⚛️ quantum physics

Handling the Cornell potential within the Lagrange-mesh method in momentum space

This paper introduces an alternative methodology within the Lagrange-mesh method in momentum space that enables the efficient and accurate computation of matrix elements for previously inaccessible potentials, such as the Cornell potential, thereby expanding the method's applicability to hadronic physics.

Original authors: Cyrille Chevalier, Joachim Viseur

Published 2026-02-27

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CC BY 4.0

Original authors: Cyrille Chevalier, Joachim Viseur

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the path of a dancer spinning on a stage. To do this, you have two main ways of looking at the problem:

The "Where" View (Position Space): You watch the dancer's feet. You see exactly where they are standing at every moment. This is like looking at a map.
The "How Fast" View (Momentum Space): You ignore the feet and look at the dancer's energy and speed. You see how fast they are spinning and in what direction. This is like looking at a speedometer and a compass.

In the world of tiny particles (like quarks inside a proton), physicists need to solve a complex math puzzle called the Schrödinger equation to understand how these particles move and stick together. Usually, they use the "Where" view (Position Space) because it's easier for some types of forces. But for other forces, especially the ones that hold quarks together, the "How Fast" view (Momentum Space) is actually much better.

The Problem: The "Broken" Calculator

For a long time, physicists had a super-powerful tool called the Lagrange-mesh method. Think of this tool as a high-tech calculator that can solve the dancer's path very quickly and accurately.

In the "Where" view: This calculator worked perfectly for all types of forces.
In the "How Fast" view: This calculator worked great for simple forces (like a gentle spring), but it broke when faced with the two most important forces in particle physics:
1. The Coulomb Force: Like the electric pull between a magnet and a paperclip (it gets very strong when things get close).
2. The Linear Force: Like a rubber band that gets harder to stretch the further you pull it.

These two forces combined are called the Cornell Potential. It's the "glue" that holds quarks together to make particles like protons and mesons. The old "How Fast" calculator would crash or give nonsense numbers when you tried to use it with the Cornell Potential.

The Solution: A New Strategy

The authors of this paper (Cyrille Chevalier and Joachim Viseur) invented a new way to use the calculator in the "How Fast" view.

Instead of trying to calculate the force directly (which caused the crash), they used a clever trick:

Step 1: They calculated the "shape" of the space the particles occupy (using a mathematical tool called a matrix).
Step 2: They rotated this shape into a new perspective where the math became simple.
Step 3: They applied the force (the Cornell Potential) to this simple shape.
Step 4: They rotated the result back to the original view.

The Analogy:
Imagine you are trying to cut a very hard, jagged rock with a dull knife (the old method). It's impossible.
The new method is like putting the rock in a blender, turning it into smooth sand, pouring the sand into a mold to make a perfect cube, and then using a laser cutter to slice the cube. You aren't cutting the rock directly; you are changing the form of the problem so the tool can handle it.

What They Discovered

It Works: They tested their new method on a simple system (like an electron orbiting a proton) and found it gave the exact same answer as the known "perfect" solution.
It's Fast: It only took a few seconds on a regular laptop to get results that used to take much longer or were impossible to get.
It's Accurate: They used it to calculate the energy levels of "mesons" (particles made of quarks). The results matched perfectly with the best existing theories and experiments.

Why Does This Matter?

This is a big deal for Hadronic Physics (the study of particles made of quarks).

Before this, if you wanted to study these particles using the "How Fast" view (which is often more accurate for high-speed particles), you were stuck because the math tools were broken.
Now, physicists can use the "How Fast" view for any type of force, including the tricky Cornell Potential.
This allows for more precise models of how the universe is built at its smallest scales, helping us understand everything from the inside of a proton to the behavior of exotic new particles.

In short: The authors fixed a broken tool in the physicist's toolbox, allowing them to solve the hardest puzzles in particle physics using the most efficient method available.

1. Problem Statement

The Lagrange-mesh method (LMM) is a highly efficient numerical technique for solving the Schrödinger equation for two-body systems, traditionally applied in configuration (position) space. While LMM in momentum space has been developed (notably by Lacroix et al., Ref. [11]), existing momentum-space implementations face a critical limitation: they cannot effectively handle long-range potentials such as the Coulomb ( $1/r$ ) and linear ( $r$ ) interactions.

