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Over the past few decades, confined quantum systems have emerged to be a subject of considerable importance in physical, chemical and biological sciences. Under such stressed conditions, they display many fascinating and notable physical and chemical properties. Here we address this situation by using two plasma models, namely a weakly coupled plasma environment mimicked by a Debye-Hückel potential (DHP) and an exponential cosine screened Coulomb potential (ECSCP). On the other hand, the endohedral confinement is achieved via a Woods-Saxon (WS) potential. The critical screening constant, dipole oscillator strength (OS) and polarizability are investigated for an arbitrary state. A Shannon entropy-based strategy has been invoked to study the phase transition here. An increase in Z leads to larger critical screening. Moreover, a detailed investigation reveals that there exists at least one bound state in such plasmas. Pilot calculations are conducted for some low-lying states ( ℓ = 1 − 5 ) using a generalized pseudo spectral scheme, providing optimal, non-uniform radial discretization. [ABSTRACT FROM AUTHOR]

Since the beginning of the twentieth century, confinement of atoms, molecules and plasma inside impenetrable sharp (hard) and smooth (soft) cavities has been studied with utmost interest. Physically, such situations introduce the trapping of a quantum system under high pressure. The internal pressure ( P n , ℓ ) of the system under multi-megabar external pressure has not yet been explored. Also, there is a lack of Hellmann–Feynman theorem (HFT) in such a scenario, which is essential to derive a concrete analytical expression of P n , ℓ under stressed condition. Here using a scaling concept we provide a general HFT in terms of expectation values of total energy, potential and kinetic energy of a given confined system. A change in boundary condition (from smooth to sharp and vice versa) does not affect this. Applying this proposed HFT, a closed-form expression of P n , ℓ and d 2 E d R 2 has been obtained for confined and shell-confined quantum systems. Based on this P n , ℓ , several virial-like equations are also modeled. This HFT and P n , ℓ are demonstrated for one- and many-electron atoms, plasmas and molecules using both analytical and numerical methods. Their applicability in several other confined systems is discussed from simple arguments. [ABSTRACT FROM AUTHOR]

Confinement of atoms inside various cavities has been studied for nearly eight decades. However, the Compton profile (CP) for such systems has not yet been investigated. Here we construct the CP for a H atom radially confined inside a hard spherical enclosure, as well as in a free condition. Some exact analytical relations for the CP's of circular or nodeless states of free atoms is presented. By means of a scaling idea, this has been further extended to the study of a H-like atom trapped inside an impenetrable cavity. The accuracy of these constructed CPs 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 ℓ, this rate diminishes with a rise in n. However, at a certain n, this rate accelerates with progress in ℓ. Similar analysis on the respective free counterpart displays an exactly opposite trend as that in a confined system. However, in both free and confined environments, the 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 atoms (and isoelectronic series) and investigated adopting an information-theoretic framework. [ABSTRACT FROM AUTHOR]

Abstract: Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective 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. Exact mathematical expressions of R r α , R p β, T r α , T p β , E r , E p are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as ( 3 5 , 3 ) in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas S r , R r α decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement. [ABSTRACT FROM AUTHOR]

Information-based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ( xc). In the symmetric case, a wide range of xc has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the 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 xc, all these properties converge to respective values of a free harmonic oscillator. In the 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. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. [ABSTRACT FROM AUTHOR]

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 phase-space 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 variation-induced exact diagonalization method. In brief, information measures and phase-space analysis can provide valuable insight toward the understanding of such potentials. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Quantum mechanical variation principle in the form of energy minimization is applicable only to ground states of systems, or, at best, states of lowest energies of given symmetries, provided the symmetry information is embedded in chosen trial functions. Thus, for bound quantum states with specified choices of trial functions involving nonlinear parameters, scope of the principle is severely restricted. A pedagogic way out is to enforce exact orthogonality of the chosen function with all exact lower energy states. In actual practice, this limits one to opt for linear variations where upper bound to each state is obtained in a single run. In this work, the motivation is to explore if there exists at all a way to determine optimized wave functions and energies for excited states via nonlinear variations but without any constraints, even for simple systems. Realizing that the major problem in excited-state nonlinear variations is concerned with the variations of nodal positions, at least for problems reducible to one dimension, we seek a route via which nodes could be fixed beforehand, so that the information gained may be subsequently utilized to construct a suitable nonlinear trial function and carry out a straightforward optimization. To achieve this end, the idea of supersymmetric quantum mechanics has been used quite profitably, yielding the nodal structure of the excited states. Workability of the strategy for several excited-state wave functions and their properties is demonstrated by choosing the problems of spherical Stark effect on hydrogen atom and anharmonic oscillator. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012 [ABSTRACT FROM AUTHOR]

Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrödinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low-lying states of some mixed oscillator and double-well potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator-level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011 [ABSTRACT FROM AUTHOR]

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 nr, 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 rc is presented, with particular attention on nonzero‐(l, m) states. Considerable new important observations are recorded. The results are quite accurate and most of these are presented for the first time. Fisher information (I) is studied in a harmonic oscillator enclosed in a spherical enclosure, in conjugate r and p spaces. A 3D confined harmonic oscillator (CHO) system may be considered as an intermediate between particle in a spherical box and free harmonic oscillator (FHO). The dependence of I on quantum numbers nr, l, m in FHO and CHO are analogous. [ABSTRACT FROM AUTHOR]