Fock's theory of hydrogen atom and quantum space

Triple differntial cross section for electron impact ionization of hydrogen atom

The electron impact ionization of atomic hydrogen is calculated for incident elrctron energy 76.46 eV. The Hartree-Fock approximation is used to calculate the initial state which includes both bound and continum wave functions. The final state continuum electron wave functions are obtained in the potential of hydrogen ion. The interaction between the two final state continuum electrons is approximated with the screening potential determined variationally.

Fock space relativistic coupled-Cluster calculations of Two-Valence Atoms

We have developed an all particle Fock-space relativistic coupled-cluster method for two-valence atomic systems. We then describe a scheme to employ the coupled-cluster wave function to calculate atomic properties. Based on these developments we calculate the excitation energies, magnetic hyperfine constants and electric dipole matrix elements of Sr, Ba and Yb. Further more, we calculate the electric quadrupole HFS constants and the electric dipole matrix elements of Sr$^+$, Ba$^+$ and Yb$^+$. For these we use the one-valence coupled-cluster wave functions obtained as an intermediate in the tw…

High-accuracy Complete Active Space multiconfiguration Dirac-Hartree-Fock calculations of hyperfine structure constants of the gold atom

The multiconfiguration Dirac-Hartree-Fock (MCDHF) model has been employed to calculate the expectation values for the hyperfine splittings of the 5d96s2 2D3/2 and 5d96s2 2D5/2 levels of atomic gold. One-, two-, and three-body electron correlation effects involving all 79 electrons have been included in a systematic manner. The approximation employed in this study is equivalent to a Complete Active Space (CAS) approach. Calculated electric field gradients, together with experimental values of the electric quadrupole hyperfine structure constants, allow us to extract a nuclear electric quadrupol…

The Hydrogen Atom of Bohr I: Own Life Time of an Excited Single Atom

In this paper it is shown that if one accept assumption of de Broglie that "unitary wave-particle" exists simultaneously and this coexistence is real, then one can find the mean life time of the hydrogen atom of Bohr (intensities).

One- and two-photon resonant spectroscopy of hydrogen and anti-hydrogen atoms in external electric fields

The resonant spectra of hydrogen and anti-hydrogen atoms in the presence of an external electric field are compared theoretically. It is shown that nonresonant corrections to the transition frequency contain terms linear in the electric field. The existence of these terms does not violate space and time parity and leads to a difference in the resonant spectroscopic measurements for hydrogen and anti-hydrogen atoms in an external electric field. The one-photon 1s-2p and the two-photon 1s-2s resonances are investigated.

Influence of external electric fields on multi-photon transitions between the 2s, 2p and 1s levels for hydrogen and anti-hydrogen atoms and hydrogen-like ions

One- and two-photon transitions in the hydrogen atom are analytically evaluated in the absence and in the presence of an external electric field. The emission probabilities are different for the hydrogen and anti-hydrogen atoms due to the existence of contributions, linear in electric field. The magnitude of these contributions is evaluated within the nonrelativistic limit. The Coulomb Green function method is applied. Different nonrelativistic "forms" for the decay probabilities in combination with different gauge choices are considered. The three-photon E1E1E1 2p-1s transition probability is…

Atomic Fock State Preparation Using Rydberg Blockade

We use coherent excitation of 3-16 atom ensembles to demonstrate collective Rabi flopping mediated by Rydberg blockade. Using calibrated atom number measurements, we quantitatively confirm the expected $\sqrt{N}$ Rabi frequency enhancement to within 4%. The resulting atom number distributions are consistent with essentially perfect blockade. We then use collective Rabi $π$ pulses to produce ${\cal N}=1,2$ atom number Fock states with fidelities of 62% and 48% respectively. The ${\cal N}=2$ Fock state shows the collective Rabi frequency enhancement without corruption from atom number fluctuatio…

Relativistic effects on information measures for hydrogen-like atoms

Position and momentum information measures are evaluated for the ground state of the \emph{relativistic} hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on…

Fock-space perturbed relativistic coupled-cluster theory for electric dipole polarizability of one-valence atomic systems: Application to Al and In

