| Coulomb potential | |
|---|---|
| Name | Coulomb potential |
| Field | Electromagnetism |
| Introduced | 18th century |
| Related | Coulomb's law, Electrostatics, Quantum mechanics |
Coulomb potential
The Coulomb potential is the classical electrostatic potential energy between two point charges, varying inversely with distance; it provides a central, long-range potential that is foundational in Electrodynamics and Quantum mechanics. In quantum physics it underpins models of atomic structure, scattering theory, and many-body interactions, with broad implications for atomic physics, condensed matter physics, and plasma physics.
The Coulomb potential arises from Coulomb's law, originally formulated by Charles-Augustin de Coulomb. For two point charges q1 and q2 separated by distance r in vacuum, the potential energy is V(r) = k q1 q2 / r, where k = 1/(4πϵ0) and ϵ0 is the vacuum permittivity. In media, the potential is modified by the dielectric constant of the material and by screening from other charges as described by Debye screening in plasmas and electrolytes. The classical formulation assumes pointlike charges and instantaneous action at a distance within the electrostatic limit of Maxwell's equations.
In quantum mechanics, the Coulomb potential is a central potential used in the Schrödinger equation, Dirac equation, and perturbative expansions in quantum electrodynamics (QED). It defines bound states, angular momentum coupling via spherical symmetry, and selection rules for transition amplitudes. The potential's 1/r form produces both discrete bound spectra and continuous scattering states, and it influences phenomena studied at institutions such as CERN and Lawrence Berkeley National Laboratory through precision spectroscopy and tests of QED. The interplay between Coulomb interactions and many-body correlations is central to work at universities like MIT, University of Cambridge, and Harvard University in atomic and condensed matter research.
Exact solutions of the nonrelativistic Schrödinger equation with a Coulomb potential yield the hydrogenic wavefunctions and energy levels first solved by Erwin Schrödinger and earlier modeled by the Bohr model. The principal quantum number n produces energies E_n = −(μ e^4)/(2ħ^2(4πϵ0)^2 n^2) for reduced mass μ, with degeneracies lifted by spin–orbit coupling in the Dirac equation and by radiative corrections in Lamb shift calculations by Willis Lamb and Robert Retherford. Precise spectroscopy of hydrogen and hydrogen-like ions like He+ and U91+ tests QED and fundamental constants studied by collaborations such as the Max Planck Institute for Quantum Optics and National Institute of Standards and Technology.
Coulomb scattering is a paradigmatic example in scattering theory, with analytic solutions given by the Rutherford scattering formula in classical physics and the quantum mechanical Coulomb phase shifts. Long-range 1/r tails violate assumptions of short-range potential scattering, requiring modified formulations such as the Møller operators and the Coulomb wave function formalism. Experimental programs in accelerator physics at SLAC National Accelerator Laboratory and low-energy electron scattering experiments at Brookhaven National Laboratory probe Coulomb-dominated processes and nuclear charge distributions.
Mathematically the Coulomb potential is singular at r = 0 for point charges, leading to ultraviolet divergences in naive quantum treatments. The 1/r potential is a Green's function of the Laplace operator in three dimensions, linking to solutions of the Poisson equation. Regularization and renormalization techniques from quantum field theory are employed when coupling Coulomb interactions with relativistic particles, as in QED computations by theorists like Richard Feynman and Sin-Itiro Tomonaga. Self-energy, operator domains, and the stability of matter proofs (e.g., work by Elliott H. Lieb and Walter Thirring) address many-body singular behavior.
Coulomb interactions determine electron-electron repulsion, screening, and collective modes in solids and plasmas. In condensed matter, Coulomb potential underlies the Hubbard model (on-site repulsion), Hartree–Fock theory, and Density Functional Theory exchange–correlation approximations developed by researchers like Walter Kohn and Lu Jeu Sham. Long-range Coulomb forces produce plasmons, charge-density waves, and influence semiconductor device physics studied at companies and labs such as Intel and IBM Research. In plasma physics, Coulomb collisions and Debye screening govern transport coefficients, as treated in the Vlasov equation and in work by Lev Landau on plasma kinetic theory.
Numerical treatments employ techniques to handle the singular 1/r kernel: Ewald summation in periodic solids, pseudopotentials and projector augmented-wave method for core electrons, and cutoff/regularization schemes in Quantum Monte Carlo and density functional theory codes like VASP and Quantum ESPRESSO. For scattering problems, the use of Coulomb-distorted waves and complex scaling addresses long-range behavior; for few-body systems, the Faddeev equations adapt to charged-particle interactions. Renormalization in QED and effective field theories isolates finite observable predictions; computational groups at Los Alamos National Laboratory and Argonne National Laboratory contribute large-scale simulations of Coulomb-dominated systems.
Category:Quantum mechanics Category:Electromagnetism Category:Atomic physics