arXiv subjects

Neetik Mukherjee

Publications and source records attributed to Neetik Mukherjee.

At least 19 recordsLinked to original sources

arXiv2025

Degeneracy and metallic character in free and confined weakly coupled plasmas: with and without electric field

Incidental degeneracy and metallic character is probed for weakly coupled plasmas in free and confined environments. The generality of incidental degeneracy in quantum mechanical systems is discussed and demonstrated. It is a fundamental property of free and confined quantum systems. In plasmas, at a given $n, \ell$state there exists $\frac{(n-\ell)(n-\ell+1)}{2}$ number of incidental degenerate states. Such degeneracy condition involves shell confinement model, where a particleis encaged inside two concentric sphere. Apart from that, Dipole oscillator strength and polarizability are examined in free and confined conditions for ground and some low-lying $\ell$ states. In excited states, negative $\alpha^{(1)}$ is recorded. Further, metallic behavior of H-like plasmas is investigated. The impact of external static electric field on these degeneracy, dipole OS, dipole polarizability are examined with utmost interest. Pilot calculation are done with, (i) Debye plasmas and, (ii) Exponential screened coulomb potentials employing the Generalized pseudo-spectral (GPS) method.

physics.plasm-ph↗

arXiv2024

Information entropy in excited states in confined quantum systems

The present contribution constitutes a brief account of information theoretical analysis in several representative model as well as real quantum mechanical systems. There has been an overwhelming interest to study such measures in various quantum systems, as evidenced by a vast amount of publications in the literature that has taken place in recent years. However, while such works are numerous in so-called \emph{free} systems, there is a genuine lack of these in their constrained counterparts. With this in mind, this chapter will focus on some of the recent exciting progresses that has been witnessed in our laboratory \cite{sen06,roy14mpla,roy14mpla_manning,roy15ijqc, roy16ijqc, mukherjee15,mukherjee16,majumdar17,mukherjee18a,mukherjee18b,mukherjee18c,mukherjee18d,majumdar20,mukherjee21,majumdar21a, majumdar21b}, and elsewhere, with special emphasis on following prototypical systems, namely, (i) double well (DW) potential (symmetric and asymmetric) (ii) \emph{free}, as well as a \emph{confined hydrogen atom} (CHA) enclosed in a spherical impenetrable cavity (iii) a many-electron atom under similar enclosed environment.

quant-ph↗

arXiv2023

Lower bound of the expressibility of ansatzes for Variational Quantum Algorithms

The expressibility of an ansatz used in a variational quantum algorithm is defined as the uniformity with which it can explore the space of unitary matrices, i.e., its covering number. The expressibility of a particular ansatz has a well-defined upper bound [1]. In this work, we show that the expressibility also has a well-defined lower bound in the hypothesis space. We provide an analytical expression for the lower bound of the covering number, which is directly related to expressibility. Further, we provide numerical analysis to support our claim. By calculating the bond length of hydrogen molecule ($H_2$) using different ansatzes in a variational quantum eigensolver (VQE) setting, we study the variation of equilibrium energy error with circuit depths. We show that in each ansatz template, a plateau exists for a range of circuit depths, which we call the set of acceptable points, and the corresponding expressibility as the best expressive region. We report that the width of this best expressive region in the hypothesis space is inversely proportional to the average error. Our analysis reveals that alongside trainability, the lower bound of expressibility also plays a crucial role in selecting variational quantum ansatzes

quant-ph↗

arXiv2022

Confined H$^-$ ion within a density functional framework

Ground and excited states of a confined negative Hydrogen ion has been pursued under Kohn-Sham density functional approach by invoking a physically motivated work-function-based exchange potential. The exchange-only results are of near Hartree-Fock quality. Local parameterised Wigner-type, and gradient- and Laplacian-dependent non-local Lee-Yang-Parr functionals are chosen to investigate the electron correlation effects. Eigenfunctions and eigenvalues are extracted by using a generalized pseudospectral method obeying Dirichlet boundary condition. Energy values are reported for 1s$^{2}$ ($^{1}$S), 1s2s ($^{3,1}$S) and 1s2p ($^{3,1}$P) states. Performance of the correlation functionals in the context of confinement is examined critically. The present results are in excellent agreement with available literature. Additionally, Shannon entropy and Onicescu energy are offered for ground and low lying singly excited 1s2s ($^{3}$S) and 1s2p ($^{3}$P) states. The influence of electron correlation is more predominant in the weaker confinement limit and it decays with an increase in confinement strength. In essence, energy and some information measures are estimated using a newly formulated density functional strategy.

