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 approximation.
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 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 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 …
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…
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…
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…
Time evolution of wave packets built from the eigenstates of the Dirac equation for a hydrogenic system is considered. We investigate the space and spin motion of wave packets which, in the non-relativistic limit, are stationary states with a probability density distributed uniformly along the classical, elliptical orbit (elliptic WP). We show that the precession of such a WP, due to relativistic corrections to the energy eigenvalues, is strongly correlated with the spin motion. We show also that the motion is universal for all hydrogenic systems with an arbitrary value of the atomic number Z.
We discuss the Klein-Gordon (KG) equation using a 5D space-time approach. We explicitly show that the KG equation in flat space-time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schrödinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space-time picture. Finally, we apply our results to the problem of hydrogenic spectra and calc…
It is shown that, in the relativistic hydrogenic approximation, expectation values of the type $\langle n\ell j|r^k|n\ell j\rangle$ can be expressed in terms of Clebsch-Gordan coefficients or $3jm$ symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1-5 (2015); Opt. Spektrosk. 118, 3-7] in the non-relativistic hydrogenic case.
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 analytical…
The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values $\langle r^α\rangle$ and $\left\langle p^α\right\rangle$, respectively. These quantities, which describe and/or are closely related to various fundamental properties of realistic systems, have not been calculated in an analytical and effective manner up until now except for a number of three-dimensional hydrogenic states. In this work we explicitly show these expectation values for all discrete stationary $D$-dimensional hydrogenic states in ter…
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）
The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schrödinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided …
Stark effect of hydrogenic ions as a test of the indefinite-metric formalism of the eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles
Center for Theoretical Physics, Faculty of Science, Parahyangan Catholic University（帕拉希扬天主教大学理学院理论物理中心）
该论文通过计算氢离子 Stark 位移验证了八分量相对论波动方程的不定度规形式体系，证明其与 Dirac 结果一致，并揭示了自旋-场耦合和负范数态的必要性。
Different operators have been suggested in the literature to describe the electron's spin degree of freedom within the relativistic Dirac theory. We compare concrete predictions of the various proposed relativistic spin operators in different physical situations. In particular, we investigate the so-called Pauli, Foldy-Wouthuysen, Czachor, Frenkel, Chakrabarti, Pryce, and Fradkin-Good spin operators. We demonstrate that when a quantum system interacts with electromagnetic potentials the various spin operators predict different expectation values. This is explicitly illustrated for the scatteri…
We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schrödinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in…
The Dirac equation for the Coulomb problem is restated by incorporating a nonlinear effective interaction into the Dirac Hamiltonian: one keeps the $1/r$ dependence for the Coulomb field, but the coupling constant is modified by a factor depending on the n (principal quantum number) power of the mean value of the Hamiltonian. In this simple context we study the Lamb shift and the hyperfine splitting of the s-levels of hydrogenic atoms. We discuss to what extent the corresponding calculations fit the energy splittings to the appropriate order in the fine structure constant.
We show that the relativistic hydrogen atom possesses an SO(4) symmetry by introducing a kind of pseudo-spin vector operator. The same SO(4) symmetry is still preserved in the relativistic quantum system in presence of an U(1) monopolar vector potential as well as a nonabelian vector potential. Lamb shift and SO(4) symmetry breaking are also discussed.
The hydrogen atom as relativistic bound-state system of a proton and an electron in the complex-mass scheme is investigated. Interaction of a proton and an electron in the atom is described by the Lorentz-scalar Coulomb potential; the proton structure is taken into account. The concept of position dependent particle mass is developed. Relativistic wave equation for two interacting spinless particles is derived; asymptotic method is used to solve the equation. % Asymptotic solution of the equation for the system in the form of %standing wave and eigenmasses of the $H$ atom are obtained. Complex…
As a serious attempt for constructing a new foundation for describing micro-entities from a causal standpoint, it was explained before in [1, 2, 3] that by unifying the concepts of information, matter and energy, each micro-entity is assumed to be composed of a probability field joined to a particle called a particle-field or PF system. The relativistic generalization of this theory and its invariance under Lorentz transformation has been proved. In this essay, based on the relativistic generalization of Schrodinger equation derived in [4], we solve the relativistic Schrodinger equation for re…
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…
The dispersion relation of ultracold atoms in variably shaped optical lattices can be tuned to resemble that of a relativistic particle, i.e. be linear instead of the usual nonrelativistic quadratic dispersion relation of a free atom. Cold atoms in such a lattice can be used to carry out quantum simulations of relativistic wave equation predictions. We begin this article by describing a Raman technique that allows to selectively load atoms into a desired Bloch band of the lattice near a band crossing. Subsequently, we review two recent experiments with quasirelativistic rubidium atoms in a bic…
