Information analysis in free and confined harmonic oscillator

Abstract

In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator (CHO). 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 (I), Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T ), Onicescu energy (E) 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.The one-dimensional confined harmonic oscillator (1DCHO), defined by v), can be classified into two forms, (a) symmetrically confined harmonic oscillator (SCHO) (when d m = 0), (b) asymmetrically confined harmonic oscillator (ACHO) (corresponding to d m = 0). Further, in latter case, confinement can be accomplished two different ways: (i) by changing the box boundary, keeping box length and d m fixed at zero; (ii) another route is to adjust d m by keeping box length and boundary fixed. SCHO can be treated as an intermediate between a particle-ina-box (PIB) and a 1DQHO. Though the Schrödinger equation for SCHO can be solved exactly, for ACHO it cannot be. We have employed two different methods for the latter, viz., (i) an imaginary time propagation (ITP), leading to minimization of an expectation value (ii) a variation-induced exact diagonalization procedure that utilizes SCHO eigenfunctions as basis. It is found that, at very low x c region, I, S, R, T, E remain invariant with change confinement length, x c . At moderate x c region S x , R x , T x progress and I x , E x decrease with rise in x c . Additionally, special attention has been paid to study relative information in SCHO and ACHO.Analogously, a 3DCHO (radically confined within an impenetrableFree and confined harmonic oscillator • • • spherical well) can act as a bridge between particle in a spherical box (PISB) and isotropic 3DQHO. The time-independent Schrödinger equation for D-dimensional harmonic oscillator in both free and confined condition can be solved exactly, within a Dirichlet boundary condition. Here a detailed exploration of information measures has been carried out in r and p spaces, for a 3DCHO. Some ex...