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arXiv:1011.3459

Quantum Potentials with q-Gaussian Ground States

Christophe Vignat, Angelo Plastino, Angelo Plastino, J. S. Dehesa

Published Nov 15, 2010• 2 citations
arXivHTMLPDF

Abstract

We determine families of spherically symmetrical $D$-dimensional quantum potential functions $V(r)$ having ground state wavefunctions that exhibit, either in configuration or in momentum space, the form of an isotropic $q$-Gaussian. These wavefunctions admit a maximum entropy description in terms of $S_q$ power-law entropies. We show that the potentials with a ground state of the $q$-Gaussian form in momentum space admit the Coulomb potential $-1/r$ as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large $r$for all values of the parameter $q$ such that $0

DOI
10.48550/arxiv.1011.3459

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We determine families of spherically symmetrical DDD-dimensional quantum potential functions V(r)V(r)V(r) having ground state wavefunctions that exhibit, either in configuration or in momentum space, the form of an isotropic qqq-Gaussian. These wavefunctions admit a maximum entropy description in terms of SqS_qSq​ power-law entropies. We show that the potentials with a ground state of the qqq-Gaussian form in momentum space admit the Coulomb potential −1/r-1/r−1/r as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large rrrfor all values of the parameter qqq such that $0
The spatial and momentum wavefunctions of a ground state quantum oscillator are Gausians in x and p respectively. A Gaussian is also the distribution which maximizes Shannon’s entropy - Sum over i P(i) ln(P(i)) for a fixed variance. Usually one would use average energy as a constraint, but for the nonrelativistic oscillator KE = pp/2m and PE=.5kxx so such a constraint is linked to the variance of x and p. In this note we consider the following questions? Why is the ground state of the quantum oscillator (i.e this form) the only quantum solution which matches a maximum entropy case (for a fixed variance)? Is there a link to a lack of information? Why do higher oscillator states represent non-maximum distributions for fixed variances? Can this also be linked to information? Here we apply the term information to actual information in the system not ln(P(i)).
The position- and momentum-space entropies of the isotropic harmonic oscillator and the hydrogen atom in D dimensions are shown to be related to some entropy integrals which involve classical orthogonal polynomials. These integrals are exactly calculated for Chebyshev polynomials and only in an approximate way for Gegenbauer polynomials. The physical entropies are explicitly obtained in the ground state and in a few low-lying excited states. Finally, the dimensionality dependence of the ground-state entropies of the two above-mentioned quantum-mechanical systems is analyzed (numerically) and the values of the entropies in a large class of excited states of the D-dimensional (D=1,2,3) harmonic oscillator and hydrogen atom are tabulated and discussed.
Energies of H, He-like (Z=2−18Z=2-18Z=2−18) ions, Li, and Be are investigated under a spherically symmetric Gaussian potential through a density functional formalism. The radial Kohn-Sham equation has been solved by invoking a work function-based exchange potential. The effect of electron correlation is analyzed by incorporating two functionals: a local parameterized Wigner functional and a non-linear gradient- and Laplacian-dependent Lee-Yang-Parr (LYP) functional. The generalized pseudospectral method is employed to provide accurate numerical eigenfunctions and eigenvalues. This allows nonuniform, optimal spatial discretization fulfilling the Dirichlet boundary conditions. This work demonstrates a possible manipulation of energy by controlling dot parameters. Apart from ground states, exploratory results are also reported for low-lying excited state 1s2s1s2s1s2s (1,3S^{1,3}S1,3S) of He atom. Companion calculations are also performed for various information-theoretic measures, such as Shannon entropy in position (SrS_{r}Sr​), momentum (SpS_{p}Sp​) spaces, and Fisher information in position space (IrI_{r}Ir​). The behavior of correlation functionals in presence of Gaussian potential is examined critically. We find that energy increases, SrS_{r}Sr​ exhibits minima, while SpS_{p}Sp​, IrI_{r}Ir​ attain maxima for a decrease in the width of potential, whereas an increase in potential depth further amplifies these effects across all properties. The Fisher-Shannon plane reveals a progressive localization as well as the compression of electronic density, and thereby indicates a weakening of relative electron-correlation effects. In the Collin's conjecture, it gives rise to a non-linear loop-like feature. Much of the results are presented here for the first time.
The radial expectation values of the probability density of a quantum system in position and momentum spaces allow one to describe numerous physical quantities of the system as well as to find generalized Heisenberg-like uncertainty relations and to bound entropic uncertainty measures. It is known that the position and momentum expectation values of the main prototype of the D-dimensional Coulomb systems, the D-dimensional hydrogenic system, can be expressed in terms of some generalized hypergeometric functions of the type p+1Fp(z) evaluated at unity with p = 2 and p = 3, respectively. In this work we determine the position and momentum expectation values in the limit of large D for all hydrogenic states from ground to very excited (Rydberg) ones in terms of the spatial dimensionality and the hyperquantum numbers of the state under consideration. This is done by means of two different approaches to calculate the leading term of the special functions 3F21 and 5F41 involved in the large D limit of the position and momentum quantities. Then, these quantities are used to obtain the generalized Heisenberg-like and logarithmic uncertainty relations, and some upper and lower bounds to the entropic uncertainty measures (Shannon, Rényi, Tsallis) of the D-dimensional hydrogenic system.
The Rényi and Shannon entropies are information‐theoretic measures which have enabled to formulate the position‐momentum uncertainty principle in a much more adequate and stringent way than the (variance‐based) Heisenberg‐like relation. Moreover, they are closely related to various energetic density‐functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment 〈r2〉 for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Rényi maximization procedure with a covariance constraint due to Costa, Hero and Vignat [1]. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given.
We calculate the single-particle momentum distribution of a quantum many-particle system in the presence of the Coulomb interaction and a confining potential. The region of intermediate momenta, where the confining potential dominates, marks a crossover from a Gaussian distribution valid at low momenta to a power-law behavior valid at high momenta. We show that for all momenta the momentum distribution can be parametrized by a qqq-Gaussian distribution whose parameters are specified by the confining potential. The real-space pair-correlation function calculated in this way can, in principle, be used to construct improved exchange-correlation functionals to solve electronic structure problems. Furthermore, we find that the functional form of the probability of transitions between the confined ground state and the nthn\mathrm{th}nth excited state is invariant under scaling of the ratio ${Q}^{2}/{\ensuremath{\nu}}_{n}$, where QQQ is the transferred momentum and ${\ensuremath{\nu}}_{n}$ is the corresponding excitation energy. Using the scaling variable ${Q}^{2}/{\ensuremath{\nu}}_{n}$ the maxima of the transition probabilities can also be expressed in terms of a qqq-Gaussian.