We have developed a Fock-space relativistic coupled-cluster theory based method for the calculation of electric dipole polarizability of one-valence atoms and ions. We employ this method to compute the ground-state and spin-orbit coupled excited state electric dipole polarizability of Al and In. To check the quality of many-electron wavefunctions, we also compute the excitation energies of some low-lying states of Al and In. The effects of Breit interaction and QED corrections from the Uehling potential and the self-energy are included to improve the accuracy of $α$ further. Our recommended va…

量级为\oldmath$m(Zα)^6(m/M)$的相对论反冲修正：类氢原子与氢分子离子

Relativistic recoil corrections of order \boldmath$m(Zα)^6(m/M)$: Hydrogen-like atom and hydrogen molecular ion

本文在绝热近似之外的两体和三体形式中，推导了类氢原子和离子自旋平均能级的反冲$mα^6(m/M)$量级相对论修正的有限算符，并给出了适合数值计算的结果。

Inner-shell excitation of open-shell atoms: A spin-dependent localized Hartree-Fock density-functional calculation

The spin-dependent localized Hartree-Fock (SLHF) density-functional approach is extended to the treatment of the inner-shell excited-state calculation of open-shell atomic systems. In this approach, the electron spin-orbitals in an electronic configuration are obtained by solving Kohn-Sham (KS) equation with SLHF exchange potential and the Slater's diagonal sum rule is used to evaluate the multiplet energy of an inner-shell excited state from the single-Slater-determinant energies of the electronic configurations involved. This approach together with the correlation potentials and energy funct…

A Fully Relativistic Treatment of Confined Hydrogen-Like Atoms

The Dirac equation is used to provide a relativistic calculation of the binding energy of a hydrogen-like atom confined within a penetrable spherical barrier. We take the potential to be Coulombic within the barrier and constant outside the barrier. Binding energies are derived for the ground state of hydrogen for various barrier heights and confining radii. In addition, it is shown that without the introduction of the principle quantum number $n$, all energy states of the confined relativistic hydrogen atom, determined by a single quantum number $k$, transfer into the known energy states of t…

The Connection between the Quantum Frequency of Radiation and Frequencies of Circling of the Electron in Atom of Hydrogen

The connection between the quantum frequency of radiation by the transition of the electron from orbit n to orbit k and frequencies of circling of electron in these orbits for the atom of hydrogen is determined.

Proton size from precision experiments on hydrogen and muonic hydrogen atoms

The "proton radius puzzle" was recently solved by reducing the four-standard deviation discrepancy between the results for electronic hydrogen ($H$) and muonic hydrogen ($μH$) atoms to $3.3$ value. The value of the root-mean-square radius of the proton ($r_p$), extracted from experiments on measuring the one-photon $2s-4p$ transition and the Lamb shift in hydrogen, is now $0.8335(95)$ fm, that is in good agreement with the muonic hydrogen experiments, $0.84087(39)$ fm. Even so, these values deviate significantly from the CODATA value, which is determined as the average using the results for va…

Relativistic dynamical polarizability of hydrogen-like atoms

Using the operator representation of the Dirac Coulomb Green function the analytical method in perturbation theory is employed in obtaining solutions of the Dirac equation for a hydrogen-like atom in a time-dependent electric field. The relativistic dynamical polarizability of hydrogen-like atoms is calculated and analysed.

Longitudinal Atomic Beam Spin Echo Experiments: A possible way to study Parity Violation in Hydrogen

We discuss the propagation of hydrogen atoms in static electric and magnetic fields in a longitudinal atomic beam spin echo (lABSE) apparatus. Depending on the choice of the external fields the atoms may acquire both dynamical and geometrical quantum mechanical phases. As an example of the former, we show first in-beam spin rotation measurements on atomic hydrogen, which are in excellent agreement with theory. Additional calculations of the behaviour of the metastable 2S states of hydrogen reveal that the geometrical phases may exhibit the signature of parity-(P-)violation. This invites for po…

Recombination process for hydrogen atom in presence of blackbody radiation

The process of recombination for the hydrogen atom in the heat bath creating the blackbody radiation is descibed within the frameworks of quantum electrodynamics. For this purpose the self-energy for unbound electron in the field of the nucleus is considered. The imaginary part of this self-energy is directly connected with the recombination cross-section. The same procedure is applied to the hydrogen atom in the field of blackbody radiation. This leads to the new thermal correction to the process of recombination for the hydrogen atom in the heat bath. This correction takes into account the f…

