Non-relativistic QED

A brief presentation is given of the construction of the theory of molecular QED. This is done by first writing a classical Lagrangian function for a collection of non-relativistic charged particles coupled to an electromagnetic field. After selecting the Coulomb gauge, Hamilton’s principle is invoked and the Lagrangian is substituted into the Euler-Lagrange equations of motion and shown to lead to the correct dynamical equations. These are Newton’s second law of motion with added Lorentz force law electric and magnetic field dependent terms, and the wave equation for the vector potential in the presence of sources. Canonically conjugate particle and field momenta are then evaluated, from which the Hamiltonian is derived. Elevation of classical variables to quantum operators finally yields the molecular QED Hamiltonian, which is expressed in minimal-coupling and multipolar forms. In the QED formulation, the electromagnetic field is described as a set of independent simple harmonic oscillators. Elementary excitations of the field, the photons, emerge automatically on quantisation.