One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory

• URL: http://arxiv.org/abs/2304.06428v2
• Date: Thu, 27 Apr 2023 08:18:18 GMT
• Title: One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory
• Authors: O. Olendski
• Abstract summary: Motion of a potential that is a combination of positive quadratic and inverse quadratic functions of the position is considered. The dependence on the particle energy and the factor $mathfraka$ describing a relative strength of its constituents is described.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Motion along semi-infinite straight line in a potential that is a combination of positive quadratic and inverse quadratic functions of the position is considered with the emphasis on the analysis of its quantum-information properties. Classical measure of symmetry of the potential is proposed and its dependence on the particle energy and the factor $\mathfrak{a}$ describing a relative strength of its constituents is described; in particular, it is shown that a variation of the parameter $\mathfrak{a}$ alters the shape from the half-harmonic oscillator (HHO) at $\mathfrak{a}=0$ to the perfectly symmetric one of the double frequency oscillator (DFO) in the limit of huge $\mathfrak{a}$. Quantum consideration focuses on the analysis of information-theoretical measures, such as standard deviations, Shannon, R\'{e}nyi and Tsallis entropies together with Fisher information, Onicescu energy and non--Gaussianity. For doing this, among others, a method of calculating momentum waveforms is proposed that results in their analytic expressions in form of the confluent hypergeometric functions. Increasing parameter $\mathfrak{a}$ modifies the measures in such a way that they gradually transform into those corresponding to the DFO what, in particular, means that the lowest orbital saturates Heisenberg, Shannon, R\'{e}nyi and Tsallis uncertainty relations with the corresponding position and momentum non--Gaussianities turning to zero. A simple expression is derived of the orbital-independent lower threshold of the semi-infinite range of the dimensionless R\'{e}nyi/Tsallis coefficient where momentum components of these one-parameter entropies exist which shows that it varies between $1/4$ at HHO and zero when $\mathfrak{a}$ tends to infinity. Physical interpretation of obtained mathematical results is provided.

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