Perturbation theory without wave functions for the Zeeman effect in hydrogen.

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems

Coxeter spectral classification of almost TP-critical one-peak posets using symbolic and numeric computations

The diamagnetic Kepler problem, or quadratic Zeeman effect in hydrogen, is converted into an asymmetric top via a transformation to Lissajous action-angle variables. These variables are especially suitable for uniform semiclassical quantization of the system for arbitrary values of the azimuthal quantum number m.

A highly optimized code for calculating atomic data at neutron star magnetic field strengths using a doubly self-consistent Hartree–Fock–Roothaan method

It is well known that in relativistic quantum theory there is not exact analytical formula for cross section of a charged particle elastic scattering on arbitrary electrostatic potential \(\varphi (\mathbf {r})\). Even in case of Coulomb field for which the exact solution of Dirac equation is known, relativistic “Coulomb wave functions”, nevertheless, because of its complex character it is impossible to obtain exact analytical expression for scattering cross section via these wave functions. The elastic scattering in relativistic domain is mainly described in approximations when the scattering potential can be considered as a perturbation (in opposite limits). These are the well-known Born and eikonal approximations corresponding to quantum perturbation theory by particle wave function (when the condition \(\left| U\right| \ll \hbar \mathrm {v}/a\) is satisfied; U is the potential energy, a is the space size of the range of effective scattering, \(\mathrm {v}\) is the particle initial velocity), and high-momentum approximation (if potential energy in the scattering field is much less than the particle initial energy: \(\left| U\right| \ll p\mathrm {v}\)), respectively. In case of Coulomb field there is also an approximation of large impact parameters (large momenta)—Farry-Sommerfeld-Maue (FSM) approximation , which describes well enough the scattering at the small angles. It is also known that the wave function in the eikonal approximation describes the particle state only in a limited space range of the scattering process (\(z\ll pa^{2}/\hbar \); z is the coordinate along the direction of particle initial momentum), i.e., the eikonal solution is not valid at large distances. The wave function of the Born approximation , in contrast to the eikonal one, describes the particle state at arbitrary point in the scattering range, particularly at asymptotically large distances. Nevertheless, the common region where both approximations under consideration are valid is very restricted. On the other hand, within the (small) potential range where both approximations are applicable, these wave functions describe the scattering by different accuracies. Practically, these wave functions describe the scattering at the opposite conditions: the eikonal wave function describes the particle state when the opposite condition of the Born approximation holds (e.g., for Coulomb field with a charge \(Z_{a}e\)—at the condition: \(Z_{a}e^{2}/\hbar \mathrm {v}\gtrsim 1\)). In these circumstances a natural question arises—is it possible to find out such an approximate solution of Dirac equation beyond the scope both perturbation theory and eikonal approximation—corresponding to more general wave function being applicable in both quantum and quasiclassical limits for relativistic potential scattering (including in particular the Born and eikonal approximations in corresponding limits—under its conditions of applicability)? Here we will try to derive such an approximate solution of the Dirac equation which satisfies the formulated requests. We will call it generalized eikonal approximation (GEA) for a spinor particle scattering on the arbitrary short-range or long-range electrostatic potential. Then we will clear up the relationship of this approximation to the known ones for electron elastic scattering on the short-range (Born and eikonal approximations) and long-range Coulomb potentials (Born, eikonal, and Farry-Sommerfeld-Maue approximations ). This GEA approximation is developed for electron inelastic scattering process too—to describe the relativistic stimulated bremsstrahlung in the field of strong and superstrong laser radiation with electrostatic potential fields of arbitrary form and finite or infinite effective radiuses (atoms, ions, etc.). The significance of GEA type wave function for description of such laser-assisted electron–atom–ion scattering processes, apart from the known restrictions of mentioned above approximations, is also conditioned with the fact that the wave functions in known approximations describe the particle state factorized by elastic scattering and induced radiation-absorption processes. In the result, we lose the phase relations at the description of electron quantum dynamics interacting with the both fields simultaneously, which have important role for induced process, in particular, for coherent part of interaction. Therefore, for description of strong and superstrong laser–matter (plasma) interaction processes, we need the electron wave function that takes into account the simultaneous influence of both scattering and radiation fields including dynamic phase relations. This induced GEA wave function may be applied specifically for the description of the above-threshold ionization (ATI) process of atoms/ions by superstrong laser fields in relativistic theory taking into account the photoelectron rescattering effect on the atomic remainder because of the action of long-range Coulomb field of atomic ion on the photoelectron final state (this process is considered in the next chapter of this book).

