Approximation Methods in Quantum Mechanics (Review)

Abstract

Approximation methods play a crucial role in quantum mechanics, enabling the study of complex systems that are analytically intractable. This paper provides an introduction to two prominent approximation methods in quantum mechanics: the Hartree-Fock approximation and atomic approximations. The Hartree-Fock approximation is a widely used method for describing the behavior of many-electron systems. It approximates the complicated many-body wavefunction by considering a single Slater determinant, where each electron occupies an independent orbital. By solving the Hartree-Fock equations, the method provides an effective mean-field description of the system, neglecting electron-electron correlations. The basic principles, mathematical formulation, and limitations of the Hartree-Fock approximation are discussed. Atomic approximations focus on modeling atomic systems, which serve as fundamental building blocks for more complex molecular structures. These approximations aim to simplify the electronic structure of atoms while retaining the essential physical characteristics. The most commonly used atomic approximations include the Thomas-Fermi model, the Thomas-Fermi-Dirac model, and the density functional theory