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.

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.

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.

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.

scholarly journals Intercombination Transitions between Levels X1Σg+ and A 3Πu in C2

1987 ◽  
Vol 120 ◽  
pp. 103-105
Author(s):  
J. Le Bourlot ◽  
E. Roueff
Keyword(s):  
Transition Probabilities ◽  
Energy Levels ◽  
Wave Functions ◽  
Spin Orbit Coupling ◽  
Energy Matrix ◽  
First Order ◽  
Emission Transition ◽  
Intercombination Transition ◽  
The One ◽  
Potential Curves

We present a new calculation of intercombination transition probabilities between levels X1Σg+ and a 3Πu of the C2 molecule. Starting from experimental energy levels, we calculate RKR potential curves using Leroy's Near Dissociation Expansion (NDE) method; these curves give us wave functions for all levels of interest. We then compute the energy matrix for the four lowest states of C2, taking into account Spin-Orbit coupling between a 3Πu and A 1Πu on the one hand and X 1Σ+g and b 3Σg− on the other. First order wave functions are then derived by diagonalization. Einstein emission transition probabilities of the Intercombination lines are finally obtained.

Perturbation Methods

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Fine Structure ◽  
Dirac Equation ◽  
Perturbation Methods ◽  
Relativistic Quantum ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Perturbation Expansions ◽  
Relativistic Corrections

Perturbation theory has been one of the most frequently used and most powerful tools of quantum mechanics. The very foundations of relativistic quantum theory—quantum electrodynamics—are perturbative in nature. Many-body perturbation theory has been used for electron correlation treatments since the early days of quantum chemistry, and in more recent times multireference perturbation theories have been developed to provide quantitative or semiquantitative information in very complex systems. In the beginnings of relativistic quantum mechanics, perturbation methods based on an expansion in powers of the fine structure constant, α = 1/c, were used extensively to obtain operators that would provide a connection with nonrelativistic quantum mechanics and permit some evaluation of relativistic corrections, in days well before the advent of the computer. This seems a reasonable approach, considering the small size of the fine structure constant—and for light elements it has been found to work remarkably well. Relativity is a small perturbation for a good portion of the periodic table. Perturbation expansions have their limitations, however, and as well as successes, there have been failures due to the highly singular or unbounded nature of the operators in the perturbation expansions. Therefore, in recent times other perturbation approaches have been developed to provide alternatives to the standard Breit–Pauli approach. This chapter is devoted to the development of perturbation expansions in powers of 1/c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy–Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit–Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties.

scholarly journals SOLUTION OF THE DIRAC EQUATION WITH NONCENTRAL THREE-VECTOR POTENTIAL

2006 ◽  
Vol 21 (07) ◽  
pp. 581-592 ◽  
Author(s):  
A. D. ALHAIDARI
Keyword(s):  
Energy Spectrum ◽  
Dirac Equation ◽  
Vector Potential ◽  
Radial Component ◽  
Wave Functions ◽  
Relativistic Energy ◽  
Dirac Oscillator ◽  
Spherical Coordinates ◽  
The One ◽  
Spinor Wave

We introduce coupling to three-vector potential in the (3+1)-dimensional Dirac equation. The potential is noncentral (angular-dependent) such that the Dirac equation separates completely in spherical coordinates. The relativistic energy spectrum and spinor wave functions are obtained for the case where the radial component of the vector potential is proportional to 1/r. The coupling presented in this work is a generalization of the one which was introduced by Moshinsky and Szczepaniak for the Dirac-oscillator problem.

Charged spinor solitons

1986 ◽  
Vol 64 (3) ◽  
pp. 232-238 ◽  
Author(s):  
P. Mathieu ◽  
T. F. Morris
Keyword(s):  
Dirac Equation ◽  
Mass Parameter ◽  
Structure Constant ◽  
Finite Energy ◽  
Stationary Solutions ◽  
Fine Structure Constant ◽  
Many Body ◽  
Nonlinear Dirac Equation ◽  
Frequency Relation ◽  
Static Energy

A nonlinear Dirac equation for which all finite-energy stationary solutions are nontopological solitons with compact support is coupled to the electromagnetic field. In a many-body situation, it is shown that the equilibrium is reached when all the solitons have the same value of the charge. This implies the de Broglie frequency relation and a relation for the fine-structure constant. In specific domains and to a very good approximation, the model reduces to the linear Dirac equation for a particle whose mass parameter is the static energy of the soliton.

scholarly journals The quantum theory of the neutron

1934 ◽  
Vol 145 (854) ◽  
pp. 344-358 ◽  
Keyword(s):  
Wave Equation ◽  
Fine Structure ◽  
Quantum Theory ◽  
Hydrogen Atom ◽  
Ad Hoc ◽  
Structure Constant ◽  
Wave Functions ◽  
Fine Structure Constant ◽  
Second Order Wave Equation ◽  
Plausible Theory

The object of this paper is to show that a plausible theory of the neutron can be developed from Dirac's wave equation without the use of any ad hoc assumptions. It is shown that the second order wave equation of the hydrogen atom, which exhibits the relativistic and spin corrections, possesses two sets of solutions "H" and "N" distinguished by their behaviour as r →0 ( r being the distance of the electron from the proton). The H-solutions are the accepted wave functions of the hydrogen atom. As r →0 these solutions tend to zero if the serial quantum number l differs from zero, and they become infinite of order r [(1 - a 2 ) 1/2 - 1] if l = 0 (α is the fine structure constant).

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.

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