The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or shape complexity (i.e., the disequilibrium times the Shannon entropic power) of hydrogenic stationary states are investigated in both position and momentum spaces. First, it is shown that not only the Fisher information and the variance (then, the Cramer-Rao measure) but also the disequilibrium associated to the quantum-mechanical probability density can be explicitly expressed in terms of the three quantum numbers (n, l, m) of the corresponding state. Second, the three composite measures mentioned above are analytically, numerically and physically discussed for both ground and excited states. It is observed, in particular, that these configuration complexities do not depend on the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to quadratically depend on the principal quantum number n. Finally, sharp upper bounds to the Fisher-Shannon measure and the shape complexity of a general hydrogenic orbital are given in terms of the quantum numbers.
By using the plane-wave expansion for the electromagnetic-field vector potential, transition matrix elements between the relativistic bound and unbound states of hydrogenic atoms were expressed explicitly in terms of finite series made of hypergeometric functions. This representation for the above mentioned matrix elements proved very convenient for direct numerical calculation of their contributions to the Lamb shift in hydrogenic atoms.
The Cramér-Rao, Fisher-Shannon and LMC shape complexity measures have been recently shown to play a relevant role to study the internal disorder of finite many-body systems (e.g., atoms, molecules, nuclei). They highlight amongst the bunch of complexities appeared in the literature by their mathematical simplicity and transparent physical interpretation. They are composed by two spreading measures (variance, Fisher information and Shannon entropies) of the single-particle probability density of the systems. Here we discuss the physico-mathematical knowledge of these two-component complexities …
The internal disorder of a D-dimensional hydrogenic system, which is strongly associated to the non-uniformity of the quantum-mechanical density of its physical states, is investigated by means of the shape complexity in the two reciprocal spaces. This quantity, which is the product of the disequilibrium or averaging density and the Shannon entropic power, is mathematically expressed for both ground and excited stationary states in terms of certain entropic functionals of Laguerre and Gegenbauer (or ultraspherical) polynomials. We emphasize the ground and circular states, where the complexity …
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 ($D \to \infty$) limit but a similar phenomenon also happens for both the Fisher-information-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 measure…
The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy and the Fisher information. As well, the complexity measures of Crámer-Rao, Fisher-Shannon and LMC types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polyno…
The complete set of operators commuting with the Dirac Hamiltonian and exact analytic solution of the Dirac equation for the two-dimensional Coulomb potential is presented. Beyond the eigenvalue $μ$ of the operator $j_{z}$, two quantum numbers $η$ and $κ$ are introduced as eigenvalues of hermitian operators $P=βσ_{z}'$ and $K=β(σ_{z}'l_{z}+1/2)$, respectively. The classification of states according to the full set of constants of motion without referring to the non-relativistic limit is proposed. The linear Paschen-Back effect is analyzed using exact field-free wave-functions as a zero-order a…
A new method for constructing of composite coherent states of the hydrogen atom, based on the dynamical group approach and various schemes of reduction to subgroups, is presented. A wide class of well-localized (Gaussian) hydrogenic wave packets for circular and elliptic orbits is found using the saddle-point method.
This article presents the generalization of a zero spin hydrogen atom to a relativistic atomic model of hydrogen with dyons using the Klein--Gordon equation. The derivation of the Klein--Gordon equation for the particle of relative motion is shown. In addition, the analytical solutions of the equation are calculated in terms of Whittaker functions and Jacobi weighted polynomials. The discrete spectrum of energy, and the charge density of the orbiting dyon are presented. For a system of positive magnetic and electric charges in the nucleus and negative charges for the orbiting particle, and con…
We derive the expansion of the irregular physical solution over the spherical solutions at negative energies, which is necessary for obtaining the $S$ matrix of the process. The relation of this expansion to the theory developed by Giannakees et al., Phys. Rev. A 94, 013419 (2016), is analyzed. In particular, we show that the expansion of the irregular solution missing in Giannakees et al.'s theory can be derived from one of their main postulates. The expansion thus obtained turns out to be numerically equivalent to our expansion up to high angular momenta. Analytical expressions for the key m…
The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system, but also the entropic uncertainty measures of Rényi type which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg's uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in $1/D$ in similar systems with a non-standard dimensiona…
The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction-towards-the-origin of the electronic charge in hydrogenic systems and the nodal disapearance (because of the raising of all the non-relativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far the nucleus. In this work we quantify these effects by means…
Technical applications using electrostatic field ionization (ESFI) need a formula for the rate-constant K_e for the free-space ESFI of a hydrogenic atom in its ground electronic state. This formula must indicate the dependence on ionization energy I. Existing formulae were derived using atomic units. However, many scientists work with ESFI as an important technical process, but are not familiar with the Gaussian or atomic units systems. In the 1970s, the present International System of Quantities (ISQ), which includes the equations behind SI units, was designated as the main system for univers…
Electrons bound to shallow donors in GaAs have orbital energy levels analogous to those of the hydrogen atom. The polarization selection rules for optical transitions between the states analogous to the 1s and 2p states of hydrogen in a magnetic field are verified using Terahertz (THz) radiation from the UCSB Free Electron Laser. A polarization-selective coherent manipulation of the quantum states is demonstrated and the relevance to quantum information processing schemes is discussed.
