Quantum uncertainties of the confined Harmonic Oscillator in position, momentum and phase‐space

Abstract

Quantum uncertainties in position, momentum and phasespace 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.

Shannon entropy: A study of confined hydrogenic-like atoms

“…For example, it was shown that in such regime the energies of confined hydrogenic-like atoms tends to the values of a particle confined in a spherical cage [32]. Recent studies for the harmonic potential showed that such analysis can be made based on Shannon entropy [33,34].The entropy sum (S t ), quantity defined as the sum of the Shannon entropies in the position and momentum spaces, has occupied a privileged place in the study of quantum systems in the information theory context. For instance, an entropic uncertainty relation that it has been treated as a stronger version of the Heisenberg uncertainty relation can be derived from the entropy sum [35].…”

Information and quantum theories: an analysis in one-dimensional systems

“…The values of S p increase with the confinement increment and affect more the states in an increasing order of energy. The behaviors of S x and S p are the same as found in reference [30].…”

“…The entropic uncertainty relation is considered as a stronger version of the Kennard's relation, in the sense that from relation (5) we can deduce the relation (6) [37]. Still, uncertainty relations in terms of S x and S p are proposed to deal with situations where the Heisenberg's uncertainty principle or Kennard's relation presents sensitivities [27,28], as for instance, in the study of separable phase-space distribution [30,49].…”

Shape effect on information theoretic measures of quantum heterostructures

“…We have used GaAs Ga Al As V o is taken to be 300 meV. Shannon entropy, Fisher information and their respective lengths, standard deviation and Onicescu energy are calculated using equations (1)- (12). In order to understand the shape effect, we have shown the energy spectrum of MQWS system for N=2, 5 and 10 in table 1.…”

“…The study of quantum systems from an information point of view is an important aspect and it has many applications in the field of quantum computation [8] and information technology [9]. Shannon entropy [10][11][12][13][14] and Fisher information [15][16][17] are the two major information entropies as they are the best estimator for uncertainty measurements. Standard deviation in position and momentum defines spreading of the wave function with respect to a particular point in the domain i.e.…”