The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p and 3d quantum states by means of the main entropy and complexity information-theoretic measures. First, the Shannon entropy and the Fisher information as well as the associated uncertainty relations are computed and discussed. Then, the Fisher-Shannon, LMC and LMC-Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces.
We study the hydrogen atom confined to a spherical box with impenetrable walls but, unlike earlier pedagogical articles on the subject, we assume that the nucleus also moves. We obtain the ground-state energy approximately by means of first--order perturbation theory and by a more accurate variational approach. We show that it is greater than the one for the case in which the nucleus is clamped at the center of the box. Present approach resembles the well-known treatment of the helium atom with clamped nucleus.
We calculate the ground--state energy and other physical properties of the hydrogen atom inside a spherical box with an impenetrable wall. We apply the variational method and perturbation theory and compare both approximate results. We show that the total, kinetic and potential energies for the moving--nucleus model are greater than those for the case in which the nucleus is clamped at the box center.
The internal disorder of the two-dimensional confined hydrogenic atom is numerically studied in terms of the confinement radius for the 1\textit{s}, 2\textit{s}, 2\textit{p} and 3\textit{d} quantum states by means of the statistical Crámer-Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer-Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the el…
Several well-known statistical measures similar to \emph{LMC} and \emph{Fisher-Shannon} complexity have been computed for confined hydrogen atom in both position ($r$) and momentum ($p$) spaces. Further, a more generalized form of these quantities with Rényi entropy ($R$) is explored here. The role of scaling parameter in the exponential part is also pursued. $R$ is evaluated taking order of entropic moments $α, β$ as $(\frac{2}{3},3)$ in $r$ and $p$ spaces. Detailed systematic results of these measures with respect to variation of confinement radius $r_c$ is presented for low-lying states suc…
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 o…
The spectral characterization of Coulomb systems confined by the homogeneous pseudo-Gaussian oscillator is investigated. This is made using the efficient computational method of generating functional. Also, the method is used for the spectral characterization of homogeneous harmonic oscillator confinement, found as a particular case of pseudo-Gaussian oscillator confinement. Finally the aspect of confinement with an impenetrable sphere of finite radius is considered by studying its conjugate effect along with harmonic oscillator upon the atom.
Fisher information (I) is investigated for confined hydrogen atom (CHA)-like systems in conjugate $r$ and $p$ spaces. A comparative study between CHA and free H atom (with respect to $I$) is pursued. In many aspects, inferences in CHA are significantly different from free counterpart; that includes its dependence on $n, l, m$. The role of atomic number and atomic radius is discussed. Further, a detailed systematic result of $I$ with respect to variation of confinement radius $r_c$ is presented, with particular emphasis on \emph{non-zero}-$(l,m)$ states. Several new interesting observations are…
The diffusion Monte Carlo method with symmetry-based state selection is used to calculate the quantum energy states of H$_2^+$ confined into potential barriers of atomic dimensions (a model for these ions in solids). Special solutions are employed permitting one to obtain satisfactory results with rather simple native code. As a test case, $^2Π_u$ and $^2Π_g$ states of H$_2^+$ ions under spherical confinement are considered. The results are interpreted using the correlation of H$_2^+$ states to atomic orbitals of H atoms lying on the confining surface and perturbation calculations. The method …
The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy and the Fisher information. As well, the complexity measures of Crámer-Rao, Fisher-Shannon and LMC types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polyno…
Shannon entropy in position ($S_{\rvec}$) and momentum ($S_{\pvec}$) spaces, along with their sum ($S_t$) are presented for unit-normalized densities of He, Li$^+$ and Be$^{2+}$ ions, spatially confined at the center of an impenetrable spherical enclosure defined by a radius $r_c$. Both ground as well as some selected low-lying singly excited states, \emph{viz.,} 1sns (n $=$ 2-4) $^3$S, 1snp (n $=$ 2-3) $^3$P, 1s3d $^3$D are considered within a density functional methodology that makes use of a work-function-based exchange potential along with two correlation potentials (local Wigner-type para…
In a recent study of the self-adjoint extensions of the Hamiltonian of a particle confined to a finite region of space, in which we generalized the Heisenberg uncertainty relation to a finite volume, we encountered bound states localized at the wall of the cavity. In this paper, we study this situation in detail both for a free particle and for a hydrogen atom centered in a spherical cavity. For appropriate values of the self-adjoint extension parameter, the bound states lo calized at the wall resonate with the standard hydrogen bound states. We also examine the accidental symmetry generated b…
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 fr…
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. Sta…
The localization of the valence electron of $H$, $Li$ and $Na$ atoms enclosed by three different fullerene molecules is studied. The structure of the fullerene molecules is used to calculate the equilibrium position of the endohedrally atom as the minimum of the classical $(N+1)$-body Lennard-Jones potential. Once the position of the guest atom is determined, the fullerene cavity is modeled by a short range attractive shell according to molecule symmetry, and the enclosed atom is modeled by an effective one-electron potential. In order to examine whether the endohedral compound is formed by a …
In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the $D$-dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ($D \to \infty$) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer-Rao and Fisher-Shannon complexities. Moreover, we show that the LMC (López-Ruiz-Mancini-Calvet) and LMC-Rényi complexity measure…
Shannon entropy ($S$), R{é}nyi entropy ($R$), Tsallis entropy ($T$), Fisher information ($I$) and Onicescu energy ($E$) have been explored extensively in both \emph{free} H atom (FHA) and \emph{confined} H atom (CHA). For a given quantum state, accurate results are presented by employing respective \emph{exact} analytical wave functions in $r$ space. The $p$-space wave functions are generated from respective Fourier transforms$-$for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. \emph{Exact} mathematical expressions of…
The hydrogen bond is a fundamental ingredient to stabilize the DNA and RNA macromolecules. The main contribution of this work is to describe quantitatively this interaction as a consequence of the quantum confinement of the hydrogen. The results for the free and confined system are compared with experimental data. The formalism to compute the energy gap of the vibration motion used to identify the spectrum lines is the Variational Method allied to Supersymmetric Quantum Mechanics.
