The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c
)2, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/
c2. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/
c3, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2
e2/3
c3)(d〈v〉/dt
)2, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.
Abstract According to classical mechanics, electron acceleration results in electromagnetic radiation while in quantum mechanics radiation is considered to be arising out of a transition of the charged particle from a higher to a lower energy state. A different narrative is presented in quantum field theory, which considers radiation as an outcome of the perturbation of zero point energy of quantum harmonic oscillator which results in a change in density of electrons in a given state. The theoretical disconnect in the phenomenological aspect of radiation in classical and quantum mechanics remains an unresolved theoretical challenge. As a charged particle changes its energy state, its wavefunction undergoes a spatial phase change, hence, we argue that the spatial phase symmetry breaking of the wavefunction is a critical aspect of radiation in quantum mechanics. This is also observed in Josephson junction, where a static voltage induces spatial phase symmetry breaking of current resulting in emission of electromagnetic waves. As temporal symmetry breaking of the magnetic vector potential generates classical radiation and a wave-function of a charged particle can always be associated with a specific magnetic vector potential; the concept of radiation under spatial phase symmetry breaking offers a novel perspective towards unifying the phenomenon of radiation in quantum mechanics and classical electromagnetism.
The mathematical expression for the electromagnetic current of a quantum particle describes an extended charge in the sense that it is non-vanishing over the volume of the wave packet or the Schroedinger wave for the particle. In the hydrodynamic model this charge current and associated density are interpreted as a physical reality. Coupling such a current to a classical electromagnetic field in the soft photon limit then results in a theory for radiation. It is shown that this predicts that bremsstrahlung is sometimes greatly suppressed when the force acting on the particle is due to a classical potential field and when the volume over which the force is active is small compared to the volume of the particle's wave packet. Solving the same problem using conventional quantum radiation theory gives a different result. Therefore it is possible to test this effect experimentally and either confirm or rule out a hydrodynamic model for Schroedinger wave mechanics. An experiment is proposed to make such a test. It involves an electron microscope with a Wien filter for producing monochromatic beam electrons and an accurate energy measurement of the particle after passing through a local force field.
INTRODUCTION A theory of classical quantum mechanics (CQM), derived from frst principles, successfully applies physical laws on all scales. The classical wave equation is solved with the constraint that a bound electron cannot radiate energy. The mathematical formulation for zero radiation based on Maxwell's equations follows from a derivation by Haus. 2 The function that describes the motion of the electron must not possess spacetime Fourier components that are synchronous with waves traveling at the speed of light. CQM gives closed form solutions for the atom, including the stability of the n=l state and the instability of the excited states, the equation of the photon and electron in excited states, the equation of the free electron, and photon which predict the wave particle duality behavior of particles and light. The current and charge density functions of the electron may be directly physically interpreted. For example, spin angular momentum results from the motion of negative
A semiclassical method of approach to the origin of radiation in a simple quantum transition is reinterpreted and extended. The resulting concepts are applied to an analysis of the Compton effect, yielding the Klein-Nishina formula as usual. More significant is the natural appearance in this case of photonlike behavior of the purely classical electromagnetic fields. The same phenomenon can be hypothesized to occur in any transition in which a quantized-photon nature of radiation appears evident. A similar classical viewpoint taken toward the Lamb shift leads to a straightforward explanation which seems to give very good agreement with experiment. The calculated frequency for the $2{P}_{\frac{1}{2}}\ensuremath{\rightarrow}2{S}_{\frac{1}{2}}$ transition in H is 1064.57 MHz = 0.0355102 ${\mathrm{cm}}^{\ensuremath{-}1}$. The argument also has some important implications for ideas about the physical nature of quantum states.