| Coulomb potential | |
|---|---|
| Name | Coulomb potential |
| Caption | Radial dependence of the Coulomb potential |
| Unit | joule |
| Symbols | V(r) |
| Parent | Electrostatic potential |
Coulomb potential
The Coulomb potential is the electrostatic potential energy function between two point charges, varying inversely with distance. In Quantum mechanics it provides the primary model for the interaction in hydrogen-like atoms and underlies bound states, scattering, and many perturbative calculations. Its singular 1/r form and long range make it central to both nonrelativistic and relativistic treatments across atomic, molecular and nuclear physics.
Classically, the Coulomb potential V(r) between two point charges q1 and q2 separated by distance r in vacuum is given by V(r) = (1/4πε0) q1 q2 / r, where ε0 is the vacuum permittivity and the constant 1/4πε0 is often denoted by k_e (Coulomb's constant). This expression follows directly from Coulomb's law for the electrostatic force, and from solving Poisson's equation for the electrostatic potential with a point-source charge density (a Dirac delta function). The potential is spherically symmetric and conservative, giving rise to conserved energy for charges moving under electrostatic forces described classically by Newtonian mechanics.
In nonrelativistic quantum theory the Coulomb potential appears in the time-independent Schrödinger equation as V(r)=−Z e^2/(4πε0 r) for an electron of charge −e interacting with a nucleus of charge Ze. The potential is central and commutes with orbital angular momentum operators, enabling separation of variables in spherical coordinates and reduction to a radial equation. The 1/r form produces discrete bound-state spectra and continuum scattering states; it also leads to characteristic degeneracies explained by additional dynamical symmetries such as the Laplace–Runge–Lenz vector. The Coulomb potential is a prototypical solvable interaction in quantum mechanics and a testing ground for approximation methods like perturbation theory, variational method, and WKB approximation.
For hydrogenic systems the Schrödinger equation with a Coulomb potential admits exact analytic solutions expressed in terms of confluent hypergeometric functions or associated Laguerre polynomials. Energy eigenvalues are given by the Rydberg formula E_n = −(μ e^4)/(2(4πε0)^2 ħ^2 n^2), where μ is the reduced mass and n is the principal quantum number; this matches spectroscopic results historically leading to the Bohr model refinement by Niels Bohr and later formalized by Schrödinger. Wavefunctions are labeled by quantum numbers (n, l, m) and display radial nodes and angular dependence via spherical harmonics. The accidental degeneracy between different l at fixed n is explained by the hidden SO(4) symmetry of the nonrelativistic Coulomb problem. Corrections from fine structure, hyperfine interactions, and Lamb shift are treated using Dirac equation, quantum electrodynamics, and radiative corrections.
Coulomb scattering (Rutherford scattering in the classical limit) is characterized quantum mechanically by the long-range 1/r potential, which produces a scattering amplitude with a singular forward peak and an infinite-range phase shift. Exact solutions of the Schrödinger equation yield the Coulomb wavefunctions (regular and irregular) and the analytic expression for the scattering matrix elements involves the Coulomb phase shifts and the Gamma function. Standard short-range scattering theory modifications are required: the usual definition of differential cross section must be combined with long-range asymptotic modifications and screening techniques. Coulomb scattering is foundational in experiments at facilities such as CERN and Lawrence Berkeley National Laboratory where charged-particle collisions test nuclear and atomic structure.
The classical 1/r singularity at r→0 gives an infinite potential energy for point charges, necessitating regularization or renormalization in some models. In nonrelativistic quantum mechanics the singularity is mild: for physically relevant angular momentum l≥1 the centrifugal barrier shields the origin, while the s-wave (l=0) remains well behaved and yields a normalizable wavefunction for attractive Coulomb potentials. In field-theoretic contexts the point-charge approximation leads to ultraviolet divergences; treatments invoke charge renormalization in quantum electrodynamics and consider finite-size form factors for composite nuclei. Numerical methods often regularize the potential at small r when solving few-body problems or implementing pseudopotentials in density functional theory and atomic structure codes (e.g., Hartree–Fock or coupled-cluster approaches).
Relativistic descriptions replace the Schrödinger equation by the Dirac equation with a Coulomb potential to account for spin and relativistic kinematics; this yields fine structure, spin–orbit coupling, and exact solutions for hydrogen-like ions up to the onset of critical charge phenomena (Zα≈1). In quantum electrodynamics (QED) the static Coulomb interaction arises from single-photon exchange in the Coulomb gauge and is modified by radiative corrections such as vacuum polarization and the Lamb shift; these effects are measured to high precision in hydrogen spectroscopy and in precision experiments at institutions like Harvard University and Max Planck Institute for Quantum Optics. Further extensions include the screened Coulomb (Yukawa-type) interactions in plasma physics and nuclear models, effective potentials in quantum chemistry and the role of Coulomb interactions in condensed matter systems, e.g., in quantum Hall effect studies and density functional theory approximations.
Category:Quantum mechanics Category:Electrostatics