A generalization of a perturbation theory without wave function is developed which applies to a hydrogen atom perturbed by magnetic and electric fields. A double perturbation expansion for the Stark–Zeeman effect for parallel fields is considered and results for the states obeying |m| = l = n − 1 in the zero field limit are presented. Other problems that can be treated this way are discussed.

This study describes the effect of hysteresis on the generation of the higher harmonics in the line voltage and current in a non-uniform transmission line when we include a quadratic non-linear term having memory to account for the hysteresis effect in the inductance part of line. The simulational analysis using MATLAB is presented which works with a non-uniform line having hysteresis. These simulations are based on a theoretical spatial Fourier analysis combined with perturbation theory which is used without bothering about source and load. The line differential equations are solved approximately in the space- time-frequency domain using perturbation theory and the numerical computations have been carried out for the voltage and current amplitude characteristics. In Appendix 1, some theoretical explanations that support our non-linear memory-based hysteresis model based on Landau’s equation of precession of a magnetic moment in a magnetic field are included. This paper also includes a brief derivation of atomic transition probability produced by e-m radiation from the transmission line falling on an atom in the far field zone. This calculation enables us to derive maximum likelihood estimation(MLE) of the hysteresis parameters by measuring which atomic transitions have occurred.

view Abstract Citations (7) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Quadratic Zeeman effect in the hydrogen Balmer lines from magnetic white dwarfs. Surmelian, G. L. ; Oconnell, R. F. Abstract Motivated by Preston's suggestion of using the quadratic Zeeman effect to detect large magnetic fields in DA white dwarfs, we use a variational method for an accurate calculation of the total Zeeman wavelength shifts of the hydrogen Balmer absorption lines (Ha-H10) for magnetic fields from 106 to 5 x 106 gauss. Interference effects between upper values of the principal quantum number n become significant in that range. The quadratic shifts are smaller than those given by perturbation theory, leading to an increase in the estimated magnetic field. Subject headings: magnetic stars - white dwarf stars - Zeeman effect Publication: The Astrophysical Journal Pub Date: November 1974 DOI: 10.1086/153209 Bibcode: 1974ApJ...193..705S Keywords: Balmer Series; H Lines; Magnetic Stars; White Dwarf Stars; Zeeman Effect; Absorption Spectra; Frequency Shift; H Alpha Line; Hydrogen Atoms; Magnetic Flux; Stellar Spectra; Astrophysics full text sources ADS |

view Abstract Citations (43) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Energy levels and oscillator strengths for the two-body problem in magnetic fields Wunner, G. ; Ruder, H. ; Herold, H. Abstract The magnetic field dependence of the physical properties derived from the solution of the two-body problem for an electron and a proton in a magnetic field is investigated. The scaling properties of the general problem of two charged particles in a magnetic field are discussed, and it is shown that the continuous magnetic field dependence of physical properties including energies and dipole matrix elements can be approximated by low-degree polynomials. The continuous magnetic field dependence of the lowest energy levels and corresponding oscillator strengths of the hydrogen atom in the limit of infinite proton mass are then examined in detail for magnetic fields from 0 to 4.7 x 10 to the 13th gauss. It is noted that the results have important applications for the hydrogen atom with a finite proton mass, positronium and the complete isoelectronic series of hydrogen in magnetic fields of intensities up to those characteristic of neutron stars. Publication: The Astrophysical Journal Pub Date: July 1981 DOI: 10.1086/159046 Bibcode: 1981ApJ...247..374W Keywords: Atomic Energy Levels; Hydrogen Atoms; Magnetic Fields; Neutron Stars; Transition Probabilities; Two Body Problem; Eigenvalues; Electrons; Isoelectronic Sequence; Matrices (Mathematics); Protons; Wave Functions; Astrophysics full text sources ADS |

