As well known, an electron will produce radiative reaction force when the electron emits electromagnetism radiation. The electron radiative effect had not been considered in Schrödinger wave equation. In this paper, we give the nonlinear quantum wave equations for the radiative electron and some dissipative systems.
The inexactness of the time-dependent Schrödinger equation of a charged particle in an external electromagnetic field is discussed in terms of the damping effect of the radiation. A possible improvement is to add a nonlinear term representing this effect to the linear Schrödinger equation. Conditions for the nonlinear term are investigated and it is demonstrated that the obtained nonlinear Schrödinger equation may present state evolutions similar to the wave-function reduction and transitions between stationary states.
The close connection between the Jacobi function for a velocity-dependent dissipative system and the nonlinear Schrödinger–Langevin equation is used to study some of the features of the latter equation. Starting with the Hamiltonian for a classical system, different forms of the Jacobi generating function are obtained. With the help of these, corresponding solutions of the nonlinear wave equation are studied. It is found that the solution of the wave equation is not unique, and that it admits quasi-stationary, decaying, and asymptotically stationary solutions. Since this nonlinear wave equation is derivable from a classical generating function, it preserves some of the characteristic properties of the linear quantum theory. Among these are: (a) the lack of any correlation between noninteracting particles moving in a viscous medium, (b) the gauge independence of the observable quantities, and (c) the constancy of the density of states in phase space. This discussion can also be extended to the problem of quantal description of a harmonically bound radiating electron.
Abstract The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is noil-relativistic only on account of the time being counted throughout as a c-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty. The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if Er is the energy of a component labelled r and θr is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each Er and θr to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the Hamiltonian H ═ ΣrEr + Ho equal to the total energy, Ho being the Hamiltonian for the atom alone, since the variables Er, θr obviously satisfy their canonical equations of motion Er ═ — ∂H/∂θr ═ 0, θr ═ ∂H/∂Er ═ 1.
A mathematical model (the Schrodinger Equation) and a physical model (Spinning Charged Ring Model) of the electron are compared for their ability to predict waves emitted by electrons. Although it is a fundamental postulate of Quantum Mechanics, the Schrodinger Equation does not allow for an interpretation of wave-functions as physical waves, cannot be a com- plete theoretical description of a micro physical system, neglects intrinsic characteristics of particle motion, and fails in several other ways to correspond to physical reality. The Ring Model of Electrons makes accurate and complete predictions of fields and waves by using the locations and internal motions of charge elements that make up the electron in order to determine the external fields of the electron. The physical Ring Model, compli- ant with the law of cause and effect, explains the meaning of Planck's Constant and accu- rately predicts the Photoelectric Effect, wavelengths of hydrogen line spectra, blackbody radiation, and the Compton Wavelength. An unbiased assessment of these facts implies that electrons are small physical particles that closely resemble the Spinning Charged Ring Model, and these electrons are a source of electromagnetic waves.
The theory of absorption and emission of electromagnetic radiation by an oscillator consisting of the atomic nucleus and one electrically charged particle is deduced using classical electrodynamics. In the steady state of an atom, emission and absorption of electromagnetic radiation are equal, so the atom is stable. In order to include reactive effects of electromagnetic radiation in the motion equations, the Newton equation is modified by adding the radiative reaction force. This paper is an introduction to the derivation of the basic assumptions of quantum mechanics.
An effective model for describing the relativistic quantum dynamics of a radiating electron is developed via a relativistic generalization of the Lindblad master equation. By incorporating both radiation reaction and vacuum fluctuations into the Dirac equation within an open quantum system framework, our approach captures the Zitterbewegung of the electron, ensuing noncommutativity of its effective spatial coordinates, and provides the quantum analogue of the Landau-Lifshitz (LL) classical equation of motion with radiation reaction. We develop the corresponding phase-space representation via the relativistic Wigner function and derive the semiclassical limit through a Foldy-Wouthuysen transformation. The latter elucidates the signature of quantum vacuum fluctuations in the LL equation, and shows its relationship with the corrected Sokolov equation. Our results offer a robust framework for investigating quantum radiation reaction effects in ultrastrong laser fields.