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.

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 Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method

BIOMATH ◽  
2017 ◽  
Vol 6 (1) ◽  
pp. 1706047 ◽  
Author(s):  
Burcu Gürbüz ◽  
Mehmet Sezer
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Equation System ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Laguerre Series ◽  
The Matrix ◽  
The One

In this work, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. Diffusion models form a reasonable basis for studying insect and animal dispersal and invasion, which arise from the question of persistence of endangered species, biodiversity, disease dynamics, multi-species competition so on. Convection diffusion problem is also a form of heat and mass transfer in biological models. The presented method is based on the Laguerre collocation method used for these problems of differential equations.In fact, the approximate solution of the problem in the truncated Laguerre series form is obtained by this method. By substituting truncated Laguerre series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Laguerre coecients can be computed. The accuracy and the efficiency of the method is showed by numerical examples and the comparisons by the other methods.

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.

Orbit–orbit interaction inn fN electron configurations

1969 ◽  
Vol 47 (17) ◽  
pp. 1885-1888 ◽  
Author(s):  
K. M. S. Saxena ◽  
G. Malli
Keyword(s):  
Orbit Interaction ◽  
Matrix Elements ◽  
Atomic Systems ◽  
Hartree Fock ◽  
The Matrix

The expressions of the matrix elements of the orbit–orbit interaction for various fN electron configurations are computed and tabulated for general usage. These expressions are used to evaluate the Hartree–Fock values of the orbit–orbit interaction in all the states for a large number of fN electron atomic systems.

scholarly journals A factorisation-aware Matrix element emulator

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
D. Maître ◽  
H. Truong
Keyword(s):  
Neural Network ◽  
Phase Space ◽  
Matrix Element ◽  
Percent Level ◽  
Matrix Elements ◽  
Leading Order ◽  
Jet Production ◽  
The Neural Network ◽  
The Matrix ◽  
The One

Abstract In this article we present a neural network based model to emulate matrix elements. This model improves on existing methods by taking advantage of the known factorisation properties of matrix elements. In doing so we can control the behaviour of simulated matrix elements when extrapolating into more singular regions than the ones used for training the neural network. We apply our model to the case of leading-order jet production in e+e− collisions with up to five jets. Our results show that this model can reproduce the matrix elements with errors below the one-percent level on the phase-space covered during fitting and testing, and a robust extrapolation to the parts of the phase-space where the matrix elements are more singular than seen at the fitting stage.

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.

Electromagnetic fields in the presence of cylindrical objects of arbitrary physical properties and cross sections

1970 ◽  
Vol 48 (15) ◽  
pp. 1789-1798 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Physical Properties ◽  
Electromagnetic Fields ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Scattering Matrix ◽  
Two Dimensional ◽  
Matrix Elements ◽  
Approximate Evaluation ◽  
Permittivity And Permeability ◽  
The Matrix

Approximate solutions for two-dimensional problems of electromagnetic fields in the presence of cylindrical objects have been found by approximate evaluation of a scattering matrix. The equations are derived for cylindrical objects of arbitrary physical properties and cross sections and a procedure for evaluation of the matrix elements is discussed. The elements of permittivity and permeability tensors are assumed to be analytic, but otherwise arbitrary functions of the transverse coordinates.

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.

Nuclear structure in a hierarchy of ideal spaces

1981 ◽  
Vol 59 (11) ◽  
pp. 1670-1673
Author(s):  
M. Banville ◽  
P.-A. Simard
Keyword(s):  
Nuclear Structure ◽  
Arbitrary Number ◽  
Ideal Space ◽  
Hamiltonian Matrix ◽  
Matrix Elements ◽  
Fractional Parentage ◽  
The Matrix ◽  
Nuclear Structure Calculations ◽  
The One ◽  
Nuclear Shell

A new method permitting nuclear structure calculations for a system with an arbitrary number of fermions in an arbitrary number of subshells is developed through a generalization of the ideal space concept used in the boson methods. The nuclear shell problem is transcribed into a hierarchy of ideal spaces; the one-to-one correspondence between the states in each ideal space permits the generation of complete bases of antisymmetric states. The Hamiltonian matrix elements for the system are given. A generalization of the fractional parentage coefficients for such systems is obtained. The symmetries of those coefficients lead to a very important reduction in the complexity of the matrix elements.

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