Direct Relativistic Formulation of Constant Magnetic and Electric Field Effects in Simple Atoms. II

1975 ◽  
Vol 53 (13) ◽  
pp. 1247-1250 ◽  
Author(s):  
R. A. Moore
Keyword(s):  
Stark Effect ◽  
Zeeman Effect ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Oscillator Strengths ◽  
Fine Structure Constant ◽  
Electric Field Effects ◽  
Relativistic Formulation ◽  
Transition Matrix Elements ◽  
Experimental Values

A previously developed method of obtaining approximate solutions to the Dirac equation for the electron is applied to the problem of constant electric and magnetic effects in simple atoms. This method makes the relativistic formulation of these problems straightforward. In all cases the standard expressions are reproduced, as one would expect, and hence illustrates the simplicity and utility of the method. The Zeeman effect is calculated to order α2, α being the usual fine structure constant, and it is seen that only the lowest order solutions are required. That is, one needs only the nonrelativistic solutions. The Stark effect and dipole transition matrix elements are calculated only to zero order in α as it is felt that that is all that is required for comparison with experiment. One concludes that the discrepancies between the theoretical values and experimental values of the oscillator strengths must be explained on the basis of nonrelativistic effects.

Reply to "A remark on Moore's new method of obtaining approximate solutions of the Dirac equation"

1981 ◽  
Vol 59 (11) ◽  
pp. 1614-1619 ◽  
Author(s):  
R. A. Moore ◽  
Sam Lee
Keyword(s):  
Dirac Equation ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Wave Functions ◽  
Fine Structure Constant ◽  
Energy Calculation ◽  
Calculation Error ◽  
First Order ◽  
The One ◽  
The Individual

This work was written to clarify the use of a recently developed procedure to obtain approximate solutions of the one-particle Dirac equation directly and in response to a recent critique on its application to lowest order. The critique emphasized the fact that when the wave functions are determined only to zero order then a first order energy calculation contains significant errors of the order of α4, α being the fine structure constant, and a matrix element calculation error of order α2. Tomishima re-affirms that higher order solutions are required to obtain accuracy of these orders. In this work the hierarchy of equations occurring in the procedure is extended to first order and it is shown that exact solutions exist for hydrogen-like atoms. It is also shown that the energy in second order contains all of the contributions of order α4. In addition, we illustrate, in detail, that the procedure can be aplied in such a way as to isolate the individual components of the wave functions and energies as power series of α2. This analysis lays the basis for the determination of suitable numerical methods and hence for application to physical systems.

scholarly journals STATUS OF STANDARD MODEL CALCULATION OF LEPTON g - 2

2014 ◽  
Vol 35 ◽  
pp. 1460405 ◽  
Author(s):  
MARC KNECHT
Keyword(s):  
Standard Model ◽  
Magnetic Moments ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Precision Determination ◽  
The Standard Model ◽  
Charged Leptons ◽  
Experimental Values ◽  
High Precision Determination

The contributions from the standard model interactions to the anomalous magnetic moments of the two lightest charged leptons, the electron and the muon, are reviewed. Comparison with the very accurately measured experimental values is made, using the most recent high-precision determination of the fine structure constant.

An Alternative Method of Obtaining Approximate Solutions to the Dirac Equation. I

1975 ◽  
Vol 53 (13) ◽  
pp. 1240-1246 ◽  
Author(s):  
R. A. Moore
Keyword(s):  
Wave Function ◽  
Dirac Equation ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Order Equation ◽  
Wave Functions ◽  
Fine Structure Constant ◽  
First Order ◽  
Alternative Method ◽  
Electric And Magnetic Fields

An alternative method of obtaining approximate solutions to the Dirac equation is presented. The method takes advantage of the fact that the wave functions can be written as an ordered series in powers of the fine structure constant, α, and that the Hamiltonian can be separated into two parts such that one part connects adjacent orders of the wave function. Energy calculations to order α2, requiring only the solution to the lowest order equation, are considered in this article. The procedure is tested by applying it to the hydrogen atom. It is seen that the lowest order equations are similar to and no more difficult to solve than the nonrelativistic equations for all systems of physical interest. The simplicity and accuracy of the method implies that full relativistic calculations are unnecessary for most situations. The inclusion of electric and magnetic fields and the solution to the first order equation will be considered in later articles.

