_{\rvec}, I_{\pvec}$
in a 3DQHO.

$\mathcal{E}_{n_r,l} (n_r =0)$ of lowest five circular states of 3DCHO, PISB at five $r_c$, from [34, 130].

$l$     $r_c=0.01$       $r_c=0.05$        $r_c=0.1$       $r_c=0.2$         $r_c=0.5$

0         49348.02202373       1973.92123372        493.48163345       123.37570844        19.77453418

1        100953.64280465       4038.14617967       1009.53830088       252.39159906        40.42827649

2        166087.30959293       6643.49293120       1660.87528919       415.22704789        66.48975653

3        244155.96823805       9753.60193720       2441.56211674       610.39965899        97.72324914

4        334771.55964446      13390.86304096       3347.71822121       836.93939916       133.97424683

0         49348.022005446        1973.92088021       493.48022005          123.37005501            19.73920880

1         100953.64278213        4038.14571128       1009.53642782         252.38410695            40.38145711

2         166087.30957134        6643.49238285       1660.87309571         415.21827392            66.43492382

3         244155.96821809        9753.60153136       2441.55968218         610.38992054            97.66238728

4         334771.55962552        13390.86238502      3347.71559625         836.92889906           133.90862385

At first we illustrate the behavior of 3DCHO at $r_c \rightarrow 0$. A diligent study reveals that, at small $r_c$ region, 3DCHO has an
energy spectrum comparable to that of a particle in a spherical box (PISB). This leaning generally holds good for all other
states as well. A cross-section of eigenvalues ($1s$-$1g$) of lowest five circular states corresponding to $l=0$-4, of 3DCHO and
PISB, given in Table 5, at five selected $r_c$, viz., $0.01, 0.05, 0.1, 0.2, 0.5$, supports this fact. However this
observation should not be misinterpreted to conclude that at $r_c \rightarrow 0$, 3DCHO leads to PISB. Because that can happen only
when both systems have nearly equal kinetic energy as well as potential energy components. This is not apparent from this table.
At this point, it is noteworthy to mention that, like 3DCHO, PISB is also exactly solvable; eigenfunctions are explicable directly
in terms of first-order Bessel function, and given as,

\begin{equation}
J_{l}(Z)=(-1)^{l}Z^{l}\left(\frac{1}{Z}\frac{d}{dZ}\right)^{l}\left(\frac{\sin{Z}}{Z}\right), \\
\end{equation}

where $Z=\sqrt{\mathcal{E}_{n_{r},l}}\ r$. At the boundary when $r = r_c$, $Z=Z_{n_r,l}$ and $J_{l}(Z_{n_r,l})=0$. Moreover,
at $r = r_c$, the energy of a $(n_r,l)$ state is written as $\mathcal{E}_{n_{r},l}=\frac{Z_{n_r,l}^{2}}{r_{c}^{2}}$
[179].  This $J_{l}(Z_{n_r,l})=0$ is a transcendental equation and at a constant $n_{r},l$, this $Z_{n_{r},l}$ is evaluated
by the help of MATHEMATICA program package. Thus all the PISB energies in lower segment of this table have been calculated
following the above procedure.

$I_{\rvec}, I_{\pvec}$ of lowest five circular states of 3DCHO and PISB at five $r_c$ at fixed $m(0)$ [130].

System      $l$     $r_c=0.01$       $r_c=0.05$        $r_c=0.1$       $r_c=0.2$         $r_c=0.5$

0         394784.1761898      15791.36986976       3947.84176           986.960440          157.913740

1         807629.1424372      32305.16943736       8076.29142          2019.072855          323.05170823

3DCHO     2        1328698.4767434      53147.9434496        13286.984765        3321.7461915         531.4794285

4        2678172.4771556      107126.9043276       26781.72476         6695.431192         1071.269013

0          394784.1760435       15791.3670417       3947.8417604       986.9604401      157.9136704

1          807629.1422570       32305.1656902       8076.2914225      2019.0728556      323.0516569

PISB     2         1328698.4765707       53147.9390628      13286.9847657      3321.7461914      531.4793906

4         2678172.4770042      107126.8990801      26781.7247700      6695.4311925     1071.2689908

0         0.0001130690       0.0028267272       0.011306900       0.0452270673       0.282533301

1         0.0001498425       0.0037460639       0.014984249       0.0599366038       0.374503742

3DCHO     2          0.0001754798       0.0043869959       0.017547979       0.0701916254       0.438623720

4         0.0002100025       0.0052500632       0.021000250       0.0840008284       0.524961467

0         0.0001130690          0.002826727     0.011306909      0.045227638      0.282672741

1         0.0001498425          0.003746064     0.014984256      0.059937024      0.374606402

PISB     2         0.0001754798          0.004386996     0.017547984      0.070191936      0.438699601

4         0.0002100025          0.005250063     0.021000253      0.084001012      0.525006325

