CONFINED HARMONIC OSCILLATOR.The aim of this study was a didactic presentation of the Shannon entropy in the quantum theory context, followed by application to the case of a one-dimensional harmonic oscillator in its ground state, both in the free case and confined case.The study of these systems allows us to highlight notions such as location or delocation of a particle, a possible interpretation that the Shannon entropy can adopt.The Shannon entropy in position (S x ) and momentum (S p ) spaces were calculated for both systems, beyond the entropic sum (S t = S x + S p ).With this procedure it was possible to identify trends in the behavior of the Shannon entropy (S x and S p ) and test the compliance of the entropic uncertainty relation (S t = S x + S p ≥ n(1 + ln(p))).
CONFINED HARMONIC OSCILLATOR.The aim of this study was a didactic presentation of the Shannon entropy in the quantum theory context, followed by application to the case of a one-dimensional harmonic oscillator in its ground state, both in the free case and confined case.The study of these systems allows us to highlight notions such as location or delocation of a particle, a possible interpretation that the Shannon entropy can adopt.The Shannon entropy in position (S x ) and momentum (S p ) spaces were calculated for both systems, beyond the entropic sum (S t = S x + S p ).With this procedure it was possible to identify trends in the behavior of the Shannon entropy (S x and S p ) and test the compliance of the entropic uncertainty relation (S t = S x + S p ≥ n(1 + ln(p))).
Quantum uncertainties in position, momentum and phase‐space are studied in the confined Harmonic Oscillator. Standard deviations and Shannon entropies are used to quantify these uncertainties and their behaviors are compared and contrasted. We observe a minimum in the momentum space Shannon entropy as the box length is increased, a feature that is not present in the momentum space standard deviation. The behaviors of the standard deviation product and the Shannon entropy sum, which form the basis of uncertainty relationships, are also analyzed. Maxima are observed in the product as the box length increases in sharp contrast to the entropy sum. The relationship between these behaviors and that of the Shannon entropy of the phase‐space Wigner function is analyzed and discussed. An analysis of the energetic components is also performed. The results reinforce the idea that the confined Harmonic Oscillator can be considered as an intermediate model which interpolates between the Particle in a Box Model and the Harmonic Oscillator and thus contains characteristics of both models. image
The time-dependent behaviour of harmonic oscillator squeezed states is examined from the viewpoint of position and momentum space Shannon entropies and their sum. Analytical expressions for these entropies are obtained from a symplectic group approach. The entropy sum returns to the entropic uncertainty bound at different time intervals, and is characterized by the presence of either a maximum (minimum) in the position space Shannon entropy and a minimum (maximum) in the momentum space one. The maximal value of the entropy sum is characterized by equality of position and momentum Shannon entropies. The transition of the entropy sum from the bound to its maximal value is studied, and relations between position and momentum space Shannon entropies are probed. The relation between the position and momentum space entropies is deformed as the value of the squeezing parameter increases. An analysis of the dependence of the Shannon entropies on the squeezing parameter shows different behaviours at different times. Both position and momentum space Shannon entropies exhibit a linear dependence on the parameter when their sum lies on the uncertainty bound. Here, the entropy sum is independent of the parameter. At other times, the relation between the entropies and squeezing parameter is non-linear, and extremal points are present. The entropy sum here is seen to be an increasing function of the parameter. Kullback–Leibler distances between squeezed and unsqueezed densities illustrate that the position or momentum space densities of the squeezed system can revert back to the unsqueezed ones during the time evolution. The phase-space Wigner function, entropy and mutual information are also examined and discussed. The statistical correlation as measured by mutual information is the entropy sum minus its value at the uncertainty bound.
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.
In this note, we examine the behaviour of Shannon’s entropy density for some examples in quantum mechanics. We start with a quasiparticle probability P(x and p) = W(x) fp exp(ipx) (where W(x) is the wavefunction given by Sum over p fp exp(ipx)) and obtain a Shannon’s entropy (after summing on p) : Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) - W(x) x d/dx W(x) ((1)) It may be noted that if f(p)=f(-p) ((1)) may also be written as: Total entropy density = -P(x) ln(P(x)) - P(p) ln(P(p)) - fp p d/dp fp We consider the cases of a particle in a box with an infinite potential at the walls and the harmonic oscillator ground state. Finally we consider the influence of u(x)= [d/dx W(x)]/W(x) on the spatial change of Shannon’s spatial entropy as W(x) and u(x) are often periodic and this leads to a periodic entropy density. In general, entropy is obtained by integrating the entropy density and one may question whether one should examine the density at all. In (1), however, entropy density is considered and it seems that there are some interesting properties of Shannon’s entropy density in quantum systems. Furthermore, there has been some recent work in the literature which makes use of entropy density to create an extra potential term for the Schrodinger equation in cases of temperature.
Abstract We pedagogically present the information theory as originally established, explaining its essential ideas and paying attention to the expression employed to measure the amount of information. We also discuss relationships between information and quantum theories. In this context we determine the information entropies on the position, S x , and momentum, S p , spaces, in addition to the entropy sum, S t . Here, we use the modified entropic expressions that are dimensionally consistent. We provide original explanations for the behaviors of the S x , S p and S t values analyzing the probability densities and by means of the normalization constants and properties of Fourier transform. We validate the entropic uncertainty relation and inspect the standard deviation and information entropies as measures of quantum uncertainty. The systems of interest are unidimensional one-particle quantum systems in the ground and excited states.
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.