Integral Equations in Quantum Mechanics: An Extensive Study of Their Role in Theoretical Physics

Abstract

Quantum mechanics heavily depends on integral equations because they supply advanced mathematical solutions for solving complex theoretical physics problems. This paper examines the development of integral equations starting from their inception through their mathematical structure until their utility in solving Schrödinger's equation and its adjoint variants. The research analyzes integral equations by combining sophisticated derivations with numerical approximation methods and case-based examinations, and multidisciplinary connections to explain their use in scattering theory and bound states, as well as quantum field theory and quantum electrodynamics and condensed matter physics and quantum information theory, and new fields of quantum computing and quantum gravity. The assessment demonstrates that integral equations continue to adapt for quantum mechanical research as they advance contemporary knowledge in this field.