physics.atom-ph↗

arXiv2022

Hydrogen-like ions in plasma environment

The behavior of H-like ions embedded in astrophysical plasmas in the form of \emph{dense, strongly and weakly coupled} plasmas are investigated. In these, the increase and decrease in temperature is impacted with a change in confinement radius $(r_{c})$. Two independent and generalized scaling ideas have been applied to modulate the effect of plasma screening constant ($λ$) and charge of ion ($Z$) on such systems. Several new relations are derived to interconnect the original Hamiltonian and two scaled Hamiltonians. In exponential cosine screened Coulomb potential (ECSCP) (dense) and weakly coupled plasma (WCP) these scaling relations have provided a linear equation connecting the critical screening constant $(λ^{(c)})$ and $Z$. Their ratio offers a state-dependent constant, beyond which, a particular state vanishes. Shannon entropy has been employed to understand the plasma effect on the ion. With increase in $λ$, the accumulation of opposite charge surrounding the ion increases leading to a reduction in number of bound states. However, with rise in ionic charge $Z$, this effect can be delayed. The competing effect of plasma charge density ($n_e$) and temperature in WCP and ECSCP is investigated. A recently proposed simple virial-like theorem has been established for these systems. Multipole ($k=1-4$) oscillator strength (OS) and polarizabilities for these are studied considering $1s, 2s$ states. As a bonus, analytical closed-form expressions are derived for $f^{(k)}$ and $α^{(k)} (k=1-4)$ involving $1s$ and $2s$ state, for \emph{free H-like ion}.

physics.plasm-ph↗

arXiv2022

Shell-confined atom and plasma: incidental degeneracy, metallic character and information entropy

Shell confined atom can serve as a generalized model to explain both \emph{free} and \emph{confined} condition. In this scenario, an atom is trapped inside two concentric spheres of inner $(R_{a})$ and outer $(R_{b})$ radius. The choice of $R_{a}, R_{b}$ renders four different quantum mechanical systems. In hydrogenic atom, they are termed as (a) free hydrogen atom (FHA) (b) confined hydrogen atom (CHA) (c) shell-confined hydrogen atom (SCHA) (d) left-confined hydrogen atom (LCHA). By placing $R_{a}, R_{b}$ at the location of radial nodes of respective \emph{free} $n,\ell$ states, a new kind of degeneracy may arise. At a given $n$ of FHA, there exists $\frac{n(n+1)(n+2)}{6}$ number of iso-energic states with energy $-\frac{Z^{2}}{2n^{2}}$. Furthermore, within a given $n$, the individual contribution of each of these four potentials has also been enumerated. This incidental degeneracy concept is further explored and analyzed in certain well-known \emph{plasma} (Debye and exponential cosine screened) systems. Multipole oscillator strength, $f^{(k)}$, and polarizability, $α^{(k)}$, are evaluated for (a)-(d) in some low-lying states $(k=1-4)$. In excited states, \emph{negative} polarizability is also observed. In this context, metallic behavior of H-like systems in SCHA is discussed and demonstrated. Additionally analytical closed-form expression of $f^{(k)}$ and $α^{(k)}$ are reported for $1s,2s,2p,3d,4f,5g$ states of FHA. Finally, Shannon entropy and Onicescu {\color{red}information} energies are investigated in ground state in SCHA and LCHA in both position and momentum spaces. Much of the results are reported here for first time.

quant-ph↗

arXiv2021

Information analysis in free and confined harmonic oscillator

In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator. 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, Shannon entropy, Renyi entropy , Tsallis entropy, Onicescu energy 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.

