NUMERICAL INVESTIGATION OF BOUND RELATIVISTIC TWO-PARTICLE SYSTEMS WITH ONE-BOSON EXCHANGE POTENTIAL Yu.A. Grishechkin and V.N. Kapshai Francisk Skorina.

Slides:



Advertisements
Similar presentations
Schrödinger Representation – Schrödinger Equation
Advertisements

The Quantum Mechanics of Simple Systems
Molecular Quantum Mechanics
Introduction to Molecular Orbitals
Quantum One: Lecture 3. Implications of Schrödinger's Wave Mechanics for Conservative Systems.
1 Cold molecules Mike Tarbutt. 2 Outline Lecture 1 – The electronic, vibrational and rotational structure of molecules. Lecture 2 – Transitions in molecules.
Photodisintegration of in three dimensional Faddeev approach The 19th International IUPAP Conference on Few-Body Problems in Physics S. Bayegan M. A. Shalchi.
The Klein-Gordon Equation
P460 - Spin1 Spin and Magnetic Moments (skip sect. 10-3) Orbital and intrinsic (spin) angular momentum produce magnetic moments coupling between moments.
P460 - Spin1 Spin and Magnetic Moments Orbital and intrinsic (spin) angular momentum produce magnetic moments coupling between moments shift atomic energies.
Overview of QM Translational Motion Rotational Motion Vibrations Cartesian Spherical Polar Centre of Mass Statics Dynamics P. in Box Rigid Rotor Spin Harmonic.
Chap 12 Quantum Theory: techniques and applications Objectives: Solve the Schrödinger equation for: Translational motion (Particle in a box) Vibrational.
Quantum Mechanics Classical – non relativistic Quantum Mechanical : Schrodinger eq.
Overview of QM Translational Motion Rotational Motion Vibrations Cartesian Spherical Polar Centre of Mass Statics Dynamics P. in Box Rigid Rotor Angular.
Physics 452 Quantum mechanics II Winter 2011 Karine Chesnel.
QM in 3D Quantum Ch.4, Physical Systems, 24.Feb.2003 EJZ Schrödinger eqn in spherical coordinates Separation of variables (Prob.4.2 p.124) Angular equation.
The Klein Gordon equation (1926) Scalar field (J=0) :
C4 Lecture 3 - Jim Libby1 Lecture 3 summary Frames of reference Invariance under transformations Rotation of a H wave function: d -functions Example: e.
Classical Model of Rigid Rotor
Bound-Free Electron-Positron Pair Production Accompanied by Coulomb Dissociation M. Yılmaz Şengül Kadir Has University & İstanbul Technical University.
The Harmonic Oscillator
P460 - Spin1 Spin and Magnetic Moments (skip sect. 10-3) Orbital and intrinsic (spin) angular momentum produce magnetic moments coupling between moments.
Density Matrix Density Operator State of a system at time t:
6. Second Quantization and Quantum Field Theory
Monte Carlo Simulation of Interacting Electron Models by a New Determinant Approach Mucheng Zhang (Under the direction of Robert W. Robinson and Heinz-Bernd.
Particles (matter) behave as waves and the Schrödinger Equation 1. Comments on quiz 9.11 and Topics in particles behave as waves:  The (most.
Atomic Orbitals, Electron Configurations, and Atomic Spectra
Cross section for potential scattering
Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel.
Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel.
A study of two-dimensional quantum dot helium in a magnetic field Golam Faruk * and Orion Ciftja, Department of Electrical Engineering and Department of.
Electronic Structure of 3D Transition Metal Atoms Christian B. Mendl M7 (Gero Friesecke) TU München DMV Jahrestagung March 10, 2010 DMV Jahrestagung March.
Maksimenko N.V., Vakulina E.V., Deryuzkova О.М. Kuchin S.М. GSU, GOMEL The Amplitude of the Сompton Scattering of the Low-Energy Photons at Quark-Antiquark.
1 Noncommutative QCDCorrections to the Gluonic Decays of Heavy Quarkonia Stefano Di Chiara A. Devoto, S. Di Chiara, W. W. Repko, Phys. Lett. B 588, 85.
1 MODELING MATTER AT NANOSCALES 4. Introduction to quantum treatments The variational method.
28.3 THE BOHR THEORY OF HYDROGEN At the beginning of the 20th century, scientists were puzzled by the failure of classical physics to explain the characteristics.
Quantum Mechanical Cross Sections In a practical scattering experiment the observables we have on hand are momenta, spins, masses, etc.. We do not directly.
1 Lattice Quantum Chromodynamics 1- Literature : Lattice QCD, C. Davis Hep-ph/ Burcham and Jobes By Leila Joulaeizadeh 19 Oct
A. Ambrosetti, F. Pederiva and E. Lipparini
1 Methods of Experimental Particle Physics Alexei Safonov Lecture #2.
1 Reading: QM course packet Ch 5 up to 5.6 SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES.
Klein-Gordon Equation in the Gravitational Field of a Charged Point Source D.A. Georgieva, S.V. Dimitrov, P.P. Fiziev, T.L. Boyadjiev Gravity, Astrophysics.
Bethe-Salper equation and its applications Guo-Li Wang Department of Physics, Harbin Institute of Technology, China.
Resonance states of relativistic two-particle systems Kapshai V.N., Grishechkin Yu.A. F. Scorina Gomel State University.
The Hydrogen Atom The only atom that can be solved exactly.
A NALYTICAL S OLUTION THE R ADIAL S CHRÖDINGER E QUATION FOR THE Q UARK - A NTIQUARK S YSTEM S. M. Kuchin The I.G. Petrovsky Bryansk State University N.
Microwave Spectroscopy and Internal Dynamics of the Ne-NO 2 Van der Waals Complex Brian J. Howard, George Economides and Lee Dyer Department of Chemistry,
ELECTROMAGNETIC PARTICLE: MASS, SPIN, CHARGE, AND MAGNETIC MOMENT Alexander A. Chernitskii.
LORENTZ AND GAUGE INVARIANT SELF-LOCALIZED SOLUTION OF THE QED EQUATIONS I.D.Feranchuk and S.I.Feranchuk Belarusian University, Minsk 10 th International.
Lecture 4 – Quantum Electrodynamics (QED)
Fundamental principles of particle physics Our description of the fundamental interactions and particles rests on two fundamental structures :
The Hydrogen Atom The only atom that can be solved exactly.
Schrödinger Representation – Schrödinger Equation
Lecture 3 The Schrödinger equation
Open quantum systems.
NGB and their parameters
Concept test 15.1 Suppose at time
Concept test 15.1 Suppose at time
Solutions of the Schrödinger equation for the ground helium by finite element method Jiahua Guo.
Spin and Magnetic Moments (skip sect. 10-3)
Quantum Two.
Peng Wang Sichuan University
Quantum Theory.
Time-Dependent Perturbation Theory
Do all the reading assignments.
Adaptive Perturbation Theory: QM and Field Theory
Quantum Theory.
Nonlinear response of gated graphene in a strong radiation field
It means anything not quadratic in fields and derivatives.
Quantum Theory.
Presentation transcript:

