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The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
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Mesh analysis becomes simpler when analyzing circuits with current sources, whether independent or dependent. The presence of current sources reduces the number of equations required for analysis. Two cases illustrate this:
Current Source in One Mesh: The analysis process is straightforward when a current source is found in only one mesh within the circuit. Mesh currents are assigned as usual, with the mesh containing the current source excluded from the analysis. Kirchhoff's voltage law...
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Mesh Analysis01:20

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Mesh analysis is a valuable method for simplifying circuit analysis using mesh currents as key circuit variables. Unlike nodal analysis, which focuses on determining unknown voltages, mesh analysis applies Kirchhoff's voltage law (KVL) to find unknown currents within a circuit. This method is particularly convenient in reducing the number of simultaneous equations that need to be solved.
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James Clerk Maxwell (1831–1879) was one of the significant contributors to physics in the nineteenth century. He is probably best known for having combined existing knowledge of the laws of electricity and the laws of magnetism with his insights to form a complete overarching electromagnetic theory, represented by Maxwell's equations. The four basic laws of electricity and magnetism were discovered experimentally through the work of physicists such as Oersted, Coulomb, Gauss, and...
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Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area...
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Related Experiment Video

Updated: Jun 26, 2025

The Mechanics of Poro-Elastic Contractile Actomyosin Networks As a Model System of the Cell Cytoskeleton
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Klein-Gordon equation on a Lagrange mesh.

Daniel Baye1

  • 1Nuclear Physics and Quantum Physics, C.P. 229, Université Libre de Bruxelles (ULB), B-1050 Brussels Belgium.

Physical Review. E
|May 17, 2024
PubMed
Summary

The Lagrange-mesh method accurately solves few-body quantum problems using the Schrödinger and Klein-Gordon equations. This efficient approach yields precise energies and wave functions with minimal computational resources.

Area of Science:

  • Computational Physics
  • Quantum Mechanics
  • Theoretical Chemistry

Background:

  • The Schrödinger equation is fundamental for describing quantum systems.
  • Solving few-body problems accurately is computationally challenging.
  • Variational methods offer approximate solutions to complex quantum equations.

Purpose of the Study:

  • To present the Lagrange-mesh method for solving the Schrödinger and Klein-Gordon equations.
  • To demonstrate the method's accuracy and efficiency for few-body problems.
  • To validate the approach using the Coulomb potential.

Main Methods:

  • The Lagrange-mesh method, an approximate variational technique.
  • Solving the stationary Klein-Gordon equation quadratically dependent on energy.

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  • Utilizing a Lagrange-Laguerre mesh for iterative solutions.
  • Testing with the exactly solvable Coulomb potential.

    • Main Results:

    • High accuracy achieved with a small number of mesh points.
    • Accurate energies and mean values obtained in few iterations.
    • Demonstrated efficiency for various potentials and energy levels.
    • Analytical wave functions are readily available.

    Conclusions:

    • The Lagrange-mesh method is a powerful and efficient tool for few-body quantum mechanics.
    • The method provides accurate solutions for bound-state and scattering problems.
    • It offers a computationally advantageous alternative to other numerical techniques.