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.

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.

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.

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.

scholarly journals Measurement of the running of the fine structure constant below 1 GeV with the KLOE detector

2019 ◽  
Vol 218 ◽  
pp. 02012
Author(s):  
Graziano Venanzoni
Keyword(s):  
Fine Structure ◽  
Energy Region ◽  
Direct Evidence ◽  
Branching Ratio ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Recent Measurement ◽  
Single Measurement ◽  
Lepton Universality ◽  
Hadronic Contribution

I will report on the recent measurement of the fine structure constant below 1 GeV with the KLOE detector. It represents the first measurement of the running of α(s) in this energy region. Our results show a more than 5σ significance of the hadronic contribution to the running of α(s), which is the strongest direct evidence both in time-and space-like regions achieved in a single measurement. From a fit of the real part of Δα(s) and assuming the lepton universality the branching ratio BR(ω → µ+µ−) = (6.6 ± 1.4stat ± 1.7syst) · 10−5 has been determined

scholarly journals Fine structure constant and the CMB damping scale

2012 ◽  
Vol 85 (10) ◽  
Author(s):  
Eloisa Menegoni ◽  
Maria Archidiacono ◽  
Erminia Calabrese ◽  
Silvia Galli ◽  
C. J. A. P. Martins ◽  
...  
Keyword(s):  
Fine Structure ◽  
Structure Constant ◽  
Fine Structure Constant

CONSTRAINTS ON SPATIAL VARIATIONS IN THE FINE-STRUCTURE CONSTANT FROMPLANCK

2014 ◽  
Vol 798 (1) ◽  
pp. 18 ◽  
Author(s):  
Jon O'Bryan ◽  
Joseph Smidt ◽  
Francesco De Bernardis ◽  
Asantha Cooray
Keyword(s):  
Fine Structure ◽  
Structure Constant ◽  
Spatial Variations ◽  
Fine Structure Constant
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