Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems

Abstract

The $D$-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large $D$. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all the $D$-dimensional harmonic oscillator states in the pseudoclassical ($D\to \infty$) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.

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Cited by in corpus (9)

• Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems
• Exact Shannon entropies for the multidimensional harmonic states
• Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
• Exact Rényi entropies of $D$-dimensional harmonic systems
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• Shannon information entropy for a quantum nonlinear oscillator on a space of non-constant curvature
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots
• Linearization and Krein-like functionals of hypergeometric orthogonal polynomials
• Complexity-like properties and parameter asymptotics of ${\mathfrak{L}}_q$-norms of Laguerre and Gegenbauer polynomials