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Lagrange-Mesh Method in Momentum Space: an Alternative Formulation

Cyrille Chevalier, Joachim Viseur

TL;DR

This work develops a momentum-space Lagrange-m mesh method (LMM) to solve two-body Schrödinger-like equations with general kinetic and long-range potentials, including Coulomb and linear forms. The core contribution is an alternative scheme for computing potential matrix elements via a transformed $r^2$ operator, allowing Gauss-Laguerre quadrature to be used directly in momentum space. The method is validated against analytical Coulomb results, yielding high accuracy as the mesh size $N$ increases, and is further illustrated on a Cornell-type meson model, showing good agreement with existing configuration-space LMM results though with slower convergence. Additionally, the authors derive expressions to represent states in configuration space, enabling momentum- and position-space densities and observables to be obtained from the same momentum-space framework, broadening the practical impact for quantum few-body and hadronic-physics problems.

Abstract

This work presents a new methodology for computing potential matrix elements within the Lagrange-mesh method in momentum space. The proposed approach extends the range of treatable potentials to include previously inaccessible cases, such as Coulomb and linear interactions. The method is validated across a variety of systems. A particular attention is given to the representation of both momentum and position probability densities.

Lagrange-Mesh Method in Momentum Space: an Alternative Formulation

TL;DR

This work develops a momentum-space Lagrange-m mesh method (LMM) to solve two-body Schrödinger-like equations with general kinetic and long-range potentials, including Coulomb and linear forms. The core contribution is an alternative scheme for computing potential matrix elements via a transformed operator, allowing Gauss-Laguerre quadrature to be used directly in momentum space. The method is validated against analytical Coulomb results, yielding high accuracy as the mesh size increases, and is further illustrated on a Cornell-type meson model, showing good agreement with existing configuration-space LMM results though with slower convergence. Additionally, the authors derive expressions to represent states in configuration space, enabling momentum- and position-space densities and observables to be obtained from the same momentum-space framework, broadening the practical impact for quantum few-body and hadronic-physics problems.

Abstract

This work presents a new methodology for computing potential matrix elements within the Lagrange-mesh method in momentum space. The proposed approach extends the range of treatable potentials to include previously inaccessible cases, such as Coulomb and linear interactions. The method is validated across a variety of systems. A particular attention is given to the representation of both momentum and position probability densities.

Paper Structure

This paper contains 13 sections, 37 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Lagrange Mesh Method
  3. Matrix Element Computation in Momentum Space
  4. Kinetic Energy Matrix Elements
  5. Potential energy matrix elements
  6. Resulting eigenvalues and eigenstates
  7. Validation with analytical results
  8. Representation of the state in configuration space
  9. Validation with analytical results
  10. Application for a physical example
  11. Conclusion
  12. Direct calculation of the LMM matrix elements in momentum representation
  13. Gaussian potential

Figures (4)

  • Figure 1: Ground-state energies of equation \ref{['eq:coul_ham']} obtained with the LMM versus the scaling parameter $h$. Different mesh sizes are compared. The x-axis is plotted on a logarithmic scale.
  • Figure 2: Representation of the probability densities $\mathcal{P}$ obtained with the LMM for the states $1S$, $2S$ and $1P$ (left, right and below, respectively) of equation \ref{['eq:coul_ham']}. For each state, different mesh sizes are compared. Exact results are also displayed as thicker black lines. The LMM curves for $N=135$ are nearly indistinguishable from the exact ones. Calculations are performed with $h=0.5$.
  • Figure 3: Representation of the probability densities $\mathcal{R}$ obtained with the LMM for the states $1S$, $2S$ and $1P$ (left, right and below, respectively) of equation \ref{['eq:coul_ham']}. For each state, different mesh sizes are compared. Exact results are also displayed as thicker black lines. The LMM curves for $N=150$ are nearly indistinguishable from the exact ones. Calculations are performed with $h=0.5$. Note that the range of the abscissa differs for the $1S$ curve.
  • Figure 4: Representation of the probability densities $\mathcal{R}$ obtained with the LMM for the states $1S$, $2S$ and $1P$ (left, right and below, respectively) of equation \ref{['eq:coul_ham']}. For each state, different mesh sizes are compared. Exact results are also displayed as thicker black lines. The LMM curves over $N=50$ are nearly indistinguishable from the exact ones. Calculations are performed with $h=0.1$. Note that the range of the abscissa differs for the $1S$ curve.