The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of particles in non-relativistic quantum systems. However, in Quantum Electrodynamics (QED), which is the theory of the electromagnetic interaction in the framework of quantum field theory, the Schrödinger equation is replaced by a different equation known as the Dirac equation.
The Dirac equation was formulated by Paul Dirac in 1928 and describes the behavior of fermionic particles, such as electrons, in a relativistic quantum framework. It incorporates special relativity and allows for the description of spin-1/2 particles.
The Dirac equation for a particle in an electromagnetic field is given by:
(iγ^μ∂_μ - eγ^μA_μ - m)Ψ = 0
In this equation, γ^μ represents the Dirac matrices, which are 4x4 matrices, ∂_μ is the four-gradient operator, A_μ is the four-potential of the electromagnetic field, e is the electric charge of the particle, m is its mass, and Ψ is the four-component Dirac spinor that represents the particle's wave function.
The Dirac equation is a relativistic wave equation that describes the behavior of particles with spin-1/2, such as electrons and positrons, in the presence of electromagnetic fields. It has been a foundational equation in the development of quantum field theory, and it forms the basis of the description of fermions and their interactions in QED.