scholarly journals Quadratic algebra approach to relativistic quantum Smorodinsky–Winternitz systems

2011 ◽  
Vol 52 (4) ◽  
pp. 042301 ◽  
Author(s):  
Ian Marquette
Keyword(s):  
Relativistic Quantum ◽  
Quadratic Algebra ◽  
Algebra Approach

Solution of the Dirac equation with some superintegrable potentials by the quadratic algebra approach

2014 ◽  
Vol 29 (06) ◽  
pp. 1450028 ◽  
Author(s):  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Quadratic Algebra ◽  
Two Dimensional ◽  
Oscillator Algebra ◽  
Quadratic Algebras ◽  
Energy Eigenvalues ◽  
Vector Potentials ◽  
Algebra Approach ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

The Dirac equation with scalar and vector potentials of equal magnitude is considered. For the two-dimensional harmonic oscillator superintegrable potential, the superintegrable potentials of E8 (case (3b)), S4 and S2, the Schrödinger-like equations are studied. The quadratic algebras of these quasi-Hamiltonians are derived. By using the realization of the quadratic algebras in a deformed oscillator algebra, the structure function and the energy eigenvalues are obtained.

scholarly journals Klein-Gordon Equation with Superintegrable Systems: Kepler-Coulomb, Harmonic Oscillator, and Hyperboloid

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
V. Mohammadi ◽  
S. Aghaei ◽  
A. Chenaghlou
Keyword(s):  
Harmonic Oscillator ◽  
Coulomb Potential ◽  
Gordon Equation ◽  
Spin Symmetry ◽  
Quadratic Algebra ◽  
Relativistic Energy ◽  
Algebra Approach ◽  
Superintegrable Systems ◽  
Klein Gordon ◽  
Klein Gordon Equation

We study the two-dimensional Klein-Gordon equation with spin symmetry in the presence of the superintegrable potentials. On Euclidean space, theSO(3)group generators of the Schrödinger-like equation with the Kepler-Coulomb potential are represented. In addition, by Levi-Civita transformation, the Schrödinger-like equation with harmonic oscillator which is dual to the Kepler-Coulomb potential and theSU(2)group generators of associated system are studied. Also, we construct the quadratic algebra of the hyperboloid superintegrable system. Then, we obtain the corresponding Casimir operators and the structure functions and the relativistic energy spectra of the corresponding quasi-Hamiltonians by using the quadratic algebra approach.

Introduction to Relativistic Quantum Chemistry

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Dirac Equation ◽  
Quantum Chemistry ◽  
Relativistic Theory ◽  
Density Functional ◽  
Relativistic Quantum ◽  
Relativistic Effects ◽  
Basis Sets ◽  
Spin Orbit ◽  
Approximate Methods ◽  
Nonrelativistic Theory

This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.

Quantum mechanics

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Mechanics ◽  
Relativistic Quantum ◽  
Transition Amplitude ◽  
External Source ◽  
Quantum Field ◽  
Damping Term ◽  
Relativistic Quantum Theory ◽  
Generating Functional

After a brief review of the operator approach to quantum mechanics, Feynmans path integral, which expresses a transition amplitude as a sum over all paths, is derived. Adding a linear coupling to an external source J and a damping term to the Lagrangian, the ground-state persistence amplitude is obtained. This quantity serves as the generating functional Z[J] for n-point Green functions which are the main target when studying quantum field theory. Then the harmonic oscillator as an example for a one-dimensional quantum field theory is discussed and the reason why a relativistic quantum theory should be based on quantum fields is explained.

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