A comparison is made between the Heisenberg equations of the quantum electron and the dynamical equations of the classical spinning and radiating electron.
For a free Dirac electron, the Heisenberg equations define an internal dynamical system in the rest frame, isomorphic to a finite three-dimensional oscillator with a compact SO(5) phase space, such that the spin of the electron is the orbital angular momentum of the internal dynamics (Barut and Bracken 1980, 1981a). In the present work, the change in this internal dynamics due to an external magnetic field is studied. In order that the internal motion can be distinguished from the centre of mass motion, the solutions of the corresponding Hamilton and Heisenberg equations for the relativistic classical motion and the relativistic quantum mechanical spinless motion are also presented. The solutions for the electron exhibit the effect of the spin terms both in the internal motion and external motion, and we are able to identify the properties of the Zitterbewegung in the external field.
A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.
Abstract This chapter reviews quantum mechanics of atoms, ions and molecules and their associated spectra. We have already invoked empirically Planck’s minimum quantum of action ℏ, and Heisenberg’s uncertainty principle. We were able to get so far, but the dynamics of electrons bound to atoms, ions and molecules requires wave or matrix mechanics of Schrödinger and Heisenberg developed in 1925. The present chapter reviews the elements of quantum theory necessary to understand the interaction of classical EM waves with electrons, bound and free. Symmetries are related to conservation equations, to find constants of the motion. The radiation field is treated mostly as a classical EM wave. Spin-orbitals and the quantum harmonic oscillator are identified as quantum building blocks. An explanation of spontaneous emission of radiation is given using the occupation number representation. Different kinds of radiative transitions found in dilute atmospheres and plasmas are discussed, quantum collisions are described, and regularities in spectra are highlighted. In typical plasmas, the important influence of electrons excited into two, not just one, spin-orbitals, is highlighted.