NUMERICAL INVESTIGATION OF BOUND RELATIVISTIC TWO-PARTICLE SYSTEMS WITH ONE-BOSON EXCHANGE POTENTIAL Yu.A. Grishechkin and V.N. Kapshai Francisk Skorina Gomel State University

Two-particle equations of quasipotential type Integral equations for bound s-states in the relativistic configurational representation (RCR) One-boson exchange potentials and their superposition Numerical method of solving Spectrum of the orthopositronium Decay width of the parapositronium Plan of the talk

Two-particle equations of quasipotential type − Logunov-Tavkhelidze equation − Kadyshevsky equation − Logunov-Tavkhelidze modified equation − Kadyshevsky modified equation 2E – two-particle system energy Integral equations for bound states in the momentum representation

Relativistic configurational representation RCR is as expansion over functions Transformations of the two-particle wave function of relative motion – from the momentum representation to the RCR – from the RCR to the momentum representation For s-states this transformation analogous the Fourier transformation where χ – is the rapidity, connected with momentum by relation

Integral two-particle equations for bound s-states in the RCR j = 1 Green functions in the RCR: j = 2 j = 3 j = 4 − Logunov-Tavkhelidze equation − Kadyshevsky equation − Logunov-Tavkhelidze modified equation − Kadyshevsky modified equation r – is radius-vector modulus in the RCR – normalization condition of wave function

Two-particle equations for bound s-states in the momentum representation Logunov-Tavkhelidze equation: Kadyshevsky equation: Normalization conditions of wave functions:

