Lagrange-mesh calculations and Fourier transform

Lacroix, Gwendolyn; Semay, Claude

doi:10.1103/physreve.84.036705

Cited by 6 publications

(21 citation statements)

References 24 publications

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“…This formula is identical to formula (6) in [13] with just the replacement of the variable r by p. The coefficients C j are linear variational parameters and the scale factor h is a nonlinear parameter aimed at adjusting the mesh to the domain of physical interest. The parameter h has here the dimension of a momentum (h plays the same role in [13] but has the dimension of a distance).…”

Section: B Matrix Equationmentioning

confidence: 99%

“…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%

“…IV, we will study the convergence of the method as a function of the scale parameter h and the number of mesh points N for nonrelativistic an semirelativistic kinematics. Let us note that an automatic determination of h has been developed for the LLM in the configuration space [13,33]. But, the generalization of such a technique to the LLM in the momentum space is very difficult due to the nonlocal nature of the interaction in (11).…”

Section: B Matrix Equationmentioning

confidence: 99%

“…Recently, it has been shown that the Fourier transform of the eigenfunctions computed in the configuration space can easily be obtained with a good accuracy in the physical domain of the momentum space [13]. Moreover, observables in this space can easily be computed with a good accuracy using only matrix elements and eigenfunctions in the configuration space.…”

Section: Introductionmentioning

confidence: 99%

“…The parameter h has here the dimension of a momentum (h plays the same role in [13] but has the dimension of a distance). Contrary to some other mesh methods, the wavefunction is also defined between mesh points by (18) and (19).…”

Section: B Matrix Equationmentioning

confidence: 99%

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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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Section: B Matrix Equationmentioning

confidence: 99%

“…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%

Section: B Matrix Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: B Matrix Equationmentioning

confidence: 99%

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Lagrange-mesh calculations in momentum space

2012

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The Lagrange-mesh method

2015

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The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the timedependent Schrödinger equation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh.

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Three theorems of quantum mechanics and their classical counterparts

Semay¹

2018

Eur. J. Phys.

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The Hellmann-Feynman, virial and comparison theorems are three fundamental theorems of quantum mechanics. For the first two, counterparts exist for classical mechanics with relativistic or nonrelativistic kinetic energy. It is shown here that these three theorems are valid for classical mechanics with a nonstandard kinetic energy. This brings some information about the connections between the quantum and classical worlds. Constraints about the functional form of the kinetic energy are also discussed.PACS numbers: 03.65. Ca, Let us consider a quantum Hamiltonian of the formwhere V is the potential energy depending only on the space variable r and T the kinetic energy depending only on the conjugate momentum p. The structure of T cannot be completely arbitrary. Some constraints are given in [1] for one-dimensional operators. But it is easy to generalize for three dimensions. We can expect that the kinetic energy is a positive quantity which is an increasing function of only the modulus of the momentum: T (p) = K(p 2 ) with p = |p|. Moreover, some degree of differentiability is desirable for the function K.If the eigensolutions are searched for in the |r representation, the action of T can be computed with the Fourier transform of the wave-function. This is the basis of the Fourier-Grid Hamiltonian method [2, 3] and a version of the Lagrange-mesh technique [4]. So, it is perfectly relevant to work with a general form T which can differ from the nonrelativistic or relativistic usual ones, and which can be useful for effective models.The Hellmann-Feynman theorem [5,6] states that, if the Hamiltonian H(λ) depends on a parameter λ, and that |λ is a normalized eigenstate with the energy E(λ), thenA demonstration of a generalized version of this theorem can be found in [7]. The general virial theorem [8] for Hamiltonians of type (1) states thatwhere the mean values are taken with an eigenstate of H, and ∂/∂a is the gradient with respect to the components of a. A simple demonstration can be obtained using the Hellmann-Feynman theorem [9]. Let us consider two ordered Hamiltonians H 1 and H 2 , that is to say φ|H 1 |φ ≤ φ|H 2 |φ for any state |φ . The comparison theorem states that each corresponding pair of eigenvalues is ordered E{α} ({α} represents a set of quantum numbers) [10]. A simple demonstration can be obtained using again the Hellmann-Feynman theorem [11]. The existence of an order is easy to check for Hamiltonians of type (1) [11].The virial and the Hellmann-Feynman theorems are also valid for classical mechanics, even if it is less well known for the latter case [12,13]. The demonstrations are made for relativistic or nonrelativistic kinetic energy operators. Before showing that the three quantum theorems described above are actually applicable to classical mechanics with a general form of T , let us study the relevance of such a mechanics. From the classical point of view, the equations of motion for Hamiltonian (1) are given by the Hamilton's equationṡ*

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Lagrange-mesh calculations and Fourier transform

Cited by 6 publications

References 24 publications

Lagrange-mesh calculations in momentum space

Lagrange-mesh calculations in momentum space

The Lagrange-mesh method

Three theorems of quantum mechanics and their classical counterparts

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