scholarly journals A SIMPLE DIRAC WAVE FUNCTION FOR A COULOMB POTENTIAL WITH LINEAR CONFINEMENT

1999 ◽  
Vol 14 (34) ◽  
pp. 2409-2411 ◽  
Author(s):  
JERROLD FRANKLIN
Keyword(s):  
Wave Function ◽  
Binding Energy ◽  
Dirac Equation ◽  
Coulomb Potential ◽  
Ground State ◽  
Linear Potential ◽  
Confining Potential ◽  
Lorentz Scalar ◽  
Dirac Wave Function ◽  
Scalar Part

A simple analytical solution is found to the Dirac equation for the combination of a Coulomb potential with a linear confining potential. An appropriate linear combination of Lorentz scalar and vector linear potentials, with the scalar part dominating, can be chosen to give a simple Dirac wave function. The binding energy depends only on the Coulomb strength and is not affected by the linear potential. The method works for the ground state, or for the lowest state with l=j-1/2, for any j.

scholarly journals EXACT SOLUTIONS OF THE DIRAC EQUATION FOR MODIFIED COULOMBIC POTENTIALS

2000 ◽  
Vol 15 (27) ◽  
pp. 4355-4360
Author(s):  
ANTONIO SOARES DE CASTRO ◽  
JERROLD FRANKLIN
Keyword(s):  
Dirac Equation ◽  
Exact Solutions ◽  
Linear Combination ◽  
Ground State ◽  
Wave Functions ◽  
Orbital State ◽  
Lorentz Scalar ◽  
Scalar Part

Exact solutions are found for the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions. The method works for the ground state or for the lowest orbital state with l=j-½, for any j.

Existence and asymptotics of ground states to the nonlinear Dirac equation with Coulomb potential

2021 ◽  
pp. 1-26
Author(s):  
Tianfang Wang ◽  
Wen Zhang ◽  
Jian Zhang
Keyword(s):  
Dirac Equation ◽  
Coulomb Potential ◽  
Ground State ◽  
Continuous Dependence ◽  
Energy Functional ◽  
Ground State Solution ◽  
State Solution ◽  
Nonlinear Dirac Equation ◽  
Relationship Of ◽  
Least Energy

In this paper we study the Dirac equation with Coulomb potential − i α · ∇ u + a β u − μ | x | u = f ( x , | u | ) u , x ∈ R 3 where a is a positive constant, μ is a positive parameter, α = ( α 1 , α 2 , α 3 ), α i and β are 4 × 4 Pauli–Dirac matrices. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions, we prove that the problem possesses a ground state solution which is exponentially decay, and the least energy has continuous dependence about μ. Moreover, we are able to obtain the asymptotic property of ground state solution as μ → 0 + , this result can characterize some relationship of the above problem between μ > 0 and μ = 0.

scholarly journals Approximate Relativistic Mass-Formulae for the Ground State Binding Energy of a Λ in Hypernuclei

2020 ◽  
Vol 1 ◽  
pp. 187
Author(s):  
C. G. Koutroulos ◽  
M. E. Grypeos
Keyword(s):  
Binding Energy ◽  
Dirac Equation ◽  
Ground State ◽  
Simplified Model ◽  
Ground State Binding Energy ◽  
Relativistic Mass

The Dirac equation with potentials having attractive and repulsive parts is con­ sidered in a simplified model and approximate semiempirical mass formulae for the ground state binding energy of a Λ in hypernuclei are derived and discussed.

Exact Numerical Procedure for the Binding Energy of Hydrogen Impurities in Quantum Dots with Parabolic Confinements

2013 ◽  
Vol 475-476 ◽  
pp. 1355-1358
Author(s):  
Arnold Abramov
Keyword(s):  
Quantum Dots ◽  
Wave Function ◽  
Binding Energy ◽  
Quantum Dot ◽  
Excited States ◽  
Ground State ◽  
Qualitative Agreement ◽  
Numerical Procedure ◽  
Impurity States ◽  
Parabolic Confinement

In this paper we present exact numerical procedure to calculate the binding energy and wave function of impurity states in a quantum dot with parabolic confinement. The developed method allows control the accuracy of obtained results, as well as calculates the characteristics of not only ground state, but also of the excited states. Comparison of our results with data obtained by other methods is in quantitative and qualitative agreement. We studied the effects of impurity position on the binding energy.

scholarly journals FOUR-VECTOR VERSUS FOUR-SCALAR REPRESENTATION OF THE DIRAC WAVE FUNCTION

2012 ◽  
Vol 09 (04) ◽  
pp. 1250026 ◽  
Author(s):  
MAYEUL ARMINJON ◽  
FRANK REIFLER
Keyword(s):  
Wave Function ◽  
Dirac Equation ◽  
Minkowski Spacetime ◽  
Bundle Theory ◽  
Curved Spacetime ◽  
Unified Framework ◽  
Tetrad Field ◽  
Dirac Equations ◽  
Dirac Wave Function ◽  
General Coordinate Transformation

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincaré group. This is not an option in a curved spacetime. Therefore, in the equation proposed independently by Fock and Weyl, the four complex components of the Dirac wave function transform as scalars under a general coordinate transformation. Recent work has shown that a covariant complex four-vector representation is also possible. Using notions of vector bundle theory, we describe these two representations in a unified framework. We prove theorems that relate together the different representations and the different choices of connections within each representation. As a result, either of the two representations can account for a variety of inequivalent, linear, covariant Dirac equations in a curved spacetime that reduce to the original Dirac equation in a Minkowski spacetime. In particular, we show that the standard Dirac equation in a curved spacetime, with any choice of the tetrad field, is equivalent to a particular realization of the covariant Dirac equation for a complex four-vector wave function.

