Lagrange‐mesh method for quantum‐mechanical problems

Baye, Daniel Jean

doi:10.1002/pssb.200541305

Cited by 109 publications

(140 citation statements)

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“…It is exact if g(x) is a polynomial of degree at most 2N − 1 times exp(−x) [16]. The regularized Lagrange functions read [12,17] …”

Section: Lagrange-mesh and Variational Methodsmentioning

confidence: 99%

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Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Baye

2012

Phys. Rev. A

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Exact analytical expressions of the dipole polarizabilities of the non-relativistic hydrogen atom in spherical coordinates are derived with the help of the Lagrange-mesh numerical method. This method can provide exact energies and wave functions for well chosen conditions of calculation.Exact dipole polarizabilities are obtained after an unambiguous rounding up to at least principal quantum numbers around n = 30. The scalar polarizability of any nl level is given by n 4 [4n 2 + 14 + 7l(l + 1)]/4 and its tensor polarizability by −n 4 [3n 2 − 9 + 11l(l + 1)]l/4(2l + 3), which allow calculating the polarizability of any hydrogen state nlm.

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“…It is exact if g(x) is a polynomial of degree at most 2N − 1 times exp(−x) [16]. The regularized Lagrange functions read [12,17] …”

Section: Lagrange-mesh and Variational Methodsmentioning

confidence: 99%

“…The Lagrange-mesh method is an approximate variational method involving a basis of Lagrange functions and using the associated Gauss quadrature for the calculation of matrix elements [10][11][12]. Lagrange functions are continuous functions that vanish at all points of a mesh but one.…”

Section: Introductionmentioning

confidence: 99%

Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Baye

2012

Phys. Rev. A

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“…The use of the LMM in configuration space is described in [2][3][4][5][6][7][8]13]. We will present here the formulation for the integral equation (11) within the LMM.…”

Section: B Matrix Equationmentioning

confidence: 99%

“…Among these techniques, the Lagrange-mesh method (LMM), which is especially easy to implement, can produce very accurate results. First created to compute eigenvalues and eigenfunctions of a two-body Schrödinger equation [2][3][4][5][6][7], it has been extended to treat semirelativistic Hamiltonian [8][9][10][11]. The trial eigenstates are developed in a basis of particular functions, the Lagrange functions, which vanish at all mesh points, except one.…”

Section: Introductionmentioning

confidence: 99%

Lagrange-mesh calculations in momentum space

2012

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The Lagrange-mesh method is a powerful method to solve eigenequations written in configuration space. It is very easy to implement and very accurate. Using a Gauss quadrature rule, the method requires only the evaluation of the potential at some mesh points. The eigenfunctions are expanded in terms of regularized Lagrange functions which vanish at all mesh points except one. It is shown that this method can be adapted to solve eigenequations written in momentum space, keeping the convenience and the accuracy of the original technique. In particular, the kinetic operator is a diagonal matrix. Observables in both configuration space and momentum space can also be easily computed with a good accuracy using only eigenfunctions computed in the momentum space. The method is tested with Gaussian and Yukawa potentials, requiring respectively a small or a great mesh to reach convergence.

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“…The parabolic quantum number k runs from −(n−|m l |−1) and (n−|m l |−1), but only the mid-Stark-manifold k = 0 Rydberg states are selected for study, as their energies are approximately field independent and provide the greatest region of crossing with the field-dependent image-state energies. The energies are calculated by diagonalization of the Hamiltonian using a DVR basis set [19]. The widths of the Rydberg curves shown represent the perturbation of the Rydberg energy due to the surface interaction over the typical range that ionization occurs, 6n 2 a 0 to 3n 2 a 0 .…”

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Resonant charge transfer of hydrogen Rydberg atoms incident at a metallic sphere

Gibbard¹,

Softley²

2016

J. Phys. B: At. Mol. Opt. Phys.

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The charge transfer (ionization) of hydrogen Rydberg atoms (n = 25 − 34) at a Cu(100) surface is investigated. Unlike fully metallic surfaces, where the Rydberg electron energy is degenerate with the conduction band of the metal, the Cu(100) surface has a projected bandgap at these energies, and only discrete image states are available through which charge transfer can take place. Resonant enhancement of charge transfer is observed for Rydberg states whose energy matches one of the image states, and the integrated surface ionization signals (signal versus applied field) show clear periodicity as a function of n as the energies come in and out of resonance with the image states. The surface ionization dynamics show a velocity dependence; decreased velocity of the incident H atom leads to a greater mean distance of ionization and a lower field required to extract the ion. The surface-ionization profiles for 'on resonance' n values show a changing shape as the velocity is changed, reflecting the finite field range over which resonance occurs.The collision of a Rydberg atom in the gas phase with a solid surface typically leads to transfer of the Rydberg electron to the surface at distances less than 5n 2 a 0 , where n is the Rydberg electron principal quantum number. This is especially true for metallic surfaces, where the Rydberg electron energy is degenerate with the conduction band so that resonant charge transfer (RCT) can occur. Experimental and theoretical studies of this phenomenon have focused on the effects of varying the n quantum number, the parabolic quantum number k, the velocity of the incoming particle and the applied fields [1,2], and observing how the rate of ionization varies as a function of distance from the surface [3]. For nonhydrogenic atoms, adiabatic and non-adiabatic passage through surface-induced energy level crossings leads to behavior that varies with the Rydberg species [4]. Thus, such studies reveal important information about the Rydberg states and their dynamics near surfaces.An equally important question for such studies is what they reveal about the nature of the surface. Experimental studies have been primarily conducted with flat-metal surfaces for which the ionization dynamics are almost independent of the material because of the generic behavior of RCT to the conduction band. However, there have also been some experimental and/or theoretical investigations of the effects of adlayers and thin insulating films [5], interaction with doped semiconductor surfaces [6] and dielectric materials [7], effects of corrugation and of patch charges [8,9]. Related theoretical calculations were used to investigate the variation of ionization rate of ground state H -with the thickness of a metal film substrate [10]. All these studies point to a degree of sensitivity of the charge transfer process to the surface characteristics. The mean radius of a hydrogenic Rydberg orbit is of order n 2 a 0 (e.g., ∼ 20 nm for n = 20) and charge transfer typically occurs at a Rydberg-surface distance of 3 − 5...

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Lagrange‐mesh method for quantum‐mechanical problems

Cited by 109 publications

References 50 publications

Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Exact nonrelativistic polarizabilities of the hydrogen atom with the Lagrange-mesh method

Lagrange-mesh calculations in momentum space

Resonant charge transfer of hydrogen Rydberg atoms incident at a metallic sphere

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