Spin-dependent localized Hartree-Fock density-functional approach for the accurate treatment of inner-shell excitation of close-shell atoms

We present a spin-dependent localized Hartree-Fock (SLHF) density-functional approach for the treatment of the inner-shell excited-state calculation of atomic systems. In this approach, the electron spin-orbitals in an electronic configuration are obtained first by solving Kohn-Sham (KS) equation with SLHF exchange potential. Then a single-Slater-determinant energy of the electronic configuration is calculated by using these electron spin-orbitals. Finally, a multiplet energy of an inner-shell excited state is evaluated from the single-Slater-determinant energies of the electronic configuratio…

Deterministic production of free-propagating Fock states of programmable photon number from a single atom-cavity system

We propose a mechanism for producing Fock states on demand leaking from a single mode optical cavity interacting with a single atom and a laser pulse. The number of photons can be chosen, as it is determined by the Zeeman substructure of the ground state of the atom and its initial state. The deterministic generation of a free-propagating Fock state of $1\leq n\leq2F$ photons is achieved, when a circularly polarized laser pulse completely transfers the atomic population between Zeeman sublevels of the ground hyperfine state $F$ through far-detuned Raman scattering thus producing linearly polar…

Hartree-Fock Critical Nuclear Charge in Two-Electron Atoms

Electron correlation effects play a key role in stabilising two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. Here, we investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts the ionisation…

Finite Size Atom in the Hartree-Fock Approximation: New Substance Quasiparticle

It is shown that, in the self-consistent quantum statistical Hartree-Fock approximation, the number of electronic states localized on one nucleus is finite. This result is obtained on the basis of the general electron-nuclear model of matter and provides convergence of the atomic statistical sum and finiteness of the "atom" size. In general approach the characteristic size of the "atom" is a function of density and temperature. However, it is shown, that in a wide range of thermodynamic parameters, for relatively low temperatures, characteristic orbits and electron energy eigenvalues are indep…

Comments on the Hartree-Fock description of the Hooke's atom and suggestion for an accurate closed-form orbital

The ground-state Hartree-Fock (HF) wavefunction of the Hooke's atom is not known in closed form, contrary to the exact solution. The single HF orbital involved has thus far been studied using expansion techniques only, leading to slightly disparate energies. Therefore, the present letter aims at proposing alternative definitions of the HF wavefunction. First, the HF limit is ascertained using a simple expansion, which makes it possible to formulate explicit expressions of HF properties. The resulting energy, 2.038 438 871 8 Eh, is found stable at the tenth digit. Second and more instructive, a…

Using Gaussian Basis-Sets with Gaussian Nuclear Charge Distribution to Solve Dirac-Hartree-Fock Equation for 83 Bi-Atom

In this paper, we consider the Dirac-Hartree-Fock equations for system has many-particles. The difficulties associated with Gaussians model are likely to be more complex in relativistic Dirac-Hartree-Fock calculations. To processing these problem, we use accurate techniques. The four-component spinors will be expanded into a finite basis-set, using Gaussian basis-set type dyall.2zp to describe 4-component wave functions, in order to describe the upper and lower two components of the 4-spinors, respectively. The small component Gaussian basis functions have been generated from large component G…

Hydrogen atom in strong magnetic field revisited

We derive in a straightforward way the spectrum of a hydrogen atom in a strong magnetic field.

Hydrogen and Helium Atoms and Molecules in an Intense Magnetic Field

We calculate the atomic structure of hydrogen and helium, atoms and molecules in an intense magnetic field, analytically and numerically with a judiciously chosen basis.

Modification of Coulomb law and energy levels of hydrogen atom in superstrong magnetic field

The screening of a Coulomb potential by superstrong magnetic field is studied. Its influence on the spectrum of a hydrogen atom is determined.