The upper portion of Table 6 portrays $I_{\rvec}$ of 3DCHO and PISB at five selected $r_c$ values introduced before; the
respective $I_{\pvec}$ of two systems are reported in lower portion. It is interesting to point out that, for 3DQHO and 3DCHO
in a state having $m=0$, $I_{\rvec}$ and $I_{\pvec}$ are directly connected to expectation values of kinetic and potential energy as;

\begin{equation}
I_{\rvec}(\omega)= 8\langle T \rangle, \ \ \ I_{\pvec}(\omega)=\frac{8}{\omega^{2}}\langle v(r)\rangle.
\end{equation}

But for a PISB, total energy is exclusively kinetic energy as the potential energy is zero. However, one can evaluate $I_{\rvec},
I_{\pvec}$ for PISB and collate the results with 3DCHO. Because, a pair of systems possessing same $I_{\rvec}, I_{\pvec}$ for all states
indicate identical physical and chemical environment. Hence, here we have exploited $I$ to investigate the characteristics of PISB and
3DCHO at $r_c \rightarrow 0$. The table clearly manifests that at $r_c \rightarrow 0$, a 3DCHO has comparable $I_{\rvec},
I_{\pvec}$ values with that of PISB, thus confirming our presumption that, at $r_c \rightarrow 0$, 3DCHO behaves like a PISB. One
also marks that, with reduction in $r_{c}$, $I_{\pvec}$ (and also potential energy) approaches zero. On the other hand, as
$r_c$ grows, the separation between $I_{\rvec}$ (also $I_{\pvec}$) values of 3DCHO and PISB tends to increase significantly. Moreover,
as expected, at $r_c \rightarrow \infty$, 3DCHO reduces to 3DQHO. In previous section it was pointed out that, a
1DCHO may be treated as a two-mode system; at smaller (approaching zero) and larger (tending infinity) confinement lengths it
behaving like a particle in a box and an 1DQHO respectively [180, 175]. Here, also we observe resembling behavior. At $r_c \rightarrow 0$ and
$\infty$, 3DCHO leads to PISB and a 3D isotropic QHO respectively. We also note that, larger the value of $\omega$ higher will be
$\langle v(r)\rangle$; as a matter of fact 3DCHO is more prone to 3DQHO in such a case. Conversely, lesser the $\omega$ value,
$\langle v(r)\rangle$ is smaller and 3DCHO, in that occasion, is inclined towards a PISB. Therefore at a fixed $r_c$, by controlling
$\omega$ values one can inquire the properties of all three systems starting from PISB to 3DQHO through 3DCHO.

$I_{\rvec}, I_{\pvec}$ for lowest five $n_r$ ($l=0$) values at six different $r_c$ [130]. See text for details.

$n_{r}$     $r_c=0.1$       $r_c=0.5$        $r_c=1$       $r_c=2$         $r_c=7$      $r_c=\infty$

0         3947.84176       157.9137401        39.48285935      10.130828577       6.00000000      6

1        15791.3670       631.654662       157.91245186        39.42241043       14.00000000      14

2        35530.57584       1421.2230213       355.3049651      88.77709457       22.00000000      22

3        63165.46816       2526.6187189       631.6541846      157.88183967      30.00000000      30

4        98696.0440      3947.8417551       986.960106         246.718681361     38.00000000      38

0        0.0113069007         0.2825333012      1.1217967676       3.9877029335        6.0000000          6

1        0.01282672989       0.32070687721     1.28512015257       5.25470219890       14.0000000         14

2        0.0131081767        0.3277292039       1.3124046596        5.34276239849      22.00000000        22

3        0.0132066828        0.3301825760       1.3216621568       5.3463257844        30.00000000        30

4        0.0132522770        0.33131731951       1.3258940221       5.3437134181       38.00000000        38

So far, we have explored the limiting trend of 3DCHO. Now we look into its behavior at intermediate $r_c$ region. For that, at
first, the dependence of $I_{\rvec}, I_{\pvec}$ on quantum number $n_r$ is recorded in Table 7. It tabulates these quantities for
lowest five $n_r$ (0-4) at six representative $r_c$ values. This clearly implies that, at fixed $m,l$ and $r_c$, both $I_{\rvec},
I_{\pvec}$ in 3DCHO get incremented as $n_{r}$ attains higher values. Henceforth, the role of $n_{r}$ on these measures is not
discussed any further.

$I_{\rvec}, I_{\pvec}$ of $1p, 1d,1f$ states of 3DCHO at six $r_c$, with varying $m$. Last column correspond
to respective 3DQHO values, computed from Eq. (77) [130].

$|m|$      $r_c=0.1$     $r_c=0.5$      $r_c=1$         $r_c=2$       $r_c=7$        $r_c=\infty$

0         8076.29142      323.0517082      80.76619765      20.39764116     10.00000000       10

1         5736.542528     229.4242339      57.21792995      13.91660078     6.00000000        6

0         0.014984249      0.374503742     1.491857857      5.577935621     10.00000000      10

1         0.01003          0.2507          0.9980           3.975           6.0000           6

0         13286.984765        531.4794285      132.8722779       33.37450170      14.00000000      14

1         10419.849672        416.7666252      104.0908347       25.74558754      10.00000000      10

2         7552.714578         302.0538220      75.3093915        18.11667339       6.00000000        6

0         0.017547979       0.43862372     1.74993894      6.70582522     14.0000000       14

1         0.0133333            0.33326            1.32905             5.0595             10.00000           10