quant-ph↗

arXiv2021

Analysis of Compton profile through information theory in H-like atoms inside impenetrable sphere

Confinement of atoms inside various cavities has been studied for nearly eight decades. However, the Compton profile for such systems has not yet been investigated. Here we construct the Compton profile (CP) for a H atom radially confined inside a \emph{hard} spherical enclosure, as well as in \emph{free condition}. Some exact analytical relations for the CP's of circular or nodeless states of free atom is presented. By means of a scaling idea, this has been further extended to the study of an H-like atom trapped inside an impenetrable cavity. The accuracy of these constructed CP has been confirmed by computing various momentum moments. Apart from that, several information theoretical measures, like Shannon entropy ($S$) and Onicescu energy ($E$) have been exploited to characterize these profiles. Exact closed form expressions are derived for $S$ and $E$ using the ground state CP in free H-like atoms. A detailed study reveals that, increase in confinement inhibits the rate of dissipation of kinetic energy. At a fixed $\ell$, this rate diminishes with rise in $n$. However, at a certain $n$, this rate accelerates with progress in $\ell$. A similar analysis on the respective free counterpart displays an exactly opposite trend as that in confined system. However, in both free and confined environments, CP generally gets broadened with rise in $Z$. Representative calculations are done numerically for low-lying states of the confined systems, taking two forms of position-space wave functions: (a) exact (b) highly accurate eigenfunctions through a generalized pseudospectral method. In essence, CPs are reported for confined H atom (and isoelectronic series) and investigated adopting an information-theoretic framework.

quant-ph↗

arXiv2019

Information entropy as a measure of tunneling and quantum confinement in a symmetric double-well potential

Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double-well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation-induced exact diagonalization procedure. The information entropy-based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase-space description provides a semiclassical explanation for this feature. Total information entropy and phase-space area show similar behavior with increasing barrier height.

quant-ph↗

arXiv2019

Information-Entropic Measures in Confined Isotropic Harmonic Oscillator

Information based uncertainty measures like R{é}nyi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency ($ω$) and confinement radius ($r_{c}$). For a given quantum state, accurate results are provided by applying respective \emph{exact} analytical wave function in $r$ space. The $p$-space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments ($α, β$) as $(\frac{3}{5}, 3)$ in $r$ and $p$ spaces. A detailed, systematic analysis is performed for confined harmonic oscillator (CHO) with respect to state indices $n_{r},l$, and $r_c$. It has been found that, CHO acts as a bridge between particle in a spherical box (PISB) and free isotropic harmonic oscillator (IHO). At smaller $r_c$, $E_{\rvec}$ increases and $R_{\rvec}^α, S_{\rvec}$ decrease with rise of $n_{r}$. At moderate $r_{c}$, there exists an interaction between two competing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of $n_{r}$ (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.

cond-mat.stat-mech↗

arXiv2019

Quantum confinement in an asymmetric double-well potential through energy analysis and information entropic measure

Localization of a particle in the wells of an asymmetric double-well (DW) potential is investigated here. Information entropy-based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phasespace area, are utilized to explain the contrasting effect of localization-delocalization and role of asymmetric term in such two-well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi-degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variationinduced exact diagonalization method. In brief, information measures and phase-space analysis can provide valuable insight toward the understanding of such potentials.

quant-ph↗

arXiv2019

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

Information-based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of \emph{symmetric and asymmetric} confinement in a quantum harmonic oscillator. Also, the transformation of Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ($x_c$). In symmetric case, a wide range of $x_{c}$ has been taken up, whereas asymmetric confinement is dealt by shifting the minimum of potential from origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary time propagation scheme. For asymmetric confinement, a variation-induced exact diagonalization procedure is also introduced, which produces very high-quality results. One finds that, in symmetric confinement, after a certain characteristic $x_{c}$, all these properties converge to respective values of free harmonic oscillator. In asymmetric situation, excited-state energies always pass through a maximum. For this potential, the classical turning-point decreases, whereas well depth increases with the strength of asymmetry. Study of these uncertainty measures reveals that, localization increases with an increase of asymmetric parameter.