Direct Relativistic Formulation of the Hyperfine Interaction in Simple Atoms. III

1975 ◽  
Vol 53 (13) ◽  
pp. 1251-1255 ◽  
Author(s):  
R. A. Moore
Keyword(s):  
Fine Structure ◽  
Hyperfine Interaction ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Fine Structure Constant ◽  
Spherically Symmetric ◽  
Numerical Work ◽  
First Order ◽  
Time Required ◽  
Nonrelativistic Calculation

We apply a previously developed procedure for obtaining approximate solutions for the Dirac equation for the electron to the formulation of the hyperfine interaction in spherically symmetric atoms. Expressions are obtained to order α2, α being the usual fine structure constant. The hydrogen 1S state is solved and seen to be correct to order α2. This result is taken to be a positive test of the procedure. Further, one sees that the first order equations are solvable, the form of the solutions, and that all the required contributions are finite. Also, in terms of numerical work the time required will not be appreciably greater than needed for a nonrelativistic calculation. This leads to the conclusion that one has a practical and valid method of solving relativistic problems to order α2.

Approximate solutions to the one-particle Dirac equation: numerical results

1986 ◽  
Vol 64 (3) ◽  
pp. 297-302 ◽  
Author(s):  
R. A. Moore ◽  
T. C. Scott
Keyword(s):  
Dirac Equation ◽  
Numerical Solutions ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Fine Structure Constant ◽  
Matrix Elements ◽  
Atomic Systems ◽  
The Matrix ◽  
The One ◽  
Significant Figures

The zero-, first-, and second-order differential equations in a previously defined hierarchy of equations giving approximate solutions to the one-particle Dirac equation and the corresponding eigenvalue contributions are each written as power series in α, the fine structure constant, for an arbitrary, spherically symmetric potential. These equations are solved numerically for the hydrogen-atom potential to obtain wave functions to order α2 and eigenvalues to order α4 for all states with n = 1–4, inclusive. The numerical solutions are then used to evaluate a number of matrix elements to order α2. A comparison with the exact expressions shows that the numerical values for the coefficients of the different powers of α have at least six significant figures in the eigenfunctions and eigenvalues and five in the matrix elements. Thus, the procedure is validated and can be applied with confidence to other atomic systems.

scholarly journals Measuring the fine structure constant on a white dwarf surface; a detailed analysis of Fe V absorption in G191-B2B

2020 ◽  
Author(s):  
J Hu ◽  
J K Webb ◽  
T R Ayres ◽  
M B Bainbridge ◽  
J D Barrow ◽  
...  
Keyword(s):  
Fine Structure ◽  
White Dwarf ◽  
Hubble Space Telescope ◽  
Zeeman Effect ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Gravitational Fields ◽  
Systematic Uncertainties ◽  
Isotopic Abundances ◽  
Comprehensive Search

Abstract The gravitational potential φ = GM/Rc2 at the surface of the white dwarf G191-B2B is 10,000 times stronger than that at the Earth’s surface. Numerous photospheric absorption features are detected, making this a suitable environment to test theories in which the fundamental constants depend on gravity. We have measured the fine structure constant, α, at the white dwarf surface, used a newly calibrated Hubble Space Telescope STIS spectrum of G191-B2B, two new independent sets of laboratory Fe V wavelengths, and new atomic calculations of the sensitivity parameters that quantify Fe V wavelength dependency on α. The two results obtained are: Δα/α0 = (6.36 ± 0.35stat ± 1.84sys) × 10−5 and Δα/α0 = (4.21 ± 0.48stat ± 2.25sys) × 10−5. The measurements hint that the fine structure constant increases slightly in the presence of strong gravitational fields. A comprehensive search for systematic errors is summarised, including possible effects from line misidentifications, line blending, stratification of the white dwarf atmosphere, the quadratic Zeeman effect and electric field effects, photospheric velocity flows, long-range wavelength distortions in the HST spectrum, and variations in the relative Fe isotopic abundances. None fully account for the observed deviation but the systematic uncertainties are heavily dominated by laboratory wavelength measurement precision.

Constructing Quantum Mechanics

2019 ◽  
Author(s):  
Anthony Duncan ◽  
Michel Janssen
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Stark Effect ◽  
Zeeman Effect ◽  
Black Body ◽  
Black Body Radiation ◽  
Wave Mechanics ◽  
Specific Heats ◽  
Old Quantum Theory ◽  
Upper Level