2         0.0091187            0.2279             0.9081              3.4132              6.00000            6

0         19532.47745          781.2991270      195.3266196       48.95155358      18.000000000      18

1         16156.649498         646.2448005      161.48315319      40.15669405      14.00000000      14

2         12780.8215           511.1904739      127.63968669      31.36183453      10.00000000      10

3         9404.993581          376.1361474      93.79622020       22.566975007     6.000000000        6

0         0.0194769435        0.486866098         1.944005783        7.551236509        18.0000000       18

1         0.0157907758        0.39471708         1.57572258          6.0995039          14.00000        14

2         0.012104608         0.30256807         1.20743938          4.6477714          10.00000         10

3         0.008418440         0.2104190          0.8391561           3.196038            6.00000         6

Now, in order to get a clear picture of the effect of magnetic quantum number, Table 8 gives $I_{\rvec}, I_{\pvec}$ of 3DCHO
for lowest three nodeless states having $l \neq 0$, i.e., $1p$-$1f$ respectively, for all allowed $m$ at 6 carefully selected
$r_c$ ($0.1, 0.5, 1, 2, 7, \infty$). Likewise, Fig. 10 portrays these for $1g$ state in panels (a) and (b), for all possible $|m|$.
It is to be noted that, the quantities in last column at $r_c = \infty$ are given from Eq. (77) considering $\omega=1$.
For this special $\omega$, Eq. (77) dictates that, $I_{\rvec}=I_{\pvec}$ for 3DQHO. One notices that, behavior of $I_{\rvec}$ and
$I_{\pvec}$ in 3DCHO is always in agreement with 3DQHO; an explicit analysis suggests identical patterns. In general, $I_{\rvec}$
decays monotonically while $I_{\pvec}$ progresses with rise in $r_c$; this is found to hold true for all $|m|$. This is
to be expected, as a growth in $r_c$ promotes delocalization in $r$ space and localization in $p$ space. At a fixed $r_c$ and
fixed quantum numbers $n_{r}$ and $m$, $I_{\rvec}$ tends to rise with $l$; this is consistent to what is observed from Eq. (77) in 3DQHO.
Because, in both 3DCHO and 3DQHO, kinetic energy grows with $n_{r},l$. Similarly, at fixed $n_{r}, l$, in both 3DCHO and 3DQHO,
$I_{\rvec}$ falls down as one descends down the table (increment in $|m|$). One striking fact is that, for all these three states
considered, both 3DCHO and 3DQHO portray comparable pattern with respect to $m$. Next, $I_{t}$ for $1p$-$1f$ states of 3DCHO are offered
in Table 9. It seems to show an inclination towards $I_{\rvec}$ in the behavioral pattern. In all instances, the lower and
upper bounds stated by Eq. (67) is satisfied. For an arbitrary state distinguished by quantum numbers $n_r,l$, change in
$I_{\rvec}, I_{\pvec}$ with $r_c$ in a 3DCHO preserves same qualitative orderings for various $m$ (general nature of the plots
remain unchanged) as depicted in these figures. This has been accomplished in a number of occasions, which are not reported here to save
space. As usual, in all cases, they all eventually reach their respective 3DCHO-limit at some sufficiently large $r_c$, which
alters from state to state.

Now to understand the dependence of $I_{\rvec}, I_{\pvec}$ on $l$, we present Table 10, where these are given for $l=1-9$ states having
$|m|=1$ and $n_{r}$ corresponding to 1, at same six chosen $r_c$ values of previous table. The last column again has same significance
as Table 8. In accordance with Eq. (77), here also for $n_rl \ (n_r=2)$ states,
$I_{\rvec}=I_{\pvec}$ in 3DQHO. Dependence of $I_{\rvec}, I_{\pvec}$ of 3DCHO on $l$ compliments our observation in 3DQHO. At a fixed
$n_{r}, m$ and $r_c$, they both enhance with rise of $l$ in 3DCHO and 3DQHO. This may occur probably because that, as $l$
advances, the density gets increasingly concentrated. Therefore, at a certain $n_{r}, m$, a state with higher $l$ undergoes
greater fluctuation. Thus all our foregoing discussion leads to a general fact that,in a 3DCHO, the qualitative variations of
$I_{\rvec}$ with all three quantum numbers remain quite analogous  to that of $I_{\pvec}$; also the patterns in 3DCHO and 3DQHO are
similar. It may be appropriate to mention a parallel work [129] along this direction for free and confined H atom
inside an impenetrable spherical environment. One finds various significant deviations in the variation pattern between two systems
there. Here we mention two of the most interesting facts, which are in complete contrast with a 3DCHO, e.g., $I_{\rvec}$ remains
unchanged with respect to changes in $l$, whereas under confinement, it reduces with $l$ (at fixed $n,m$). Besides, for a given
state having fixed $n,m$, $I_{\pvec}$ enhances when the atom is compressed, whereas declines in a free H atom.

Variation of $I_{\rvec}$ and $I_{\pvec}$ in 3DCHO, with $r_c$, for all allowed $|m|$ values of $1g$ state, in panels
(a) and (b) respectively [130]. See text for details.