One-boson exchange potentials in the momentum representation One of the first one-boson exchange potentials for two scalar particles obtained in framework quasipotential approach is This potential was obtained on the basis of diagram technique of the quantum field theory Hamiltonian formulation [1]. Mass of exchange boson is equal to zero. For s-states this potential has the form [1] Kadyshevsky V.G. Quasipotential type equation for the relativistic scattering amplitude / Nucl. Phys. – 1968.– V.B6, №1. – P

In article [2] one-boson exchange potentials were obtained on the basis of retarded and causal Green functions calculation. where [2] Капшай В.Н., Саврин В.И., Скачков Н.Б. О зависимости квазипотенциала от полной энергии двухчастичной системы / – ТМФ, – Т.69, №3. – С

Potentials of two fermions interaction in the cases of different total spin and total angular momentum values were obtained in article [3] as coefficient of general three-dimensional potential for two fermions [4] expansion into the spherical spinors. Total spin of system is equal to zero: [3] Двоеглазов В.В, Скачков Н.Б., Тюхтяев Ю.Н., Худяков С.В. Релятивистские парциальные интегральные уравнения для волновой функции системы двух фермионов / – ЯФ, – Т.54, №3. – С [4] Архипов А.А. Приближение одноглюонного обмена для квазипотенциала взаимодействия двух кварков в квантовой хромодинамике / – ТМФ, – Т.83, №3. – С

Total spin of system is equal to one: where

Potentials in the RCR One-boson exchange potential Parameter α is associated with mass of exchange boson µ Mass of exchange boson is equal to zero Superposition of two one-boson exchange potentials with masses are equal to zero and 2m Coulomb potential in the RCR

Numerical method of equations solving in the momentum representation For solving of equations the rectangles quadrature method was used after replaced the variable Nonlinear energy eigenvalue problem for algebraic equation systems. For solving this problem the iteration method was used [5]. Initial value energy 2E (0) [5] T.M. Solov’eva Numerical Calculation of the Energy Spectrum of a Two-Fermion System / Comp. Phys. Comm., 136 (2001), p To find the eigenvalues of the matrices one can use standard methods. Process has to be continued until the condition holds, ε – accuracy. Then, the Richardson extrapolation process is applied to energy values 2E, and normalized wave functions, obtained on two grids N and 2N.

Numerical method of equations solving in the RCR To find solutions of integral equations in the RCR we used the composite Gauss quadrature method. Nonlinear energy eigenvalue problem for algebraic equation systems. For solving this problem the iteration method was used too. Initial value w (0) (energy 2E (0) =2m cos(w (0) ) ) Process has to be continued until the condition holds, ε – accuracy.

Results of equations solving in the momentum representation Wave functions for Logunov-Tavkhelidze equation for potential Coupling constant is equal to the fine structure constant

Results for spectrum

Convergence of the energy spectrum for scalar particles

Experimental date for frequency of transition from the ground state to the first excited for orthopositronium: (3.2)МHz.

Results for spectrum Method was tested for solving modified Logunov-Tavkhelidse equation in the case of potential Exact expression for quantization condition of energy Results of equations solving in the RCR

Experimental date for frequency of transition from the ground state to the first excited for orthopositronium: (3.2)МHz. Frequency of transition obtained by solving the Schrödinger equation with the Coulomb potential: MHz. Frequency of transition obtained from the quantization condition 2E n =(4m 2 -λ 2 /n 2 ) 1/2 : MHz. The solution of the Kadyshevsky equation with the potential V(r)=tanh(πmr/2)/r gives the best agreement with experimental data.

Decay width for two-particle systems The decay width for systems of two scalar particles into two photons [6]: [6] Г.А. Козлов О распаде связанного состояния µ + µ - -пары в e + e - далитц-пару и γ - квант / ТМФ, 60 (1984), №1, с

The substitution of non-relativistic wave function for Coulomb potential into this formula gives value: The substitution of exact wave function for potential gives value: The experimental value for parapositronium:

Conclusions Our methods allows to solve effectively two-particle integral equations in the momentum representation and in the RCR for slowly decreasing potentials like Coulomb potential The wave functions in the RCR allow to calculate decay width simply The results of numerical solution for energy spectrum and decay width in the case of one-boson exchange potential and superposition of such potentials give good agreement with the experimental values for positronium. Herewith results for phenomenological potentials coincide with experimental values better then results for potentials which derived strongly.