Relativistic Fock space coupled cluster study of bismuth electronic structure to extract the Bi nuclear quadrupole moment

We report the value of the electric quadrupole moment of $^{209}$Bi extracted from the atomic data. For this, we performed electronic structure calculations for the ground $^4S^o_{3/2}$ and excited $^2P^o_{3/2}$ states of atomic Bi using the Dirac-Coulomb-Breit Hamiltonian and the Fock space coupled cluster method with single, double, and full triple amplitudes for the three-particle Fock space sector. The value of the quadrupole moment of $^{209}$Bi, $Q(^{209}$Bi$)=-418(6)$~mb, derived from the resulting electric field gradient values and available atomic hyperfine splittings is in excellent …

Ion-atom-atom three-body recombination in cold hydrogen and deuterium plasmas

We present a detailed study about ion-atom-atom three-body recombination in hydrogen and deuterium plasmas based on classical trajectory calculations in hyperspherical coordinates. Our results, due to the predominant role of the long-range charged-induced dipole interaction, indicate that H$_2^+$ and D$_2^+$ are the main reaction products in the case of hydrogen and deuterium plasmas, respectively. Besides, we find a more steep energy-dependent reaction rate when the collision energy surpasses the dissociation energy of the molecular ion, thus entering a new dynamical regime dominated by short…

Optical deceleration of atomic hydrogen

High-precision hydrogen spectroscopy is an active field which helps to determine the Rydberg constant and proton charge radius, tests bound-state QED, and can search for Beyond Standard Model (BSM) Physics. Additionally, with recent demonstrations of anti-hydrogen trapping and spectroscopy, a new line of investigation is possible whereby hydrogen can be compared to its antimatter counterpart. The next generation of precision hydrogen spectroscopy will likely require additional motional control of the atomic sample - similar to what is possible with heavier elements. Unfortunately, laser coolin…

Parity Violation in Hydrogen and Longitudinal Atomic Beam Spin Echo I

We discuss the propagation of hydrogen atoms in static electric and magnetic fields in a longitudinal atomic beam spin echo (lABSE) apparatus. There the atoms acquire geometric (Berry) phases that exhibit a new manifestation of parity-(P-)violation in atomic physics. We provide analytical as well as numerical calculations of the behaviour of the metastable 2S states of hydrogen. The conditions for electromagnetic field configurations that allow for adiabatic evolution of the relevant atomic states are investigated. Our results provide the theoretical basis for the discussion of possible measur…

Interaction of a slow monopole with a hydrogen atom

The electric dipole moment of the hydrogen-like atom induced by a monopole moving outside the electron shell is calculated. The correction to the energy of the ground state of the hydrogen atom due to this interaction is calculated.

Inside the Hydrogen Atom

We apply the non-linear Euler-Heisenberg theory to calculate the electric field inside the hydrogen atom. We will demonstrate that the electric field calculated in the Euler-Heisenberg theory can be much smaller than the corresponding field emerging from the Maxwellian theory. In the hydrogen atom this happens only at very small distances. This effect reduces the large electric field inside the hydrogen atom calculated from the electromagnetic form-factors via the Maxwell equations. The energy content of the field is below the pair production threshold.

库仑定律修正对氢原子Lamb移位的贡献

Input of the Coulomb law modification to the Lamb shift of the hydrogen atom

本文基于库仑势Dirac方程精确解的新方法，计算了导致库仑定律修正的辐射修正，进而获得了氢原子能谱并计算了最低能态的Lamb移位。

Hydrogen Atom in the Cosmic Microwave Background

The cosmic microwave background covers the entire universe, which suggests the absence of ay closed system, except the universe itself. In this paper, I consider the effect of the cosmic microwave background on the hydrogen atom, which must be very small, otherwise, changes in energy levels would have been measurable. But how small is small? This I compute by considering a system in an environment or bath. I derived the bath's, (in this case the CMB) effect on the hydrogen atom in the Feynman-Vernon approach to an open system. The effect is small and quantified in terms of a correction to the …

Lamb shift in hydrogen-like muonic atoms to test fractal extra dimensions

From the comparison between the experimental and theoretical results for the energy difference between 2s1/2 and 2p1/2 states in hydrogen and hydrogen-like ions with different values of Z, we estimate bounds for the presence of an extra fractional dimension. These bounds are higher when we work with muonic hydrogen and muonic deuterium, which open the possibility that future measurements of the Lamb shift in muonic atoms with one electron and higher values of Z may show evidence of the extra dimension.