At this juncture, a few words may be devoted to the influence of $\omega$ on $I_{\rvec}, I_{\pvec}$. In order to establish this,
Fig. 11 depicts plots of $I_{\rvec}$ and $I_{\pvec}$ against $r_c$, for 5 selected $\omega^{2}$ ($1, 2, 4, 8, 32$), in bottom
((a), (b)) and top ((c), (d)) panels. These are provided for $1p$ state; left and right panels characterize $|m|=0$ and 1
respectively. Evidently at a fixed $r_c$, $I_{\rvec}$ grows and $I_{\pvec}$ decays with increment in $\omega$. At a certain
$\omega$, dependence of these measures on $n_{r},l,m$ is similar to that in 3DQHO. As $\omega$ goes up, there is more localization,
hence compactness in single-particle density increases with oscillation strength.

An in-depth analysis of $I$ reveals that, a gradual increase in $r_c$ should lead to a delocalization of the system in such a
way that, at $r_c \rightarrow \infty$ it should approach towards an 3DQHO. On the contrary, when $r_c \rightarrow 0$ the impact of
confinement is maximum. It suffices for us to vary $r_c$ from $0.1$ to $8$. This parametric increase in $r_c$ elicits the system
from an extremely confined environment to a free 3DQHO scenario.

$I_{t}$ for $1p$-$1f$ $(n_{r}=0)$ orbitals at six different $r_c$ [130]. See text for details.

$|m|$      $r_c=0.1$     $r_c=0.5$      $r_c=1$         $r_c=2$       $r_c=7$        $r_c=\infty$

$0^{\dag}$       121.01515     120.98286      120.491051      113.776614     100.00000000      100

1                57.53752      57.5166        57.1034         55.318         36.0000            36

$0^{\dag}$     233.15973      233.117506     232.518248     223.80357     196.000000000      196

1             138.93098      138.89164      138.3419       130.2598      100.00000          100

2             68.870938       68.8380       68.3884         61.8358      36.00000            36

$0^{\dag}$     380.432960       380.388051      379.716077     369.644758     324.0000000      324

1              255.126029       255.083860      254.452650     244.935911     196.00000        196

2              154.706834       154.669915      154.117184     145.762637     100.00000        100

3              79.1753741       79.1461919      78.7096703     72.1249096      36.00000         36

$^\dag$ These also correspond to upper bounds, given in Eq. (66).

$^a$ Lower bounds Eq. (11), for $1p$ at 6 $r_c$ are: 10.70940, 10.71226, 10.75598,
11.39074,12.96, 12.96.

$^b$ Lower bounds Eq. (11), for $1d$ at 6 $r_c$ are: 5.55842, 5.559428, 5.573756,
5.79079,6.61224, 6.61224.

$^c$ Lower bounds Eq. (11), for $1f$ at 6 $r_c$ are: 3.40664, 3.40704, 3.41307,
3.506068, 4, 4.

Next, Table 11 shows evaluated $R_{\rvec}^{\alpha}, R_{\pvec}^{\beta}, R_{t}^{(\alpha, \beta)}$ for first two ($n_r=1,2$)
$s,p$ and $d$ states of 3DCHO at a selected set of $r_c$ values. $R_{\rvec}^{\alpha}$, starting from a $(-)$ve value at very low
$r_c$, continuously increases, before finally reaching to the respective 3DQHO limit. This pattern in $R_{\rvec}^{\alpha}$ distinctly
expresses delocalization of the system with growth in $r_c$. In contrast, $R_{\pvec}^{\beta}$, for all states generally tend
to diminish monotonically with $r_c$, again converging to 3DQHO in the end. The analysis of $R_{t}^{(\alpha, \beta)}$ shows
that, for $l=0$ states it lowers with $r_c$ to reach the unconfined result. However, for $1p$ state, it advances with $r_c$ to
attain the 3DQHO limit. In case of $2p,1d$ and $2d$ states it initially increases with $r_c$, then passes through a maximum, and
finally convenes to limiting 3DQHO result. At small $r_c$ (up to $ \approx 2$), $n_r=1$ states have higher $R_{\rvec}^{\alpha}$
compared to their $n_r=2$ counterparts. But at moderate $r_c$ ($ > 2$) region the situation alters; the $n_r=2$ states now
possess larger $R_{\rvec}^{\alpha}$. Moreover this crossover region appears at larger $r_c$ with rise in $l$. This observation
concludes that, at smaller $r_c$ region the effect of confinement is more pronounced for higher $n_r$ states (as $n_r=1$ states
possess higher $R_{\rvec}^{\alpha}$ values than that of respective $n_r=2$ states). There is no such crossover in
$R_{\pvec}^{\beta}$ and $R_{t}^{(\alpha, \beta)}$ in any of these reported states. The above observations are graphically demonstrated in Fig. 12,
where in segments (a)-(c),
$R_{\rvec}^{\alpha}, R_{\pvec}^{\beta}, R_{t}^{(\alpha, \beta)}$ of first five circular states are presented as function of $r_c$.
Panel (a) leads to the fact that, for all of them, $R_{\rvec}^{\alpha}$'s quite steadily progress with $r_c$ and finally converge
to 3DQHO. Similarly, from panel (b) it is clear that, $R_{\pvec}^{\beta}$ shows reverse pattern with $r_c$, before reaching
3DQHO-limit. From the last panel, it appears that, for $l=0$ state, $R_{t}^{(\alpha, \beta)}$ abates with $r_c$, whereas an
opposite trend occurs otherwise. Also one finds that major changes in $R_{t}^{(\alpha, \beta)}$ occurs in the range of $r_c$
varying from 2.65-5.30, with the higher state showing most predominant effect (see Figs. 15 and 17 later for $S_t$ and $E_t$, for
similar effects). Some of these may not be evident from the figure as such, but becomes clear upon magnification of the plot, in
consultation with the data given in Table 11.

