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.
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.
This book gathers state-of-the-art advances on harmonic oscillators including their types, functions, and applications. In Chapter 1, Neetik and Amlan have discussed 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. \n \nThe 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 physic-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. \n \nIn Chapter 2, Nabakumar, Subhasree, and Paulami have revisited classical-quantum correspondence in the context of linear Simple Harmonic Oscillator (SHO). According to Bohr’s correspondence principle, quantum mechanically calculated results match with the classically expected results when quantum number is very high. Classical quantum correspondence may also be visualized in the limit when the action integral is much greater than Planck’s constant. When de-Broglie wave length associated with a particle is much larger than system size, then quantum mechanical results also match with the classical results. In the context of dynamics, Ehrenfest equation of motion is used in quantum domain, which is analogous to classical Newton’s equation of motion. SHO is one of the most important systems for several reasons. It is one of the few exactly solvable problems. Any stable molecular potential can be approximated by SHO near the equilibrium point. This builds the foundation for the understanding of complex modes of vibration in large molecules, the motion of atoms in a solid lattice, the theory of heat capacity, vibration motion of nuclei in molecule etc. The authors have revisited the common solution techniques and important properties of both classical and quantum linear SHO. Then they focused on probability distribution, quantum mechanical tunneling, classical and quantum dynamics of position, momentum and their actuations, viral theorems, etc. and also analyzed how quantum mechanical results finally tend to classical results in the high quantum number limit. \n \nIn Chapter 3, Neeraj has discussed the nature of atomic motions, sometimes referred to as lattice vibrations. The lattice dynamics deals with the vibrations of the atoms inside the crystals. In order to write the dynamic equations of the motion of crystal atoms, we need to describe an inter-atomic interaction. Therefore, it is natural to start the study of the lattice dynamics with the case of small harmonic vibrations. The dynamics of one-dimensional and two-dimensional vibrations of monatomic and diatomic crystals can be understood by using the simple model forces based on harmonic approximation. This harmonic approximation is related to a simple ball-spring model. According to this model, each atom is coupled with the neighboring atoms by spring constants. The collective motion of atoms leads to a distinct traveling wave over the whole crystal, leading to the collective motion, so-called phonon. The simple ball-spring model enlightens us some of the significant common features of lattice dynamics that have been discussed throughout this chapter. Further, this chapter helps in understanding the quantization energy of a harmonic oscillation and the concept of phonon.
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 ( x c ). In the symmetric case, a wide range of x c 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 x c , 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. image
With the development of quantum technology and information science, significant attention is focused on studying the quantum information properties in confined systems. In this work, we present a systematic investigation of quantum information measures – including position-momentum uncertainty (Δx,Δp), Shannon entropy (Sx, Sp), and Onicescu energy (Ox, Op) in a confined Robin harmonic oscillator (CRHO) with symmetric boundary conditions λL = λR = λ. It is interesting to find that the Heisenberg uncertainty relation ΔxΔp≥ℏ/2, which holds under the Dirichlet boundary condition (λ = 0), is violated for the ground state of the CRHO. In contrast, the entropic uncertainty relation (Sx + Sp ≥ 1 + lnπ) remains satisfied at any Robin length λ and quantum state, demonstrating its fundamental advantage and broader applicability in quantum systems. For a fixed Robin length, we observe that as the quantum well width increases within a finite range, the position-space Shannon entropy Sₓ increases, while the Onicescu energy Oₓ decreases. Despite their opposing tendencies, both Shannon entropy and Onicescu energy reflect the same physical insight regarding the localisation-delocalisation property for the CRHO system. In addition, the dependence of these information measures on the Robin length is discussed, and their physical implications are highlighted. This study demonstrates that boundary conditions play a critical role in quantum information metrics, thereby informing the design and optimisation of quantum devices such as quantum dots, nanoscale resonators and atom-based sensors.
Abstract Information‐based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) are employed to understand the influence of radial confinement in an isotropic harmonic oscillator (IHO). The transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (). For a given quantum state, accurate results are provided by applying the respective exact analytical wave function in the r‐space. The p‐space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments () as in r‐ and p‐spaces. A detailed, systematic analysis is performed for a 3D confined harmonic oscillator (CHO) with respect to state indices , and . It has been found that the CHO acts as a bridge between particles in a spherical box and the 3D free IHO. In the low‐ region, increases while decrease with the rise in . At moderate , there exists an interaction between two competing factors: i) radial confinement (localization) and ii) accumulation of radial nodes with growth of (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
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 , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as 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 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.