Bound electron states in highly charged ions are strongly influenced by the effects of relativity and quantum electrodynamics (QED). These effects induce shifts of the binding energies as well as corrections to observables related to atomic processes. In this work a numerical procedure is described and implemented in which the QED effects are treated as corrections to relativistic bound-state wavefunctions. This approach, which is based on the recently developed covariant evolution-operator formalism, allows for a merging of QED with the standard methods of many-body perturbation theory. In particular, it enables an evaluation of the combined effect of QED and electron correlation in few-electron systems. Numerical results for this effect are presented for the ground state energy of heliumlike ions. A detailed analysis of the contribution from the electron self-energy is carried out in both the Feynman and Coulomb gauge. It is found that the Feynman gauge suffers from large numerical cancellations and aquires significant contributions from terms involving multiple interactions with the nuclear potential (the so-called many-potential terms), while the Coulomb gauge is well suited for an approximate treatment based on terms involving only freely propagating electrons (the zero-potential terms). With the help of QED-corrected wavefunctions it is also possible to compute corrections to observables in basic atomic processes. In this work some of the one-loop QED corrections (those derivable from perturbed wavefunctions and energies) to the differential cross section and distribution of polarization in radiative recombination of initially bare uranium nuclei are evaluated, as well as the corresponding corrections to the ratio of the electric dipole and magnetic quadrupole transition amplitudes in the 2p3/2 → 1s radiative decay of hydrogenlike uranium. The results from these calculations are all of the expected magnitude, namely on the order of the fine-structure contant.

Context. The second solar spectrum resulting from coherent scattering is a main tool for diagnostics of turbulent magnetic fields on the Sun. Scattering on diatomic molecules plays an important role in forming this spectrum and even dominates in some spectral regions. Aims. In a magnetic field electronic states of a molecule are often perturbed via the Paschen-Back effect. Sometimes this perturbation can completely change the spectrum, not only quantitatively, but even qualitatively. Here we calculate molecular scattering properties taking into account the Paschen-Back effect. Methods. Starting with the Hund's case (a) wave functions as a basis we obtain with the perturbation theory wave functions of the intermediate Hund's case (a-b) in a magnetic field. Using new, perturbed values of the Lande factors and transition amplitudes we calculate the Mueller matrix for coherent scattering at diatomic molecules in the intermediate Hund's case (a-b) and look for the effects that can be caused by the Paschen-Back effect. Results. We have found a considerable deviation from the Zeeman regime and discuss here the quantitative and qualitative effects on observed polarization signals for the CN B 2 Σ - X 2 Σ and MgH B' 2 Σ - X 2 Σ systems as examples.

A quasi-classical approach is proposed for describing the motion of a charged particle in a magnetic field, taking into account irreversible losses due to the macroscopic viscosity of the medium. The wave function of a charged particle corresponding to its quasi-classical coherent state is found. It is shown that viscosity leads to an irreversible collapse of the wave function in directions perpendicular to the magnetic field. At the same time along the magnetic field the wave function experiences irreversible spreading up to a certain static limit. Thus, in the transverse directions a viscous medium and a magnetic field behave like a classical measuring device. In the longitudinal directions the signs of the quantum Zeno effect are visible. As a result of such anisotropic quantum dynamics the wave packet of the probability density takes the form of a thin filament wound around a magnetic field. The length of the filament is determined by the limiting value of the uncertainty of the longitudinal coordinate of the particle. In turn, this asymptotic uncertainty contains the information about its initial value, about the mass of the particle, and about the properties of the viscous medium.