$I_{\rvec}, I_{\pvec}$ for $2p$-$2m$ ($|m|=1$) states in 3DCHO, at six selected $r_c$ [130].

$l$      $r_c=0.1$      $r_c=0.5$        $r_c=1$       $r_c=2$         $r_c=7$       $r_c=\infty$

1       19544.75146        781.7758499        195.3902144       48.61008407       14.000000000        14

2       28201.56829        1128.0488516       281.9599554       70.272141773      18.000000000       18

3       37976.00550        1519.027385        379.70865634      94.73419843       22.0000000000        22

4       48838.86407        1953.5428786       488.3419262       121.91517166      26.0000000000        26

5       60766.49285        2430.649110        607.62258270      151.75456390      30.0000000001       30

6       73739.4732         2949.5692932       737.3562778       184.204055587     34.0000000743       34

7       87741.58879        3509.6547640       877.38084537      219.22364350      38.00000037         38

8       102759.07974       4110.3551463       1027.5587357      256.77944910      42.0000017797        42

9       118780.106607      4751.196872        1187.77161130     296.84230573      46.0000074341       46

1       0.01221          0.3054            1.222            4.917           14.000          14

2       0.0133333        0.333337          1.33356          5.34750         18.00000        18

3       0.014439157      0.36097833        1.4438781        5.7743588       21.999997       22

4       0.0154743767     0.38685628        1.54723822       6.1782594       26.0000000      26

5       0.016426161      0.4106495         1.6423270        6.55303527      30.000000       30

6       0.017296672      0.4324115849      1.7293332625     6.898114452     34.00000        34

7       0.0180927445     0.4523131126      1.808922468      7.21517631      38.00000        38

8       0.0188222240     0.470550071       1.88186836       7.50667165      41.9999991      42

9       0.019492648      0.487310795       1.94891804       7.775182        45.99998        46

To gain further insight, Fig. 13 delineates $R^{\alpha}_{\rvec}, R^{\beta}_{\pvec},  R_{t}^{(\alpha, \beta)}$, in left (a),
middle (b), right (c) panels, for lowest five circular states with changes in $n_r$ (maximum being 9). Four different $r_c$'s are
chosen, namely, $0.1, 3, 5, \infty$ in segments (A)-(D) from bottom to top. At the lowest $r_c$, for all $l$,
$R^{\alpha}_{\rvec}$'s gradually falls off with advancement in $n_r$, providing highest values for $l=0$. But for non-zero
$l$, it increases with $l$ for smaller $n_r$. However, at $n_r>6$, there seems to be a crossover between $l=1$ and $l=2$ states.
Hence it can be concluded that, effect of confinement is maximum, minimum for $l=1$, $l=0$ states respectively. At a fixed $l$ and
$r_c$ (lower region), higher $n_r$-states encounter the confinement to a greater extent. On the other hand, both
$R^{\beta}_{\pvec}$ (b) and $R_{t}^{(\alpha, \beta)}$ (c) show inverse trend. For smaller $n_r$, both these quantities follow the
sequence $1g>1f>1d>1p>1s$, which however, gets reversed ($1s>1p>1d>1f>1g$) at higher $n_r$. These results infers that, at lower
$r_c$ region quantum effect gets amplified as information content decreases, whereas total information increases with $n_r$. The
first column (a) interestingly shows the appearance of maximum in $R^{\alpha}_{\rvec}$ at some moderate $r_c$, whose positions
get right shifted as $r_c$ enhances. Apparently there exists an interplay between two opposing effects: (i) radial confinement
(localization) (ii) accumulation of nodes with $n_r$ (delocalization). As $r_c$ progresses, delocalization prevails for lower
$n_r$. Hence with gradual relaxation in confinement, states with larger $n_r$ get delocalized. At $r_c \rightarrow \infty$,
the second effect predominates; so the system behaves like 3DQHO. In second and third columns, one sees that, both
$R^{\beta}_{\pvec}, R_{t}^{(\alpha, \beta)}$ enhance with $n_r$. At $r_c \rightarrow \infty$, these two quantities approach the
respective 3DQHO-limit, as predicted.

Plots of $I_{\rvec}$ and $I_{\pvec}$ in bottom ((a), (b)) and top ((c),(d)) panels in 3DCHO with $r_c$, of
$1p$ state at five selected $\omega^{2} (1, 2, 4, 8, 32)$. Left, right columns correspond to $|m|=0$, 1 respectively [130].