TWO-CENTER PROBLEM FOR THE DIRAC EQUATION WITH A COULOMB AND SCALAR POTENTIAL

2007 ◽  
Vol 22 (18) ◽  
pp. 3123-3130
Author(s):  
V. V. BONDARCHUK ◽  
I. M. SHVAB ◽  
A. V. KATERNOGA
Keyword(s):  
Wave Function ◽  
Dirac Equation ◽  
Ground State ◽  
Scalar Potential ◽  
Scalar Coupling ◽  
Center Problem ◽  
Energy Term ◽  
Coupling Constant ◽  
Scalar Coupling Constant ◽  
Scalar Potentials

The ground state wave function and the energy term of a relativistic electron moving in the field of two fixed centers, when interaction of this particle with centers is described by two Coulomb and two Coulomb-like scalar potentials are calculated analytically by the LCAO method. Dependence of electron binding energy from value of scalar coupling constant was investigated using obtained analytic results.

scholarly journals THE EFFECT OF SCREENED COULOMB INTERACTION ON THE OPTICAL PROPERTIES OF EuS/PbS/EuS FINITE CONFINING POTENTIAL QUANTUM WELL

2012 ◽  
Vol 15 ◽  
pp. 224-231 ◽  
Author(s):  
K. H. AHARONYAN ◽  
E. M. KAZARYAN
Keyword(s):  
Optical Properties ◽  
Binding Energy ◽  
Quantum Well ◽  
Coulomb Potential ◽  
Band Edge ◽  
Lead Salt ◽  
Confining Potential ◽  
Mass Approximation ◽  
Excitonic Effects ◽  
Pair Energy

We theoretically investigate excitonic effects on the optical properties of quasi-two dimensional realistic EuS/PbS/EuS finite confining potential quantum well. The strong contrast of material parameter’s across the well interfaces and the characteristic band-nonparabolicity effect of lead salt semiconductor are taken into account. In presence of medium polarization a feasibility of the screened Coulomb potential in quantum well is examined and appropriate screening radius is revealed for the first time. The screened binding energy investigated while a variational approach has been used. Depend on the realistic nanostructure specifics a monotonic decrease of the enhanced binding energy is received when in-plane carrier density/temperature ratio parameter increases. Exciton absorption spectra near the band edge within the effective-mass approximation is examined. Coulomb potential having a cutoff type is used, that efficiently represented the potential in the realistic quantum well. Screened exciton factor, which is the absorption intensity enhancement of the unbound (continuum) exciton states that are located above the band edge, is used. Plot of the dependence of the exciton factor on the exciton pair energy is given.

MAGNETIC MOMENT μ OF Ω--BARYON USING RICHARDSON POTENTIAL IN A RELATIVISTIC HARTREE–FOCK (RHF) METHOD

2000 ◽  
Vol 15 (10) ◽  
pp. 683-693
Author(s):  
KANAD RAY ◽  
JISHNU DEY ◽  
MIRA DEY
Keyword(s):  
Wave Function ◽  
Magnetic Moment ◽  
Ground State ◽  
Detailed Structure ◽  
Hartree Fock ◽  
Free Part ◽  
Dirac Wave Function ◽  
Good Agreement

A RHF method is used to calculate magnetic moment μ of Ω- using the Richardson potential. Unlike the mass of baryon, μ is very sensitive to the detailed structure of the model since it depends on the wave function. Given any quark–quark (qq) potential we have the RHF wave function for the ground state. We have developed a code for expanding this Dirac wave function in terms of two sets of oscillators. To fit both the mass and magnetic moment simultaneously we find that it is essential to separate out the confining part from the asymptotically free part of the potential and use different parameters for each part. The best fitted results from our model are in good agreement with experiment.

BOUND STATES AND KLEIN PARADOX OF GRAPHENE SYSTEMS IN CONFINING POTENTIALS

2012 ◽  
Vol 26 (17) ◽  
pp. 1250108
Author(s):  
HAI HUANG ◽  
XIA HUANG
Keyword(s):  
Dirac Equation ◽  
Bound States ◽  
Bound State ◽  
Linear Potential ◽  
Two Dimensional ◽  
Potential Well ◽  
Klein Paradox ◽  
Confining Potential ◽  
Potential Depth

Two-dimensional massive Dirac equation in both potential well and linear potential is discussed. We find that in the case of potential well, the bound states disappear from the spectrum for large enough potential depth. With the linear confining potential, we show that the Dirac equation presents no bound state. Both these results can be identified as fine examples of the Klein paradox. Applications to graphene systems are also discussed.

PHENOMENOLOGICAL RELATIVISTIC STUDY OF THE ENERGY OF A Λ IN ITS GROUND AND EXCITED STATES IN HYPERNUCLEI

1994 ◽  
Vol 03 (03) ◽  
pp. 939-951
Author(s):  
G.J. PAPADOPOULOS ◽  
C.G. KOUTROULOS ◽  
M.E. GRYPEOS
Keyword(s):  
Wave Function ◽  
Binding Energy ◽  
Dirac Equation ◽  
Least Squares ◽  
Excited States ◽  
Energy Eigenvalue ◽  
Bound State ◽  
Small Component ◽  
Repulsive Potentials ◽  
Analytic Expressions

The binding energy BΛ of a Λ-particle in hypernuclei is studied by means of the Dirac equation with attractive and repulsive potentials of rectangular shapes and of the same radius. The energy eigenvalue equation in this case is obtained analytically for every bound state, as well as the large and small component of the wave function (for given BΛ). A first attempt is also made to investigate the possibility of deriving in particular cases approximate analytic expressions of BΛ for the excited states. Using various least squares fittings, numerical results for the binding energy of the Λ are given and comparisons and comments are also made.

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