Helium-like atoms. The Green's function approach to the Fock expansion calculations

The renewed Green's function approach to calculating the angular Fock coefficients, $ψ_{k,p}(α,θ)$ is presented. The final formulas are simplified and specified to be applicable for analytical as well as numerical calculations. The Green's function formulas with the hyperspherical angles $θ=0,π$ (arbitrary $α$) or $α=0,π$ (arbitrary $θ$) are indicated as corresponding to the angular Fock coefficients possessing physical meaning. The most interesting case of $θ=0$ corresponding to a collinear arrangement of the particles is studied in detail. It is emphasized that this case represents the gener…

Mesoscopic Superposition States Generated by Synthetic Spin-orbit Interaction in Fock-state Lattices

Mesoscopic superposition states of photons can be prepared in three cavities interacting with the same two-level atom. By periodically modulating the three cavity frequencies around the transition frequency of the atom with $2π/3$ phase difference, the time reversal symmetry is broken and an optical circulator is generated with chiralities depending on the quantum state of the atom. A superposition of the atomic states can guide photons from one cavity to a mesoscopic superposition of the other two cavities. The physics can be understood in a finite spin-orbit-coupled Fock-state lattice where …

Stochastic Ionization of Relativistic Hydrogen-Like Atom

Stochastic ionization of highly excited relativistic hydrogenlike atom in the monochromatic field is considered. A theoretical analisis of chaotic dynamics of the electron based on Chirikov's criterion is given. Critical value of the external field is evaluated analytically.

Diffusive Ionization of Relativistic Hydrogen-Like Atom

Stochastic ionization of highly excited relativistic hydrogenlike atom in the monochromatic field is investigated. A theoretical analisis of chaotic dynamics of the relativistic electron based on Chirikov criterion is given for the cases of one- and three-dimensional atoms. Critical value of the external field is evaluated analitically. The diffusion coefficient and ionization time are calculated.

超原子氢：在低于金属氢的压力下实现氢原子的有效聚集

Superatomic hydrogen: achieving effective aggregation of hydrogen atoms at pressures lower than that of metallic hydrogen

针对金属氢需数百GPa以上极端压力的难题，提出超原子氢模型，经高精度从头算验证H13体系成超原子态的压力比金属氢低约两个数量级且具备金属氢电子离域特性，为核聚变等领域应用开辟新前景。

Convergent variational calculation of positronium-hydrogen-atom scattering lengths

We present a convergent variational basis-set calculational scheme for elastic scattering of positronium atom by hydrogen atom in S wave. Highly correlated trial functions with appropriate symmetry are needed for achieving convergence. We report convergent results for scattering lengths in atomic units for both singlet ($=3.49\pm 0.20$) and triplet ($=2.46\pm 0.10$) states.

The influence of the electric polarization of hydrogen atoms on the red shift of its spectral lines

The Lorentz oscillator system is studied to interpret the spectral lines of hydrogen atoms. The dielectric constant of this system is analyzed, which takes into account the electrical polarization of hydrogen atoms. This dielectric constant gives the red shift of the spectral line and the appearance of the optical spectrum dip. This dip is on the blue side of the spectral position of the shifted line. The value of this red shift and the width of this dip strongly depend on the hydrogen atom concentration and the spectral position of the not shifted line. This red shift increases with an increa…

On the q-analogue of the hydrogen atom

The discrete spectrum of a q-analogue of the hydrogen atom is obtained from a deformation of the Pauli equations. As an alternative, the spectrum is derived from a deformation of the four-dimensional oscillator arising in the application of the Kustaanheimo-Stiefel transformation to the hydrogen atom. A model of the 2s-2p Dirac shift is proposed in the context of q-deformations.

Hénon-Heiles Interaction for Hydrogen Atom in Phase Space

Using elements of symmetry, as gauge invariance, several aspects of a Schrödinger equation represented in phase-space are introduced and analyzed under physical basis. The Hydrogen atom is explored in the same context. Then we add a Hénon-Heiles potential to the Hydrogen atom in order to explore chaotic features.

B-Spline basis Hartree-Fock method for arbitrary central potentials: atoms, clusters and electron gas

An implementation of the Hartree-Fock (HF) method capable of robust convergence for well-behaved arbitrary central potentials is presented. The Hartree-Fock equations are converted to a generalized eigenvalue problem by employing a B-spline basis in a finite-size box. Convergence of the self-consistency iterations for the occupied electron orbitals is achieved by increasing the magnitude of the electron-electron Coulomb interaction gradually to its true value. For the Coulomb central potential, convergence patterns and energies are presented for a selection of atoms and negative ions, and are …