Next we move to $S$ in Table 12, where $S_{\rvec}, S_{\pvec}$ and $S_{t}$ are demonstrated for $1s,2s,1p,2p,1d,2d$ states of 3DCHO
at same set of $r_c$ of the previous table. Akin to
$R^{\alpha}_{\rvec}$, $S_{\rvec}$ also yield $(-)$ve values for all six states at very low $r_c$, and then continuously
increase until reaching the limiting 3DQHO value. This trend of $S_{\rvec}$ also interprets the delocalization of 3DCHO with
progress in $r_c$. Like $R^{\beta}_{\pvec}$, $S_{\pvec}$ also offers a reverse nature (continuous decrease with $r_c$) of
$S_{\rvec}$ (and $R^{\alpha}_{\rvec}$); from an initial $(+)$ve value, consistently falls off to reach 3DQHO. $S_{t}$'s for both
$1s,1p$ states decrease to attain 3DQHO result. Interestingly, for $1d$ state it first declines, attains a minimum value, then rises
to a maximum and ultimately reaches to 3DQHO values. But for $2s,2p,2d$ states it first increases, then reaches a maximum and
finally merges to borderline 3DQHO values.

$R_{\rvec}^{\alpha},R_{\pvec}^{\beta}$ and $R_{t}^{\alpha,\beta}$ for lowest three $l$ (having $n_r=1,2$) states in 3DCHO
at eight selected values of $r_c$.  $R_{t}^{\alpha,\beta}$ for all these states obey the lower bound given in Eq. (57) [181].

$r_c$         $R_{\rvec}^{\alpha}$         $R_{\pvec}^{\beta}$       $R_{t}^{\alpha,\beta}$
$r_c$         $R_{\rvec}^{\alpha}$         $R_{\pvec}^{\beta}$       $R_{t}^{\alpha,\beta}$

1-4 5-8
4c$1s$             4c$2s$

1-4 5-8
0.1          $-$6.0366917844     12.247171      6.210480     0.1       $-$6.0653334752     14.251561     8.186227

0.2          $-$3.9572613535     10.167740      6.210479     0.2       $-$3.9858896952     12.172113     8.186223

0.5          $-$1.2088403559     7.4192722      6.210431     0.5       $-$1.2369267194      9.422993     8.186067

1.0          0.8636306014       5.3460818      6.209712     1.0          0.8438971150      7.339578     8.183475

5.0          3.6326806767       2.5410652      6.173745     5.0          4.5764993107      2.616203     7.192702

8.0          3.6326909163       2.5410540      6.173744     8.0          4.5767695172      2.614825     7.191594

$\infty$     3.6326909163       2.5410540      6.173744    $\infty$      4.5767695172      2.614825     7.191594

1-4 5-8
4c$1p$             4c$2p$

1-4 5-8
0.1         $-$6.1671363542     12.806502     6.639365       0.1          $-$6.29011971     14.178463      7.888343

0.2         $-$4.0876997330     10.727065     6.639366       0.2          $-$4.21067738     12.099025      7.888348

0.5         $-$1.3390273792     7.978413      6.639386       0.5          $-$1.46177311      9.350298      7.888525

1.0          0.7373200936       5.902381      6.639701       1.0          0.61816051        7.273091      7.891251

5.0          3.8830108378       2.834951      6.717917       5.0          4.57588095        2.982808      7.558689

8.0          3.8830566606     2.834947      6.718004       8.0          4.57683522         2.982008      7.558843

$\infty$     3.8830566606     2.834947      6.718004      $\infty$      4.57683522         2.982008      7.558843

1-4 5-8
4c$1d$             4c$2d$

1-4 5-8
0.1        $-$6.1181191683     13.228954      7.110835     0.1       $-$6.2478841627       14.295117     8.047233

0.2        $-$4.0386800101     11.149516      7.110836     0.2       $-$4.1684423307       12.215678     8.047236

0.5        $-$1.2899046230      8.400780      7.110875     0.5       $-$1.4195583633       9.466934     8.047376

1.0         0.7880364400        6.323450      7.111487     1.0         0.6600654534        7.389472     8.049538

5.0         4.2284239258        3.048859      7.277282     5.0         4.8017869530        3.233068     8.034855

8.0         4.2285908429       3.047026      7.275616     8.0         4.8046032500      3.230664     8.035267

$\infty$    4.2285908429       3.047026      7.275616    $\infty$     4.8046032500      3.230664     8.035267

Figure 14 delineates behavioral trends of $S_{\rvec}$, $S_{\pvec}$, $S_{t}$ with $r_c$, in segments (a)-(c), for same five states of
Fig. 12 (done for $R$). One notices that, both Figs. 12 and 14 offer qualitatively analogous changes with respect to $r_c$
variation; thus signalling comparable trends in $R$ and $S$. For all these states $S_{\rvec}$'s tend to progress with $r_c$ and
then reach the corresponding $r$-space 3DQHO, while $S_{\pvec}$'s reduce before attaining that.  The last panel portrays that, for
$1s,1p$ states $S_{t}$ decreases with $r_c$, while for the remaining three states, $S_t$ passes through a minimum. It is worth
noting that, in the $1p$ state, $R_t$ progresses and $S_t$ regresses after $r_c=2.65$. In both occasions, however, as usual they
all gain the limiting 3DQHO behavior at a sufficiently large $r_c$.