This is a significant gap in hadronic physics, where the Cornell potential (a combination of Coulomb and linear terms, $V(r) = -\kappa/r + ar + C$ ) is the standard model for describing quark-antiquark interactions in mesons. The previous momentum-space methodology relied on calculating the Fourier transform of the potential, which leads to mathematical divergences (specifically involving Legendre functions of the second kind) when applied to Coulomb or linear potentials. Consequently, researchers were forced to use configuration space for these specific problems, losing the advantages of a momentum-space description (e.g., compatibility with quantum field theories and exotic kinematics).

2. Methodology

The authors propose an alternative methodology to compute potential matrix elements within the momentum-space LMM. Instead of relying on the Fourier transform of the potential (which fails for long-range interactions), they adapt a strategy originally used for handling semi-relativistic kinematics in configuration space.

The core of the new approach involves the following steps:

Trial States: The system is expanded using Lagrange mesh trial states defined in momentum space, utilizing Gauss-Laguerre quadrature points ( $x_k$ ) and weights ( $\lambda_k$ ).
Kinetic Energy: The kinetic energy matrix elements are computed directly at the mesh points, as is standard in LMM.
Potential Energy (The Innovation):
- Instead of transforming $V(r)$ to momentum space, the authors compute the matrix elements of the operator $r^2$ directly in momentum space.
- Since $r^2$ acts as a differential operator in momentum space (analogous to $p^2$ in position space), its matrix elements can be derived using known formulas for Lagrange functions.
- The resulting $r^2$ matrix is diagonalized to find its eigenvalues and eigenvectors.
- The potential function $V(r)$ is then applied directly to the eigenvalues of $r^2$ (since $V(r)$ is a function of $r$ , and $r^2$ is diagonal in this new basis).
- Finally, the potential matrix is transformed back to the original Lagrange basis using the diagonalization transition matrix.

This approach bypasses the need for the Fourier transform of the potential, thereby avoiding the divergences associated with Coulomb and linear terms.

3. Key Contributions

Extension of Momentum-Space LMM: The paper successfully extends the applicability of the momentum-space LMM to include singular and linear potentials, specifically the Cornell potential.
Analytical Validation: The method is rigorously tested against analytical solutions for the Coulomb system (hydrogen-like atoms). The authors demonstrate that the method reproduces exact energy eigenvalues and probability densities with high precision.
Position Space Reconstruction: The authors derive formulas to transform the momentum-space wavefunctions back into configuration space (position probability densities). They address the issue of unphysical oscillations in the position representation for small mesh sizes, showing that these can be mitigated by adjusting the scaling parameter $h$ or increasing the mesh size $N$ .
Hadronic Phenomenology Application: The method is applied to a semi-relativistic constituent quark model for mesons using the Cornell potential. The results are compared with both the original phenomenological model (Ref. [3]) and previous configuration-space LMM calculations (Ref. [10]).

4. Results

Accuracy: For the Coulomb system, the method achieves high accuracy (5+ significant digits) with a mesh size of $N=300$ . The energy eigenvalues converge rapidly, and the probability densities match analytical results almost perfectly for $N \geq 135$ .
Cornell Potential: When applied to the meson spectrum (using parameters from Ref. [3]), the momentum-space LMM reproduces the eigenenergies with an accuracy of up to four significant digits.
- Comparison: While the convergence is slightly slower than the configuration-space LMM (likely due to the use of a fixed scaling parameter $h$ ), the results are fully consistent with established benchmarks.
Efficiency: The calculations are computationally efficient, taking only a few seconds on a standard laptop for the tested cases.
Oscillations: The paper identifies that small mesh sizes in momentum space can lead to oscillations in the reconstructed position-space densities. However, these are shown to be artifacts that vanish as $N$ increases or $h$ is optimized.

5. Significance

This work resolves a long-standing limitation in the application of the Lagrange-mesh method to hadronic physics. By enabling the treatment of Cornell potentials directly in momentum space, the authors provide a powerful tool for:

Hadron Spectroscopy: Accurately modeling meson and baryon spectra without switching to configuration space.
Quantum Field Theory (QFT) Compatibility: Momentum space is the natural language for QFT. This method allows for the integration of potential models with momentum-dependent interactions and exotic kinematics (e.g., spinless Salpeter equation) more naturally than configuration-space methods.
Versatility: The methodology is general and can be applied to any potential $V(r)$ , not just the Cornell potential, making it a robust alternative for solving two-body Schrödinger-like equations where momentum-dependent descriptions are preferred.

In summary, the paper provides a mathematically sound and numerically efficient bridge between the Lagrange-mesh method and the complex potentials required for modern hadronic physics, all while maintaining the computational advantages of a momentum-space formulation.