Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThree Body Bound State Calculations without Angular Momentum Decomposition
209
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The Faddeev equations for the three body bound state are solved directly as three dimensional integral equation without employing partial wave decomposition. The numerical stability of the algorithm is demonstrated. The three body binding energy is calculated for Malfliet-Tjon type potentials and compared with results obtained from calculations based on partial wave decomposition. The full three body wave function is calculated as function of the vector Jacobi momenta. It is shown that it satisfies the Schrodinger equation with high accuracy. The properties of the full wave function are displayed and compared to the ones of the corresponding wave functions obtained as finite sum of partial wave components. The agreement between the two approaches is essentially perfect in all respects.
rate research
Read More
A method to calculate the bound states of three-atoms without resorting to an explicit partial wave decomposition is presented. The differential form of the Faddeev equations in the total angular momentum representation is used for this purpose. The method utilizes Cartesian coordinates combined with the tensor-trick preconditioning for large linear systems and Arnoldis algorithm for eigenanalysis. As an example, we consider the He$_3$ system in which the interatomic force has a very strong repulsive core that makes the three-body calculations with standard methods tedious and cumbersome requiring the inclusion of a large number of partial waves. The results obtained compare favorably with other results in the field.
In this paper, we study the relativistic effects in a three-body bound state. For this purpose, the relativistic form of the Faddeev equations is solved in momentum space as a function of the Jacobi momentum vectors without using a partial wave decomposition. The inputs for the three-dimensional Faddeev integral equation are the off-shell boost two-body $t-$matrices, which are calculated directly from the boost two-body interactions by solving the Lippmann-Schwinger equation. The matrix elements of the boost interactions are obtained from the nonrelativistic interactions by solving a nonlinear integral equation using an iterative scheme. The relativistic effects on three-body binding energy are calculated for the Malfliet-Tjon potential. Our calculations show that the relativistic effects lead to a roughly 2% reduction in the three-body binding energy. The contribution of different Faddeev components in the normalization of the relativistic three-body wave function is studied in detail. The accuracy of our numerical solutions is tested by calculation of the expectation value of the three-body mass operator, which shows an excellent agreement with the relativistic energy eigenvalue.
Firstly, a systematic procedure is derived for obtaining three-dimensional bound-state equations from four-dimensional ones. Unlike ``quasi-potential approaches this procedure does not involve the use of delta-function constraints on the relative four-momentum. In the absence of negative-energy states, the kernels of the three-dimensional equations derived by this technique may be represented as sums of time-ordered perturbation theory diagrams. Consequently, such equations have two major advantages over quasi-potential equations: they may easily be written down in any Lorentz frame, and they include the meson-retardation effects present in the original four-dimensional equation. Secondly, a simple four-dimensional equation with the correct one-body limit is obtained by a reorganization of the generalized ladder Bethe-Salpeter kernel. Thirdly, our approach to deriving three-dimensional equations is applied to this four-dimensional equation, thus yielding a retarded interaction for use in the three-dimensional bound-state equation of Wallace and Mandelzweig. The resulting three-dimensional equation has the correct one-body limit and may be systematically improved upon. The quality of the three-dimensional equation, and our general technique for deriving such equations, is then tested by calculating bound-state properties in a scalar field theory using six different bound-state equations. It is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four-dimensional equations significantly better than the corresponding quasi-potential equations do.
Recently a formalism for a direct treatment of the Faddeev equation for the three-nucleon bound state in three dimensions has been proposed. It relies on an operator representation of the Faddeev component in the momentum space and leads to a finite set of coupled equations for scalar functions which depend only on three variables. In this paper we provide further elements of this formalism and show the first numerical results for chiral NNLO nuclear forces.
We use a sequential $R$-matrix model to describe the breakup of the Hoyle state into three $alpha$ particles via the ground state of $^8mathrm{Be}$. It is shown that even in a sequential picture, features resembling a direct breakup branch appear in the phase-space distribution of the $alpha$ particles. We construct a toy model to describe the Coulomb interaction in the three-body final state and its effects on the decay spectrum are investigated. The framework is also used to predict the phase-space distribution of the $alpha$ particles emitted in a direct breakup of the Hoyle state and the possibility of interference between a direct and sequential branch is discussed. Our numerical results are compared to the current upper limit on the direct decay branch determined in recent experiments.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
