Abstract 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 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.
Abstract 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 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.
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.
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
We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius rc with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.
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.
Abstract In this work, we study an electron subjected to a harmonic oscillator potential confined in a circle of radius r0 r 0 and in the presence of a constant electric field. We obtain energies and eigenfunctions for three different confinement radii as a function of the electric field strength. We have used the linear variational method by constructing the trial function as a linear combination of two-dimensional confined harmonic oscillator wave functions. We calculate the radial standard deviation as a measure of the dispersion of the probability density. We also computed the Shannon entropy and Fisher information, in configuration and momentum spaces, as localization-delocalization measures for three different confinement radii and as a function of the electric field strength. We find that Shannon entropy and Fisher information are more reliable than variance in determining electron location. The behaviour of Shannon entropy and Fisher information curves is shown to depend on each specific state under study. Graphical abstract