In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D-dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical (D → ∞) limit but a similar phenomenon also happens for both the Fisherinformation-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer-Rao and Fisher-Shannon complexities. Moreover, we show that the LMC (López-Ruiz-Mancini-Calvet) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.
The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work, the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers (n, μ 1 , μ 2 , …, μ D−1 ), the system's dimensionality D and the nuclear charge Z in an analytical, compact form.Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform to a relevant class of specific states of the D-dimensional hydrogenic system characterized by the hyperquantum numbers μ 1 = μ 2 … = μ D −1 = n − 1, including the ground state. K E Y W O R D S dimensional dependence of Shannon entropy, entropic uncertainty, Shannon entropy of multidimensional hydrogenic states 1 | INTRODUCTIONThe entropic uncertainty measures have been shown to be much more adequate than the Heisenberg ones (ie, those based on the SD and its moment-type generalizations [1] ) not only to formulate the fundamental uncertainty principle of quantum mechanics [2][3][4][5] and for the development of quantum information and computation, [6][7][8][9][10][11] but also to describe the physical and chemical properties of multidimensional quantum systems. The latter is specially so in quantum chemistry, where D-dimensional notions and techniques play a very relevant role as the chemists Dudley R. Herschbach (Nobel Prize 1986) [12] and Gerhard Ertl (Nobel Prize 2007) and their collaborators have shown. For example, in chemistry, two dimensions are often better than three, since surface-bound reactions can be proved in greater detail than those in a liquid solution, as Ertl et al. illustrated in their pioneering contributions to the field of surface chemistry. [13][14][15] Most efforts have been focused on the entropic information entropies of Rényi and Shannon types and its variations, [16][17][18][19][20] basically because they have underpinned fundamental progress in a great diversity of scientific and technological fields such as applied mathematics, theoretical physics, quaand information technologies, including electronic correlations, complexity theory, statistical mechanics, quantum information theory, telecommunications, biochemical phenomena, and quantum information processing. [6,8,[21][22][23][24][25][26] The analytical determination of these entropic quantities is a serious task, [10] basically because the quantum probability density of these systems is not generally known, except for some extreme cases and/or some special states (high-dimensional and...
The recently proposed density functional theory for steady-state transport (i-DFT) is extended to include temperature gradients between the leads. Within this framework, a general and exact expression is derived for the linear Seebeck coefficient which can be written as the sum of the Kohn-Sham coefficient and an exchange-correlation contribution. The formalism is applied to the singleimpurity Anderson model for which approximate exchange-correlation functionals are suggested for temperatures both above and below the Kondo temperature. A certain structural property of the exchange-correlation potentials in the Coulomb blockade regime allows to recover an earlier result expressing the Seebeck coefficient in terms of quantities of equilibrium density functional theory. The numerical i-DFT results are compared to calculations with the numerical renormalization group over a wide range of temperatures finding a reasonable agreement while i-DFT comes at a much lower computational cost.
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