scholarly journals ACCURACY OF APPROXIMATE EIGENSTATES

2000 ◽  
Vol 15 (20) ◽  
pp. 3221-3235 ◽  
Author(s):  
WOLFGANG LUCHA ◽  
FRANZ F. SCHÖBERL
Keyword(s):  
Eigenvalue Problem ◽  
Mass Spectra ◽  
Bound States ◽  
Main Tool ◽  
Bound State ◽  
Variational Techniques ◽  
The Matrix ◽  
Reasonable Choice ◽  
Unperturbed Solution ◽  
New Criteria

Besides perturbation theory, which requires the knowledge of the exact unperturbed solution, variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by the variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to that Hamiltonian the eigenvalue problem of which defines the spinless Salpeter equation. This bound-state wave equation may be regarded as the most straightforward relativistic generalization of the usual nonrelativistic Schrödinger formalism, and is frequently used to describe, e.g. spin-averaged mass spectra of bound states of quarks.

scholarly journals Energy levels in muonic helium

2019 ◽  
Vol 222 ◽  
pp. 03009
Author(s):  
A.V. Eskin ◽  
V.I. Korobov ◽  
A.P. Martynenko ◽  
V.V. Sorokin
Keyword(s):  
Hyperfine Structure ◽  
Bound States ◽  
Energy Levels ◽  
Computer Code ◽  
Bound State ◽  
Relativistic Correction ◽  
Matrix Elements ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The energy spectrum of bound states and hyperfine structure of muonic helium is calculated on the basis of stochastic variational method. The basis wave functions of muonic helium are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation a computer code is written in the MATLAB system. As a result, numerical values of bound state energies and hyperfine structure are obtained. We calculate also correction to the structure of the nucleus, vacuum polarization and relativistic correction.

An approach to heavy quarkonium spectra

2014 ◽  
Vol 29 (32) ◽  
pp. 1450170 ◽  
Author(s):  
Y. Cançelik ◽  
B. Gönül
Keyword(s):  
Eigenvalue Problem ◽  
Coulomb Potential ◽  
Mass Spectra ◽  
Bound State ◽  
Considerable Effect ◽  
Heavy Quarkonia ◽  
Heavy Quarkonium ◽  
Related Literature ◽  
Exactly Solvable ◽  
Math Phys

An application of the recently introduced method [M. Çapak et al., J. Math. Phys. 52, 102102 (2011)] to the bound-state eigenvalue problem in the elementary quarkonium potential V(r) = -a/r + br + cr2 is described, proved and illustrated for [Formula: see text] and [Formula: see text] systems. The quasi- and conditionally-exactly solvable spin-averaged mass spectra of heavy quarkonia are obtained in compact forms. The comparison of the present predictions with those of other theories in the related literature, together with the available data, has shown the success of the model used in this work and also revealed that the use of different confinings in the perturbed Coulomb potential descriptions has no considerable effect on the mass spectra of such systems.

scholarly journals Closed form bound-state perturbation theory

1980 ◽  
Vol 3 (2) ◽  
pp. 351-368 ◽  
Author(s):  
Ollie J. Rose ◽  
Carl G. Adler
Keyword(s):  
Perturbation Theory ◽  
Eigenvalue Problem ◽  
Closed Form ◽  
Bound States ◽  
Integral Form ◽  
Approximate Solutions ◽  
Perturbation Parameter ◽  
Bound State ◽  
The Given ◽  
Natural Way

The perturbed Schrödinger eigenvalue problem for bound states is cast into integral form using Green's Functions. A systematic algorithm is developed and applied to the resulting equation giving rise to approximate solutions expressed as functions of the given perturbation parameter. As a by-product, convergence radii for the traditional Rayleigh-Schrödinger and Brillouin-Wigner perturbation theories emerge in a natural way.

scholarly journals INSTANTANEOUS BETHE–SALPETER EQUATION: IMPROVED ANALYTICAL SOLUTION

2002 ◽  
Vol 17 (16) ◽  
pp. 2233-2240 ◽  
Author(s):  
WOLFGANG LUCHA ◽  
FRANZ F. SCHÖBERL
Keyword(s):  
Analytical Solution ◽  
Ground State ◽  
Bound States ◽  
Rest Frame ◽  
Matrix Representation ◽  
Bound State ◽  
High Accuracy ◽  
The Matrix