quant-ph↗

arXiv2019

Information entropy and complexity measure in generalized Kratzer potential

Shannon entropy ($S$), Fisher information ($I$) and a measure equivalent to Fisher-Shannon complexity $(C_{IS})$ of a ro-vibrational state of diatomic molecules (O$_2$, O$_2^+$, NO, NO$^+$) with generalized Kratzer potential is analyzed. \emph{Exact} analytical expression of $I_{\rvec}$ is derived for the arbitrary state, whereas the same could be done for $I_{\pvec}$ with $\{n,\ell,m=0\}$ state. It is found that shifting from neutral to the cationic system, $I_{\rvec}$ increases while $S_{\rvec}$ decreases, consistent with the interpretation of a localization in the probability distribution. Additionally, this study reveals that $C_{IS}$ increases with the number of nodes in a system.

quant-ph↗

arXiv2019

Fisher information in confined isotropic harmonic oscillator

Fisher information ($I$) is investigated in a confined harmonic oscillator (CHO) enclosed in a spherical enclosure, in conjugate $r$ and $p$ spaces. A comparative study between CHO and a free quantum particle in spherical box (PISB), as well as CHO and respective free harmonic oscillator (FHO) is pursued with respect to energy spectrum and $I$. This reveals that, a CHO offers two exactly solvable limits, namely, a PISB and an FHO. Moreover, the dependence of $I$ on quantum numbers $n_{r}, l, m$ in FHO and CHO are analogous. The role of force constant is discussed. Further, a thorough systematic analysis of $I$ with respect to variation of confinement radius $r_c$ is presented, with particular attention on \emph{non-zero}-$(l,m)$ states. Considerable new important observations are recorded. The results are quite accurate and most of these are presented for the first time.

quant-ph↗

arXiv2019

Relative Fisher information in some central potentials

Relative Fisher information (IR), which is a measure of correlative fluctuation between two probability densities, has been pursued for a number of quantum systems, such as, 1D quantum harmonic oscillator (QHO) and a few central potentials namely, 3D isotropic QHO, hydrogen atom and pseudoharmonic potential (PHP) in both position ($r$) and momentum ($p$) spaces. In the 1D case, the $n=0$ state is chosen as reference, whereas for a central potential, the respective circular or node-less (corresponding to lowest radial quantum number $n_{r}$) state of a given $l$ quantum number, is selected. Starting from their exact wave functions, expressions of IR in both $r$ and $p$ spaces are obtained in closed analytical forms in all these systems. A careful analysis reveals that, for the 1D QHO, IR in both coordinate spaces increase linearly with quantum number $n$. Likewise, for 3D QHO and PHP, it varies with single power of radial quantum number $n_{r}$ in both spaces. But, in H atom they depend on both principal ($n$) and azimuthal ($l$) quantum numbers. However, at a fixed $l$, IR (in conjugate spaces) initially advance with rise of $n$ and then falls off; also for a given $n$, it always decreases with $l$.

quant-ph↗

arXiv2019

Quantum mechanical virial-like theorem for confined quantum systems

Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schrödinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.

quant-ph↗

arXiv2019

Information-entropic measures for non-zero l states of confined hydrogen-like ions

quant-ph↗

arXiv2019

Some complexity measures in confined isotropic harmonic oscillator

Various well-known statistical measures like \emph{López-Ruiz, Mancini, Calbet} (LMC) and \emph{Fisher-Shannon} complexity have been explored for confined isotropic harmonic oscillator (CHO) in composite position ($r$) and momentum ($p$) spaces. To get a deeper insight about CHO, a more generalized form of these quantities with Rényi entropy ($R$) is invoked here. The importance of scaling parameter in the exponential part is also investigated. $R$ is estimated considering order of entropic moments $α, β$ as $(\frac{2}{3},3)$ in $r$ and $p$ spaces respectively. Explicit results of these measures with respect to variation of confinement radius $r_c$ is provided systematically for first eight energy states, namely, $1s,~1p,~1d,~2s,~1f,~2p,~1g$ and $2d$. Detailed analysis of these complexity measures provides many hitherto unreported interesting features.

quant-ph↗