On the variational methods for bound-state and scattering problems

Abdel‐Raouf, M. A.

doi:10.1016/0370-1573(82)90051-5

Cited by 46 publications

(6 citation statements)

References 334 publications

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“…The coexistence of antihydrogen with hydrogen, deuterium and tritium was mathematically proved by Abdel-Raouf and Ladik [2] based on the theory of four-body system (see [3], [4] and [5]). A computational code was developed for performing quite elaborate calculations for four-body systems using Ritz' variational method, (for a review, see Abdel-Raouf [6]). The code was employed by Abdel-Raouf et al [7] in order to confirm the existence of exotic four-body molecules.…”

Section: Introductionmentioning

confidence: 99%

Novel Consequences of Coexistence of Matter and Antimatter in Nature

Abdel‐Raouf¹

2020

JHEPGC

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In the present letter we discuss novel possible applications and implications of the formation of exotic atomic and molecular structures composed of particles and antiparticles. Particularly, we argue that huge amount of energy could be produced as a result of matter-antimatter cold fusion. Crucial questions raised concerning the fate of particles and antiparticles produced by the big bang are addressed. Assumptions of possible existence of two kinds of gravity and masses of different signs are proposed.

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Section: Introductionmentioning

confidence: 99%

Novel Consequences of Coexistence of Matter and Antimatter in Nature

Abdel‐Raouf¹

2020

JHEPGC

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“…is usually solved variationally [24][25][26][27][28]. The approach is straightforward in the case of the ground (n ¼ 0) state, or the state of lowest energy of a given symmetry, be the variation linear, nonlinear, or coupled.…”

Section: Introductionmentioning

confidence: 99%

“…For approximations to stationary states and energies, the Schrödinger energy eigenvalue problem is usually solved variationally24–28. The approach is straightforward in the case of the ground ( n = 0) state, or the state of lowest energy of a given symmetry, be the variation linear, nonlinear, or coupled.…”

Section: Introductionmentioning

confidence: 99%

Near‐exact supersymmetric partner potentials: Construction and exploitation

Mukherjee

Pathak²,

Bhattacharyya

2010

Int J of Quantum Chemistry

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Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrö dinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low-lying states of some mixed oscillator and doublewell potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator-level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations.

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“…The MEM varies the wave function Ψ to minimize the error F , while making sure that Ψ also satisfies the scattering boundary conditions. ,− We have experimented with two methods for minimizing the error. In one we take as variables the values of the wave function at the grid points and the R -matrix and vary them until F is minimized.…”

Section: Introductionmentioning

confidence: 99%

Minimum-Error Method for Scattering Problems in Quantum Mechanics: Two Stable and Efficient Implementations

2006

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We examine here, by using a simple example, two implementations of the minimum error method (MEM), a least-squares minimization for scattering problems in quantum mechanics, and show that they provide an efficient, numerically stable alternative to Kohn variational principle. MEM defines an error-functional consisting of the sum of the values of (HPsi - EPsi)2 at a set of grid points. The wave function Psi, is forced to satisfy the scattering boundary conditions and is determined by minimizing the least-squares error. We study two implementations of this idea. In one, we represent the wave function as a linear combination of Chebyshev polynomials and minimize the error by varying the coefficients of the expansion and the R-matrix (present in the asymptotic form of Psi). This leads to a linear equation for the coefficients and the R-matrix, which we solve by matrix inversion. In the other implementation, we use a conjugate-gradient procedure to minimize the error with respect to the values of Psi at the grid points and the R-matrix. The use of the Chebyshev polynomials allows an efficient and accurate calculation of the derivative of the wave function, by using Fast Chebyshev Transforms. We find that, unlike KVP, MEM is numerically stable when we use the R-matrix asymptotic condition and gives accurate wave functions in the interaction region.

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On the variational methods for bound-state and scattering problems

Cited by 46 publications

References 334 publications

Novel Consequences of Coexistence of Matter and Antimatter in Nature

Novel Consequences of Coexistence of Matter and Antimatter in Nature

Near‐exact supersymmetric partner potentials: Construction and exploitation

Minimum-Error Method for Scattering Problems in Quantum Mechanics: Two Stable and Efficient Implementations

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