An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius $r_0$, in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz and Grabert : The centrifugal term is expanded perturbatively (in powers of $\hbar$), decomposing it into 2 terms -- the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energie…
In this paper the Diffusion Monte Carlo (DMC) method is applied to the confined hydrogen atom with different confinement geometries. This approach is validated using the much studied spherical and cylindrical confinements and then applied to cubical and squared ones, for which data are not available, as new applications of the method relevant to solid state physics. The energy eigenvalues of the ground state and one low-lying excited state are reported as a function of the characteristic confinement length.
When confined to small regions quantum systems exhibit electronic and structural properties different from their free space behavior. These properties are of interest, for example, for molecular insertion, hydrogen storage and the exploration of new pathways for chemical and nuclear reactions. Here, a confined three-body problem is studied, with emphasis on the study of the "quantum scars" associated to dynamical collisions. For the particular case of nuclear reactions it is proposed that a molecular cage might simply be used as a confining device with the collision states accessed by quantum …
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 measur…
We report the identification and construction of raising and lowering operators for the complete eigenfunctions of isotropic harmonic oscillators confined by dihedral angles, in circular cylindrical and spherical coordinates; as well as for the hydrogen atom in the same situation of confinement, in spherical, parabolic and prolate spheroidal coordinates. The actions of such operators on any eigenfunction are examined in the respective coordinates, illustrating the possibility of generating the complete bases of eigenfunctions in the respective coordinates for both physical systems. The relatio…
Computational codes based on the Diffusion Monte Carlo method can be used to determine the quantum state of two-electron systems confined by external potentials of various nature and geometry. In this work, we show how the application of this technique in its simplest form, that does not employ complex analytic guess functions, allows to obtain satisfactory results and, at the same time, to write programs that are readily adaptable from one type of confinement to another. This adaptability allows an easy exploration of the many possibilities in terms of both geometry and structure of the syste…
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 …
Similarity between quantum mechanics and thermodynamics is discussed. It is found that if the Clausius equality is imposed on the Shannon entropy and the analogue of the heat quantity, then the value of the Shannon entropy comes to formally coincide with that of the von Neumann entropy of the canonical density matrix, and pure-state quantum mechanics apparently transmutes into quantum thermodynamics. The corresponding quantum Carnot cycle of a simple two-state model of a particle confined in a one-dimensional infinite potential well is studied, and its efficiency is shown to be identical to th…
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 trans…
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$^{2}$ ($^{1}$…
In this paper we investigate some entanglement properties for the Hydrogen molecule considered as a two interacting spin 1/2 (qubit) model. The entanglement related to the $H_{2}$ molecule is evaluated both using the von Neumann entropy and the Concurrence and it is compared with the corresponding quantities for the two interacting spin system. Many aspects of these functions are examinated employing in part analytical and, essentially, numerical techniques. We have compared analogous results obtained by Huang and Kais a few years ago. In this respect, some possible controversial situations ar…
We introduce a systematic framework to calculate the bipartite entanglement entropy of a spatial subsystem in a one-dimensional quantum gas which can be mapped into a noninteracting fermion system. To show the wide range of applicability of the proposed formalism, we use it for the calculation of the entanglement in the eigenstates of periodic systems, in a gas confined by boundaries or external potentials, in junctions of quantum wires and in a time-dependent parabolic potential.
We consider the Aharonov-Bohm effect on entanglement entropy for one interval in (1+1) dimensional conformal field theory on a one dimensional ring. The magnetic field is confined inside the ring, i.e. there is a Wilson loop on the ring. The Aharonov-Bohm phase factor which is proportional to the Wilson loop is represented as insertion of twist operators. We compute exactly the Renyi entropy from a four point function of twist operators in a free charged scalar field.