In Fig. 15, $S_{\rvec}$ (a), $S_{\pvec}$ (b), $S_{t}$ (c) of $l=0-4$ states are plotted with respect to $n_r$ at
same four values of $r_c$ in Fig. 13, in panels marked with (A)-(D) from bottom to top. Again, these graphs produce resembling
shape and propensity as in Fig. 13. Thus in harmony with $R^{\alpha}_{\rvec}$ (for all five $l$ at the lowest $r_c=0.1$),
$S_{\rvec}$ gets lowered in A(a), while $S_{\pvec}$s and $S_{t}$s accelerate with $n_r$ in A(b) and A(c), respectively. This
supports our previous observation (of $R$ in Fig. 13) that, at very low $r_c$, the effect of confinement is more prominent in
high-lying states, signifying an intensification of quantum nature in such a scenario. As usual, like $R^{\alpha}_{\rvec}$ here
also, the first column (a) shows the appearance of maximum in $S_{\rvec}$ plot with gradual progress of $r_c$ from the bottom to
top. Their positions switch to right as $r_c$ enhances. This observation indicates that, at $r_c \rightarrow \infty$ the system
behaves like 3DQHO.

At this point we proceed for the last measure ($E$) of this study in Table 13. A cross-section of $E_{\rvec}, E_{\pvec}, E_{t}$
for $1s,2s,1p,2p,1d,2d$ states of 3DCHO (same set of $r_c$ as used in previous table) is offered. One notices that, $E_{\rvec}$
reduces (as opposed to $R^{\alpha}_{\rvec}, S_{\rvec}$) while $E_{\pvec}$ progresses (in contrast to $R^{\beta}_{\pvec},
S_{\pvec}$) with rise in $r_c$. However, the nature of variation of $E_{t}$ with $r_c$ varies from state to state. For
$1s,1p,1d$ states, it enhances with $r_c$, while $2s,2p, 2d$ states, it goes through a minimum. These changes in $E_{\rvec},
E_{\pvec}$ and $E_{t}$ with $r_c$ are demonstrated pictorially in Fig. 16, in left (a), middle (b) and right (c) panels for first five
node-less states. It is found that, $E_{t}$ for $1s,1p,1d$ states advances with $r_c$, while for $1f,1g$ states it decreases.
Interestingly, at large $r_c$, both  $E_{\rvec}, E_{\pvec}$ decline with growth in $l$. Finally, Fig. 17, depicts $E_{\rvec},
E_{\pvec}, E_{t}$ (in columns (a),(b),(c)) for $l=0-4$ states as functions of $n_r$ at four different $r_c$ (in segments A-D). At
the lowest $r_c$ considered, $E_{\rvec}$ advances with $n_r$. However, the first column (a) suggests that, a minimum appears in
$E_{\rvec}$ graphs as $r_c$ is increased. Also the positions of these minima shift towards right with rise in $r_c$. On the
contrary, for all concerned $r_c$, both $E_{\pvec}$, $E_{t}$ decay with $n_r$.

All the discussed measures of Eq. (72) have been investigated with change of $r_c$, choosing two different and widely used values
of $b$ ($1,\frac{2}{3}$). Note that, for $b=1$, $C_{ES}^{(2)}$ transforms to $C_{LMC}$; similarly $C_{IS}^{(1)}$ coincides with
$C_{IS}$ at $b=\frac{2}{3}$. In order to simplify the discussion, a few words may be devoted to the notation used. A uniform symbol
$C_{order_{s}, disorder_{s}}^b$ is used; where the two subscripts refer to two order ($E, I$) and disorder ($S, R$) parameters.
Another subscript $s$ is used to specify the space; viz., $r, p$ or $t$ (total). Two scaling parameters $b=\frac{2}{3}, 1$
are identified with superscripts (1), (2). These measures are examined systematically for $1s,~1p,~1d,~2s,~1f,~2p,~1g$ and $2d$
states in conjugate spaces. In this part we will discuss the outcome extracted from a recent complexity analysis undertaken by us.
The detailed results with relevant figures and tables will be published elsewhere separately.

