In this paper it is shown that the Lyman-$α$ transition of a single hydrogen-like system driven by a laser exhibits macroscopic dark periods, provided there exists an additional constant electric field. We describe the photon-counting process under the condition that the polarization of the laser coincides with the direction of the constant electric field. The theoretical results are given for the example of $^4{He}^+$. We show that the emission behavior depends sensitively on the Lamb shift (W.E. Lamb, R.C. Retherford, Phys. Rev. 72, 241 (1947)) between the $2s_{1/2}$ and $2p_{1/2}$ energy levels. A possibly realizable measurement of the mean duration of the dark periods should give quantitative information about the above energy difference by using the proposed photon-counting process.
Modern quantum theory introduces quantum structures (decompositions into subsystems) as a new discourse that is not fully comparable with the classical-physics counterpart. To this end, so-called Entanglement Relativity appears as a corollary of the universally valid quantum mechanics that can provide for a deeper and more elaborate description of the composite quantum systems. In this paper we employ this new concept to describe the hydrogen atom. We offer a consistent picture of the hydrogen atom as an open quantum system that naturally answers the following important questions: (a) how do t…
Non-Markovian evolution of an open quantum system can be `unraveled' into pure state trajectories generated by a non-Markovian stochastic (diffusive) Schrödinger equation, as introduced by Diósi, Gisin, and Strunz. Recently we have shown that such equations can be derived using the modal (hidden variable) interpretation of quantum mechanics. In this paper we generalize this theory to treat jump-like unravelings. To illustrate the jump-like behavior we consider a simple system: A classically driven (at Rabi frequency $Ω$) two-level atom coupled linearly to a three mode optical bath, with a cent…
The method and status of a study to provide numerical, high-precision values of the self-energy level shift in hydrogen and hydrogen-like ions is described. Graphs of the self energy in hydrogen-like ions with nuclear charge number between 20 and 110 are given for a large number of states. The self-energy is the largest contribution of Quantum Electrodynamics (QED) to the energy levels of these atomic systems. These results greatly expand the number of levels for which the self energy is known with a controlled and high precision. Applications include the adjustment of the Rydberg constant and…
Using the modified factorization method introduced by Mielnik, we construct a new class of radial potentials whose spectrum for l=0 coincides exactly with that of the hydrogen atom. A limiting case of our family coincides with the potentials previously derived by Abraham and Moses
It is shown that the radial part of the Hydrogen Hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant epsilon. The 1-susy approach is used to construct non-hermitian Hamiltonians with hydrogen spectra. Other non-hermitian Hamiltonians are shown to admit an extra `complex energy' at epsilon. New self-adjoint hydrogen-like Hamiltonians are also derived by using a 2-susy transformation with complex conjugate pairs epsilon, (c.c) epsilon.
A set of factorization energies is introduced, giving rise to a generalization of the Schrödinger (or Infeld and Hull) factorization for the radial hydrogen-like Hamiltonian. An algebraic intertwining technique involving such factorization energies leads to derive $n$-parametric families of potentials in general almost-isospectral to the hydrogen-like radial Hamiltonians. The construction of SUSY partner Hamiltonians with ground state energies greater than the corresponding ground state energy of the initial Hamiltonian is also explicitly performed.
The well known hypervirial perturbation method (HPM)\ based on hypervirial relations and the Hellmann-Feynman theorem is suitable for the calculation of perturbation corrections of large order for the two-dimensional hydrogen-like atom in a uniform magnetic field. We show analytical results in terms of the quantum numbers and large order corrections for particular states. This approach appears to be simpler and more efficient than the recently proposed one based on Green functions.
The relativistic theory of above-threshold ionization (ATI) of hydrogen-like atoms in ultrastrong radiation fields, taking into account the photoelectron induced rescattering in the continuum spectrum is developed. It is shown that the contribution of the latter in the multiphoton ionization probability even in the Born approximation by Coulomb field is of the order of ATI probability in the scope of Keldysh-Faisal-Reiss ansatz.
The relativistic nuclear recoil corrections to the energy levels of low-laying states of hydrogen-like and high $Z$ lithium-like atoms in all orders in $αZ$ are calculated. The calculations are carried out using the B-spline method for the Dirac equation. For low $Z$ the results of the calculation are in good agreement with the $αZ$ -expansion results. It is found that the nuclear recoil contribution, additional to the Salpeter's one, to the Lamb shift ($n=2$) of hydrogen is $-1.32(6)\,kHz$. The total nuclear recoil correction to the energy of the $(1s)^{2}2p_{\frac{1}{2}}-(1s)^{2}2s$ transiti…
We present a calculation scheme for the two-loop vacuum polarization correction of order $α^2$ to the Lamb shift of hydrogen-like high-Z atoms. The interaction with the external Coulomb field is taken into account to all orders in $(Zα)$. By means of a modified potential approach the problem is reduced to the evaluation of effective one-loop vacuum polarization potentials. An expression for the energy shift is deduced within the framework of partial wave decomposition performing appropriate subtractions. Exact results for the two-loop vacuum polarization contribution to the Lamb shift of K- an…
The relativistic nuclear recoil corrections to the energy of the $2p_{\frac{3}{2}}$ state of hydrogen-like and the $(1s)^{2}2p_{\frac{3}{2}}$ state of high $Z$ lithium-like atoms in all orders in $αZ$ are calculated. The calculations are carried out using the B-spline method for the Dirac equation. For low $Z$ the results of the calculation are in good agreement with the $αZ$ -expansion results. It is found that the total nuclear recoil contribution to the energy of the $(1s)^{2}2p_{\frac{3}{2}}- (1s)^{2}2s$ transition in lithium-like uranium constitutes $-0.09\,eV$.
