Abstract
The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state’s angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies–Thakkar, Lieb–Thirring, Redheffer–Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.
Highlights
The central field approximation has been successfully applied to investigate the natural systems in the three-dimensional world, and in multidimensional physics
Multidimensional central potentials with a specific analytical form have been used to interpret a great deal of physical phenomena and chemical processes [7,8,9,10,11,12,13,14,15]. They have been applied to study the behaviour of nanotechnological systems and to explain the experiments of dilute systems in magnetic traps at extremely low temperatures [16,17,18], which has allowed for a fast development of a densityfunctional theory of independent particles in multidimensional central potentials [11,19]
We investigate the combined balance of the space and spin dimensionality effects on the Heisenberg-like uncertainty relation of general fermionic systems, and the space, spin and spherical effects on the Fisher-information-based uncertainty relation of fermionic systems with arbitrary central potentials. This is done by use of integral inequalities of Daubechies–Thakkar and Lieb– Thirring type and variational methods based on the extremization of the entropy-like measures
Summary
The central field approximation has been successfully applied to investigate the natural systems in the three-dimensional world, and in multidimensional physics. They quantify the separation of the region(s) of the probability concentration with respect to a specific point of the system’s domain (usually, the origin or the mean value); so, they are misleading and not adequate quantifiers for the quantum uncertainty of numerous physical systems with heavy-tailed and oscillating, multimodal densities [30,31,32,33] To avoid such drawbacks and since the quantum probability densities of the potential’s bound states are strongly oscillating, except for a few ones such as e.g., the S states, the entropy-like measures are much more natural and appropriate to quantify the position-momentum uncertainty of the system. This is done by use of integral inequalities of Daubechies–Thakkar and Lieb– Thirring type and variational methods based on the extremization of the entropy-like measures.
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