In the January 2002 edition of SIAM News, Nick Trefethen announced the '$100, 100-Digit Challenge'. In this note he presented ten easy-to-state but hard-to-solve problems of numerical analysis, and challenged readers to find each answer to ten-digit accuracy. Trefethen closed with the enticing comment: 'Hint: They're hard! If anyone gets 50 digits in total, I will be impressed.' This challenge obviously struck a chord in hundreds of numerical mathematicians worldwide, as 94 teams from 25 nations later submitted entries. Many of these submissions exceeded the target of 50 correct digits; in fact, 20 teams achieved a perfect score of 100 correct digits. Trefethen had offered $100 for the best submission. Given the overwhelming response, a generous donor (William Browning, founder of Applied Mathematics, Inc.) provided additional funds to provide a $100 award to each of the 20 winning teams. Soon after the results were out, four participants, each from a winning team, got together and agreed to write a book about the problems and their solutions. The team is truly international: Bornemann is from Germany, Laurie is from South Africa, Wagon is from the USA, and Waldvogel is from Switzerland. This book provides some mathematical background for each problem, and then shows in detail how each of them can be solved. In fact, multiple solution techniques are mentioned in each case. The book describes how to extend these solutions to much larger problems and much higher numeric precision (hundreds or thousands of digit accuracy). The authors also show how to compute error bounds for the results, so that one can say with confidence that one's results are accurate to the level stated. Numerous numerical software tools are demonstrated in the process, including the commercial products Mathematica, Maple and Matlab. Computer programs that perform many of the algorithms mentioned in the book are provided, both in an appendix to the book and on a website. In the process, the authors take the reader on a wide-ranging tour of modern numerical mathematics, with enough background material so that even readers with little or no training in numerical analysis can follow. Here is a list of just a few of the topics visited: numerical quadrature (i.e., numerical integration), series summation, sequence extrapolation, contour integration, Fourier integrals, high-precision arithmetic, interval arithmetic, symbolic computing, numerical linear algebra, perturbation theory, Euler-Maclaurin summation, global minimization, eigenvalue methods, evolutionary algorithms, matrix preconditioning, random walks, special functions, elliptic functions, Monte-Carlo methods, and numerical differentiation.

In the present paper, by using the method of the effective Hamiltonian, we study Bloch's electrons in the presence of a uniform electric and magnetic field. We show that all wave functions are periodic in time. This result is the fully quantum-mechanical analog of the well-known Bloch oscillations predicted by quasi-classical dynamics. Furthermore, using the githt-binding method for a single band of an electron in a simple cubic lattice in the presence of a uniform electric and magnetic field, we obtain an equation of Harper's form. From this equation we take exactly the points at which Bragg reflection occurs for the subbands of Landau.

Certain aspects of Rayleigh-Schr\"odinger (RS) perturbation theory and of the variational principle are combined in a new perturbationally oriented variational approach for obtaining approximate solutions to Schr\"odinger's time-independent equation. This perturbational-variational (PV) procedure is applicable for all perturbed Hamiltonian operators which can be expanded as $H(\ensuremath{\lambda})=\ensuremath{\Sigma}{H}_{t}{\ensuremath{\lambda}}^{t}$, where $\ensuremath{\lambda}$ is an external parameter governing the strength of the perturbing terms; in particular, the theory is developed here for the important special form, $H={H}_{0}+{H}_{1}\ensuremath{\lambda}$. It is shown how the application of the PV procedure to an arbitrary analytic approximate wave function yields the variational parameters embedded in the wave function, the wave function itself, the corresponding approximate energy, and other expectation values, all in the form of energetically optimized PV expansions in powers of $\ensuremath{\lambda}$ to any desired order. A variational wave-operator formalism is introduced to deal with the multivariant expansions required to form the PV expansions. The remainder theorem and the variational Hellmann-Feynman theorem are derived and their role in the PV methodology is discussed. The principal advantages of the new method are: (1) Unlike RS perturbation theory, there is no requirement that the exact solution of a simpler unperturbed system be known; (2) the explicit $\ensuremath{\lambda}$ dependence of the optimum quantities is analytically derived; (3) the numerical aspects of a given variational problem are simplified because the variational equations generated by the PV procedure are almost completely linearized in the variational parameters; and (4) the PV expansions form a natural framework for the systematic classification and evaluation of all perturbed variational wave functions in respect to their quality. The PV classification scheme, previously presented in bare outline, is rigorously derived. The insight obtained from PV theory is further illustrated by the derivation and application of the variational integral Hellmann-Feynman theorem, the virial theorem, and a theorem relating to the expectation value of the unperturbed potential. The a priori PV classification of perturbed variational wave functions is extended to the perturbed variational parameters embedded in the wave functions; it is shown that the anomalous PV expansions of some open-shell parameters and orbitals can be simply explained in terms of PV theory.