A calculation of the simplest part of the second-order electron self-energy (loop after loop irreducible contribution) for hydrogen-like ions with nuclear charge numbers $3 \leq Z \leq 92$ is presented. This serves as a test for the more complicated second-order self-energy parts (loop inside loop and crossed loop contributions) for heavy one-electron ions. Our results are in strong disagreement with recent calculations of Mallampalli and Sapirstein for low $Z$ values but are compatible with the two known terms of the analytical $Zα$-expansion.
We analyze a special class of 1-D quantum walks (QWs) realized using optical multi-ports. We assume non-perfect multi-ports showing errors in the connectivity, i.e. with a small probability the multi- ports can connect not to their nearest neighbor but to another multi-port at a fixed distance - we call this a jump. We study two cases of QW with jumps where multiple displacements can emerge at one timestep. The first case assumes time-correlated jumps (static disorder). In the second case, we choose the positions of jumps randomly in time (dynamic disorder). The probability distributions of po…
There exists a Klein-Gordon-like equation for a spin-1/2 particle in an electromagnetic field with 2-spinors as wave functions that is a direct consequence of the corresponding Dirac equation. Thus, it reproduces the same binding energies for an electron in a hydrogen atom as the Dirac equation. There is also a square-root equation for 2-spinors which can give the same binding energies up to the forth order of the fine structure constant, inclusively, what will be shown by means of a comparison of the non-relativistic limit plus a first relativistic correction of both equations for 2-spinors w…
We show that, in the relativistic Bohm model of a Dirac-like particle, the electron in the ground state of the hydrogen atom is moving, unlike the prediction for the case of a Schrödinger-like particle, where the electron is stationary. This accounts for the empirically observed dilation of the decay time of the muon in the ground state of muonium.
To describe a relativistic hydrogen atom we used the Poincare-covariant model of a two particle system with gauge invariant potential. The kernel of the radial integral equation is obtained which describes a system of two fermions with electromagnetic interaction.
By applying algebraic techniques, we construct a two-parametric family of strictly isospectral Hydrogen-like potentials as well as some of its one-parametric limits. An additional one-parametric almost isospectral family of Hydrogen-like potentials is also investigated. It is argued that the construction of a SUSY partner Hamiltonian using a factorization energy $δ$ less than the ground state energy of the departure Hamiltonian is unnecessarily restrictive.
Being comparable in quantum systems makes it possible for spaces with varying dimensions to attribute each other using special conversions can attribute schrodinger equation with like-hydrogen atom potential in defined dimensions to a schrodinger equation with other certified dimensions with isotropic oscillator potential. Applying special transformation provides a relationship between different dimensions of two quantum systems. The result of the quantized isotropic oscillator can be generalized to like-hydrogen atom problem in different dimensions. The connection between coordinate spaces in…
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…
Fisher information (I) is investigated for confined hydrogen atom (CHA)-like systems in conjugate $r$ and $p$ spaces. A comparative study between CHA and free H atom (with respect to $I$) is pursued. In many aspects, inferences in CHA are significantly different from free counterpart; that includes its dependence on $n, l, m$. The role of atomic number and atomic radius is discussed. Further, a detailed systematic result of $I$ with respect to variation of confinement radius $r_c$ is presented, with particular emphasis on \emph{non-zero}-$(l,m)$ states. Several new interesting observations are…
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 …
The spectral characterization of Coulomb systems confined by the homogeneous pseudo-Gaussian oscillator is investigated. This is made using the efficient computational method of generating functional. Also, the method is used for the spectral characterization of homogeneous harmonic oscillator confinement, found as a particular case of pseudo-Gaussian oscillator confinement. Finally the aspect of confinement with an impenetrable sphere of finite radius is considered by studying its conjugate effect along with harmonic oscillator upon the atom.
The Rydberg formula along with the Ritz quantum defect ansatz has been a standard theoretical tool used in atomic physics since before the advent of quantum mechanics, yet this approach has remained limited by its non-relativistic foundation. Here I present a long-distance relativistic effective theory describing hydrogen-like systems with arbitrary mass ratios, thereby extending the canonical Ritz-like approach. Fitting the relativistic theory to the hydrogen energy levels predicted by bound-state QED indicates that it is superior to the canonical, nonrelativistic approach. An analytic analys…