Toroidal Transitions in Hydrogenic and Alkali Atoms
Centre for Quantum Technologies, Nanyang Technological University（南洋理工大学量子技术中心）；Centre for Disruptive Photonic Technologies, SPMS, The Photonics Institute, Nanyang Technological University（南洋理工大学光波研究中心、理学院光子学研究所）；MajuLab, International Joint Research Unit IRL 3654, CNRS, Université Côte d’Azur, Sorbonne Université, National University of Singapore, Nanyang Technological University（马久实验室，国际联合研究单位IRL 3654，法国国家科学研究中心，蔚蓝海岸大学，索邦大学，新加坡国立大学，南洋理工大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
We describe the calculation of hydrogenic (one-loop) Bethe logarithms for all states with principal quantum numbers n <= 200. While, in principle, the calculation of the Bethe logarithm is a rather easy computational problem involving only the nonrelativistic (Schroedinger) theory of the hydrogen atom, certain calculational difficulties affect highly excited states, and in particular states for which the principal quantum number is much larger than the orbital angular momentum quantum number. Two evaluation methods are contrasted. One of these is based on the calculation of the principal value…
We consider a fractional generalization of two-dimensional (2D) quantum-mechanical Kepler problem corresponding to 2D hydrogen atom. Our main finding is that the solution for discreet spectrum exists only for $μ>1$ (more specifically $1 < μ\leq 2$, where $μ=2$ corresponds to "ordinary" 2D hydrogenic problem), where $μ$ is the Lévy index. We show also that in fractional 2D hydrogen atom, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number $n$ but also on orbital $m$. To solve the spectral problem, we pass to the momentum representat…
Supported by results obtained with semi-classical quantization techniques, and with a quantum mechanical "square-root Klein-Gordon" operator, it is argued that Positronium (Ps) may exhibit a proper quantum-mechanical ground state whose energy level lies $\approx 2m c^2$ below its "hydrogenic (pseudo-) ground state" energy, where $m$ is the empirical rest mass of the electron. While the familiar hydrogenic pseudo-ground state of Ps is caused by the Coulomb attraction of electron and anti-electron, modified by small spin-spin and radiative QED corrections, the proper ground state is caused by th…
We realize a non-invasive electrometer based on state engineering in a Rydberg hydrogenic manifold. A quantum interference process involving states with very different dipoles measures directly the time correlation of a stochastic electric field, with a 200~ns time resolution and with a $83.2\pm1.4$~mV/m single-atom sensitivity, beyond the Standard Quantum Limit in this context. This quantum-enabled correlation electrometer opens the way to applications in mesoscopic physics, e.g. for the study of individual charge transits in nano-structures.
The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p and 3d quantum states by means of the main entropy and complexity information-theoretic measures. First, the Shannon entropy and the Fisher information as well as the associated uncertainty relations are computed and discussed. Then, the Fisher-Shannon, LMC and LMC-Rényi complexity measures are examined and mutually compar…
Shell confined atom can serve as a generalized model to explain both \emph{free} and \emph{confined} condition. In this scenario, an atom is trapped inside two concentric spheres of inner $(R_{a})$ and outer $(R_{b})$ radius. The choice of $R_{a}, R_{b}$ renders four different quantum mechanical systems. In hydrogenic atom, they are termed as (a) free hydrogen atom (FHA) (b) confined hydrogen atom (CHA) (c) shell-confined hydrogen atom (SCHA) (d) left-confined hydrogen atom (LCHA). By placing $R_{a}, R_{b}$ at the location of radial nodes of respective \emph{free} $n,\ell$ states, a new kind o…
The internal disorder of the two-dimensional confined hydrogenic atom is numerically studied in terms of the confinement radius for the 1\textit{s}, 2\textit{s}, 2\textit{p} and 3\textit{d} quantum states by means of the statistical Crámer-Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer-Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the el…
The complex processes leading to the collisional population of ultra-long-lived Rydberg states with very high angular momentum can be explained surprisingly well using classical mechanics. In this article, we explain the reason behind this striking agreement between classical theory and experiment by showing that the classical and quantum dynamics of Rydberg electrons in weak, slowly varying external fields agree beyond the mandates of Ehrenfest's Theorem. In particular, we show that the expectation values of angular momentum and Runge-Lenz vectors in hydrogenic eigenstates obey exactly the sa…