Studying the Bethe–Salpeter formalism for interactions instantaneous in the rest-frame of the bound states described, we show that, for bound-state constituents of arbitrary masses, the mass of the ground state of a given spin may be calculated almost entirely analytically with high accuracy, without the (numerical) diagonalization of the matrix representation obtained by expansion of the solutions over a suitable set of basis states.

scholarly journals Bound states of dtμ, pdμ and t pμ mesomolecules

2019 ◽  
Vol 204 ◽  
pp. 05006 ◽  
Author(s):  
A. V. Eskin ◽  
V. I. Korobov ◽  
A. P. Martynenko ◽  
V. V. Sorokin
Keyword(s):  
Numerical Calculation ◽  
Bound States ◽  
Computer Code ◽  
Bound State ◽  
Wave Functions ◽  
Matrix Elements ◽  
Muonic Molecule ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The energy spectrum of excited bound states of muonic molecules ptμ, pdμ, and dtμ is calculated on the basis of the stochastic variational method. The basis wave functions of the muonic molecule are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation, a computer code was written in the MATLAB system. As a result, the numerical values of bound state energies for excited P-states of muonic molecules ptμ, pdμ and dtμ were obtained.

SEARCH FOR K−pn AND K−pp BOUND STATES AT SPRING-8/LEPS

2008 ◽  
Vol 23 (27n30) ◽  
pp. 2544-2547 ◽  
Author(s):  
◽  
J. D. PARKER
Keyword(s):  
Mass Spectra ◽  
Bound States ◽  
Bound State ◽  
Liquid Hydrogen ◽  
Missing Mass ◽  
Upper Limits ◽  
State Production

K−pn and K−pp bound states are searched for at SPring-8/LEPS in the d(γ, K+) and d(γ, K+π−) reactions with photon energies from 1.5 to 2.4 GeV. The data was collected during the 2002-2003 run period utilizing liquid hydrogen and liquid deuterium targets. Using the K+ and K+π− missing mass spectra from the deuteron, we hope to determine upper limits for K−pn and K−pp bound state production.

scholarly journals Bound states of a Klein–Gordon particle in the presence of a smooth potential well

2020 ◽  
Vol 35 (23) ◽  
pp. 2050140
Author(s):  
Eduardo López ◽  
Clara Rojas
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Bound State ◽  
Potential Well ◽  
One Dimensional ◽  
Smooth Potential ◽  
Klein Gordon ◽  
The One ◽  
Potential Parameters ◽  
Klein Gordon Equation

We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.

RETARDATION EFFECTS IN COVARIANT THREE-DIMENSIONAL BOUND STATE WAVE FUNCTIONS

2005 ◽  
Vol 14 (06) ◽  
pp. 931-947 ◽  
Author(s):  
F. PILOTTO ◽  
M. DILLIG
Keyword(s):  
Bound States ◽  
Three Dimensional ◽  
Bound State ◽  
Relative Energy ◽  
Wave Functions ◽  
Three Dimensions ◽  
State Wave ◽  
Scalar Diquark ◽  
Bound State Wave Function ◽  
Retardation Effects

We investigate the influence of retardation effects on covariant 3-dimensional wave functions for bound hadrons. Within a quark-(scalar) diquark representation of a baryon, the four-dimensional Bethe–Salpeter equation is solved for a 1-rank separable kernel which simulates Coulombic attraction and confinement. We project the manifestly covariant bound state wave function into three dimensions upon integrating out the non-static energy dependence and compare it with solutions of three-dimensional quasi-potential equations obtained from different kinematical projections on the relative energy variable. We find that for long-range interactions, as characteristic in QCD, retardation effects in bound states are of crucial importance.

THE NEXT STEP IN LEPTONIC DECAYS OF ETA AND KL BOUND STATES

1992 ◽  
Vol 07 (09) ◽  
pp. 1935-1951 ◽  
Author(s):  
G.A. KOZLOV
Keyword(s):  
Field Theory ◽  
Transition Form Factor ◽  
Bound States ◽  
Three Dimensional ◽  
Bound State ◽  
Quantum Field ◽  
Relative Momentum ◽  
Transition Form ◽  
Structure Transition ◽  
Leptonic Decays

A systematic discussion of the probability of eta and KL bound-state decays—[Formula: see text] and [Formula: see text](l=e, μ)—within a three-dimensional reduction to the two-body quantum field theory is presented. The bound-state vertex function depends on the relative momentum of constituent-like particles. A structure-transition form factor is defined by a confinement-type quark-antiquark wave function. The phenomenology of this kind of decays is analyzed.

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