A generalization of Numerov's method for the numerical solution of the Schrödinger equation in two dimensions

Avdelas, G.; Konguetsof, A.; Simos, T. E.

doi:10.1016/s0097-8485(99)00096-0

Cited by 9 publications

(12 citation statements)

References 22 publications

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“…But, the Hamiltonian is diagonal, and there are some numerical method to solve it. We use a well-known Numerov numerical methods to solve ordinary differential equations of second order [19], [20]. In this method, we first organize the square matrix Hamiltonian…”

Section: D Potential In Cylindrical Coordinatesmentioning

confidence: 99%

Two-Center Gaussian Potential Well for Studying Light Nucleus in Cluster Structure

Roshanbakht

Shojaei

2017

Advances in High Energy Physics

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The clustering phenomena is very important to determine structure of light nuclei and deformation of spherical shape is inevitable. Hence, we calculated the energy levels of twocenter Gaussian potential well including spin-orbit coupling by solving the Schrödinger equation in the cylindrical coordinates. This model could predict the spin and parity of the light nucleus that have cluster structure consist of two identical clusters.

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Section: D Potential In Cylindrical Coordinatesmentioning

confidence: 99%

Two-Center Gaussian Potential Well for Studying Light Nucleus in Cluster Structure

Roshanbakht

Shojaei

2017

Advances in High Energy Physics

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“…Several other variants in order to increase both stability and accuracy while calculating the resonance problems and high−lying bound states were proposed. Generalization of the algorithm to an error of arbitrary order [15][16][17][18][19][20][21][22][23][24][25] and extension it for solution of differential equations with more than one−dimension [8,[26][27][28][29][30][31][32][33] form the main framework of the studies. A Numerov−type exponentially fitted method was suggested [18-20, 32, 34-43], accordingly.…”

Section: Introductionmentioning

confidence: 99%

An efficient approximation for accelerating convergence of the numerical power series. Results for the 1D Schrödinger's equation

Bagci¹,

Guneş²

2021

Preprint

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The numerical matrix Numerov method is used to solve the stationary Schrödinger equation for central Coulomb potentials. An efficient approximation for accelerating the convergence is proposed. The Numerov method is error−prone if the magnitude of grid−size is not chosen properly. A number of rules so far, have been devised. The effectiveness of these rules decrease for more complicated equations. The grid−size that is used for discretization of the Schrödinger's equation is allowed to be variationally for the given boundary conditions. In order to test efficiency of the technique used for accelerating the convergence, the grid-sizes are determined via the optimization procedure performed for each energy state. The results obtained for energy eigenvalues and corresponding eigen−vectors are compared with the literature. It is observed that, once the values of grid−sizes for hydrogen energy eigenvalues are obtained, they can simply be determined for the hydrogen iso−electronic series as, hε(Z) = hε(1)/Z.

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“…One of the reasons for the 1D case has earned that much attention lies partially in the fact that many higher dimensional problems have spherically symmetric potentials which then allows separation of variables and the study of the radial equation only. Numerical methods for obtaining bound state as well as scattering state solutions can be broadly classified into two types: numerical integration or in other words the shooting methods [3,4] and matrix methods [2,5,6,7,8,9,10]. Also, there are analytical approaches too like perturbative treatments or the Rayleigh-Ritz variational principle, see Ref.…”

Section: Introductionmentioning

confidence: 99%

“…This is because the first approximation already yields very accurate eigenvalues with error at most O(h 4 ). This is also true in two and higher dimensions, though the procedure is slightly complicated by the fact that the recurrence relation involves second-neighbors as well and so some additional information or physically motivated approximation is needed close to the boundary in order to start the algorithm [9,14]. Though its performance, in the sense of accuracy, is obviously not as good as the classical heptadiagonal approximation, with higher order approximations for the derivatives [13,17,18] or with extrapolation techniques [7,8,9] (Richardson, Aitken) it can be refined easily up to O(h 6 ) or higher.…”

Section: Introductionmentioning

confidence: 99%

Interacting particles in two dimensions: numerical solution of the four-dimensional Schrödinger equation in a hypercube

Vanyolos,

Varga

2008

Preprint

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We study numerically the Coulomb interacting two-particle stationary states of the Schrödinger equation, where the particles are confined in a two-dimensional infinite square well. Inside the domain the particles are subjected to a steeply increasing isotropic harmonic potential, resembling that in a nucleus. For these circumstances we have developed a fully discretized finite difference method of the Numerov-type that approximates the four-dimensional Laplace operator, and thus the whole Schrödinger equation, with a local truncation error of O(h 6 ), with h being the uniform step size. The method is built on a 89-point central difference scheme in the four-dimensional grid. As expected from the general theorem by Keller [Num. Math. 7, 412 (1965)], the error of eigenvalues so obtained are found to be the same order of magnitude which we have proved analytically as well. Based on this difference scheme we have obtained a generalized matrix Schrödinger equation by vectorization. In the course of its numerical solution group theoretical methods were applied extensively to classify energy eigenvalues and associated two-particle wave functions. This classification scheme inherently accounts for the symmetry property of the two-particle state under permutation, thereby making it very easy to fully explore the completely symmetric and antisymmetric subspaces of the full Hilbert space. We have obtained the invariance group of the interacting Hamiltonian and determined its irreducible representations. With the help of these and Wigner-Eckart theorem, we derived an equivalent block diagonal form of the eigenvalue equation. In the low-energy subspace of the full Hilbert space we numerically computed the ground state and many (≈ 200) excited states for the noninteracting as well as the interacting cases. Comparison with the noninteracting exact results, which we have found analytically too, reveals indeed that already at a modest resolution of h ≈ 1/15 our numerical data for the eigenvalues are accurate globally up to three or more digits of precision. Having obtained the energy eigenvalues and eigenstates we calculated some relevant physical quantities with and without interaction. These are the static two-particle densities labeled by irreducible representations, oneand two-particle density of states and some different measures of entanglement, including the reduced density matrix and the von Neumann entropy. All these quantities signal concordantly that the symmetric states (with respect to permutation of particles) as well as their energies are affected by the interaction more dramatically then their antisymmetric counterparts.

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A generalization of Numerov's method for the numerical solution of the Schrödinger equation in two dimensions

Cited by 9 publications

References 22 publications

Two-Center Gaussian Potential Well for Studying Light Nucleus in Cluster Structure

Two-Center Gaussian Potential Well for Studying Light Nucleus in Cluster Structure

An efficient approximation for accelerating convergence of the numerical power series. Results for the 1D Schrödinger's equation

Interacting particles in two dimensions: numerical solution of the four-dimensional Schrödinger equation in a hypercube

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