Now we lay our focus on $C_{E_{\rvec}S_{\rvec}}^{(1)},C_{E_{\pvec}S_{\pvec}}^{(1)}$ and $C_{E_{\rvec}S_{\rvec}}^{(2)},
C_{E_{\pvec}S_{\pvec}}^{(2)}$ respectively. $C_{E_{\rvec}S_{\rvec}}^{(1)}$ decreases and $C_{E_{\pvec}S_{\pvec}}^{(1)}$ increases
with rise of $r_c$, before reaching the borderline 3DQHO result. As mentioned earlier, the 3DCHO model acts as an intermediate
between the PISB and 3DQHO [130]. That means, an enhancement in $r_c$ should lead the system towards delocalization
(equilibrium), which is supported by the declining nature of $C_{E_{\rvec}S_{\rvec}}^{(1)}$ with progress in $r_c$. At a
fixed $l$, $C_{E_{\rvec}S_{\rvec}}^{(1)}$ progresses with $n_r$. It may be noted that in a 3DCHO, number of nodes varies with $n_r$
only (in contrast to, say the H atom, where it depends on both $n=n_r$ and $l$). Hence in this case, the increment in number of
nodes takes the system towards the side of ordering. But no such pattern is observed for
$C_{E_{\pvec}S_{\pvec}}^{(1)}$. The variation of $C_{E_{\rvec}S_{\rvec}}^{(2)},~C_{E_{\pvec}S_{\pvec}}^{(2)}$ with change of $r_c$
reveals that, former advances with growth of $r_c$ indicating that this is more inclined towards order. On the other
hand, at first $C_{E_{\pvec}S_{\pvec}}^{(2)}$ decreases with $r_c$, attains a minimum and finally merges to 3DQHO. The ordering of
$C_{E_{\rvec}S_{\rvec}}^{(2)}$ regarding $n_r$ quantum number is akin to $C_{E_{\rvec}S_{\rvec}}^{(1)}$. This observation
complements our previous conjecture that, an increase in number of nodes takes the system towards disequilibrium. It is noticed
that, after a certain $r_c$ ($\approx 3$), both $C_{E_{\rvec}S_{\rvec}}^{(2)},C_{E_{\pvec}S_{\pvec}}^{(2)}$ show analogous
nature (increase to reach their respective limiting value). Here, exploring these two sets of complexity measure, namely,
$C^{(1)}_{ES}$ and $C^{(2)}_{ES}$, it is evident that, $C_{E_{\rvec}S_{\rvec}}^{(1)}$ and
$C_{E_{\pvec}S_{\pvec}}^{(1)}$ complement each other better (as former decreases, later increases with $r_c$) and more tends
towards disorder (equilibrium), whereas the other set is inclined towards the order-direction. Thus, $C_{ES}^{(1)}$ with
$b=\frac{2}{3}$ explains the system better. It is important to point out that, when 3DCHO approaches to 3DQHO,
$C_{E_{\rvec}S_{\rvec}}^{(1)}$ becomes equal to $C_{E_{\pvec}S_{\pvec}}^{(1)}$.

Plots of $R_{\rvec}^{\alpha}$, $R_{\pvec}^{\beta}$, $R_{t}^{(\alpha, \beta)}$ against $r_c$ for first five circular
states of 3DCHO in panels (a), (b), (c) respectively [181]. See text for details.

In the same direction, now we interpret the pattern changes of $C_{E_{\rvec}R_{\rvec}}^{(1)},C_{E_{\pvec}R_{\pvec}}^{(1)}$ with $r_{c}$.
At first, $C_{E_{\rvec}R_{\rvec}}^{(1)}$ declines with growth of $r_c$, then passes through a minimum and finally converges to respective
3DQHO result. This minimum gets flatter with rise of both $n_r,~l$. Here also $C_{E_{\rvec}R_{\rvec}}^{(1)}$ shows similar trend
as observed in $C_{E_{\rvec}S_{\rvec}}^{(1)}$, viz., (i) like $C_{E_{\rvec}S_{\rvec}}^{(1)}$,
$C_{E_{\rvec}R_{\rvec}}^{(1)}$ is more aligned towards disorder (ii) at a particular $l$, they progress with $n_r$. The first point
suggests that, like $C_{E_{\rvec}S_{\rvec}}^{(1)}$, $C_{E_{\rvec}R_{\rvec}}^{(1)}$ also interprets that, with increase in $r_c$
the system progresses towards 3DQHO from PISB via 3DCHO. The second point says that, an increase/decrease in number of nodes leads
towards the direction of order/disorder respectively. Like $C_{E_{\pvec}S_{\pvec}}^{(1)}$, $C_{E_{\pvec}R_{\pvec}}^{(1)}$ increases with
growth in $r_c$. Moreover, at a definite $l$, $C_{E_{\pvec}R_{\pvec}}^{(1)}$ falls off with $n_r$. At smaller $r_c$ region ($\approx 3$),
both $C_{E_{\rvec}R_{\rvec}}^{(2)},C_{E_{\pvec}R_{\pvec}}^{(2)}$ vary slowly. At around same $r_c$, the former jumps and latter drops to
attain the borderline 3DQHO limit. For both $C_{E_{\rvec}R_{\rvec}}^{(2)}$ and $C_{E_{\pvec}R_{\pvec}}^{(2)}$ absolute values of the
slope of the curve
enhance as $n_r$ grows (fixed $l$) and decrease with growth of $l$ (fixed $n_r$). The dependence of $C_{E_{\rvec}R_{\rvec}}^{(2)}$
on $n_r$ is similar to $C_{E_{\rvec}R_{\rvec}}^{(1)}$. This again validates the dependency of this complexity measure on number of
nodes of the system. Once again one may conclude that, out of $C^{(1)}_{ER}$ and $C^{(2)}_{ER}$, the former offers a more
complete knowledge about the characteristics of 3DCHO. Apart from that, $C_{E_{\pvec}R_{\pvec}}^{(1)}$ shows more lucid trend than
$C_{E_{\pvec}S_{\pvec}}^{(1)}$ with respect to dependence of quantum number $n_r$. Hence, in practice, $C^{(1)}_{ER}$ may possibly