This is the first of two volumes on the genesis of quantum mechanics. It covers the key developments in the period 1900–1923 that provided the scaffold on which the arch of modern quantum mechanics was built in the period 1923–1927 (covered in the second volume). After tracing the early contributions by Planck, Einstein, and Bohr to the theories of black‐body radiation, specific heats, and spectroscopy, all showing the need for drastic changes to the physics of their day, the book tackles the efforts by Sommerfeld and others to provide a new theory, now known as the old quantum theory. After some striking initial successes (explaining the fine structure of hydrogen, X‐ray spectra, and the Stark effect), the old quantum theory ran into serious difficulties (failing to provide consistent models for helium and the Zeeman effect) and eventually gave way to matrix and wave mechanics. Constructing Quantum Mechanics is based on the best and latest scholarship in the field, to which the authors have made significant contributions themselves. It breaks new ground, especially in its treatment of the work of Sommerfeld and his associates, but also offers new perspectives on classic papers by Planck, Einstein, and Bohr. Throughout the book, the authors provide detailed reconstructions (at the level of an upper‐level undergraduate physics course) of the cental arguments and derivations of the physicists involved. All in all, Constructing Quantum Mechanics promises to take the place of older books as the standard source on the genesis of quantum mechanics.

scholarly journals New Limit on Space-Time Variations in the Proton-to-Electron Mass Ratio from Analysis of Quasar J110325-264515 Spectra

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 344
Author(s):  
T. D. Le
Keyword(s):  
Fine Structure ◽  
Structure Constant ◽  
Space Time ◽  
Fine Structure Constant ◽  
Electron Mass ◽  
Mass Ratio ◽  
Quasar Spectra ◽  
Time Variations ◽  
Using Data ◽  
The Universe

Astrophysical tests of current values for dimensionless constants known on Earth, such as the fine-structure constant, α , and proton-to-electron mass ratio, μ = m p / m e , are communicated using data from high-resolution quasar spectra in different regions or epochs of the universe. The symmetry wavelengths of [Fe II] lines from redshifted quasar spectra of J110325-264515 and their corresponding values in the laboratory were combined to find a new limit on space-time variations in the proton-to-electron mass ratio, ∆ μ / μ = ( 0.096 ± 0.182 ) × 10 − 7 . The results show how the indicated astrophysical observations can further improve the accuracy and space-time variations of physics constants.

Systematic calculations of energy levels and transitions rates in Mo XXVIII

2020 ◽  
Vol 75 (8) ◽  
pp. 739-747
Author(s):  
Feng Hu ◽  
Yan Sun ◽  
Maofei Mei
Keyword(s):  
Energy Levels ◽  
Oscillator Strengths ◽  
Atomic Data ◽  
Data Sets ◽  
Electron Systems ◽  
Excitation Energies ◽  
Experimental Values ◽  
Qed Effects ◽  
Hyperfine Structures ◽  
Good Agreement

AbstractComplete and consistent atomic data, including excitation energies, lifetimes, wavelengths, hyperfine structures, Landé gJ-factors and E1, E2, M1, and M2 line strengths, oscillator strengths, transitions rates are reported for the low-lying 41 levels of Mo XXVIII, belonging to the n = 3 states (1s22s22p6)3s23p3, 3s3p4, and 3s23p23d. High-accuracy calculations have been performed as benchmarks in the request for accurate treatments of relativity, electron correlation, and quantum electrodynamic (QED) effects in multi-valence-electron systems. Comparisons are made between the present two data sets, as well as with the experimental results and the experimentally compiled energy values of the National Institute for Standards and Technology wherever available. The calculated values including core-valence correction are found to be in a good agreement with other theoretical and experimental values. The present results are accurate enough for identification and deblending of emission lines involving the n = 3 levels, and are also useful for modeling and diagnosing plasmas.

scholarly journals Challenges for a QCD axion at the 10 MeV scale

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jia Liu ◽  
Navin McGinnis ◽  
Carlos E. M. Wagner ◽  
Xiao-Ping Wang
Keyword(s):  
Higgs Boson ◽  
Standard Model ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Multiple Sources ◽  
Standard Model Extension ◽  
Mixing Angles ◽  
Quark Masses ◽  
Model Extension ◽  
The One

Abstract We report on an interesting realization of the QCD axion, with mass in the range $$ \mathcal{O} $$ O (10) MeV. It has previously been shown that although this scenario is stringently constrained from multiple sources, the model remains viable for a range of parameters that leads to an explanation of the Atomki experiment anomaly. In this article we study in more detail the additional constraints proceeding from recent low energy experiments and study the compatibility of the allowed parameter space with the one leading to consistency of the most recent measurements of the electron anomalous magnetic moment and the fine structure constant. We further provide an ultraviolet completion of this axion variant and show the conditions under which it may lead to the observed quark masses and CKM mixing angles, and remain consistent with experimental constraints on the extended scalar sector appearing in this Standard Model extension. In particular, the decay of the Standard Model-like Higgs boson into two light axions may be relevant and leads to a novel Higgs boson signature that may be searched for at the LHC in the near future.

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