scholarly journals The quantum theory of the emission and absorption of radiation

1927 ◽  
Vol 114 (767) ◽  
pp. 243-265 ◽  
Keyword(s):  
Quantum Theory ◽  
Radiation Field ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Correct Treatment ◽  
Principle Of Relativity ◽  
Satisfactory Theory ◽  
General Method

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 E r is the energy of a component labelled r and θ r is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each E r and θ r 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 ═ Σ r E r + H o equal to the total energy, H o being the Hamiltonian for the atom alone, since the variables E r , θ r obviously satisfy their canonical equations of motion E r ═ — ∂H/∂θ r ═ 0, θ r ═ ∂H/∂E r ═ 1.

scholarly journals Semi-classical approximations based on Bohmian mechanics

2020 ◽  
Vol 35 (14) ◽  
pp. 2050070 ◽  
Author(s):  
Ward Struyve
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Expectation Values ◽  
Usual Approach ◽  
Classical Approximation

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

scholarly journals Quaternionic wave function

2019 ◽  
Vol 34 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Pavel A. Bolokhov
Keyword(s):  
Wave Function ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Nonrelativistic Limit ◽  
Natural Choice ◽  
Complex Wave ◽  
Spinor Formalism ◽  
Function Language

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

scholarly journals A minimalist pilot-wave model for quantum electrodynamics

2007 ◽  
Vol 463 (2088) ◽  
pp. 3115-3129 ◽  
Author(s):  
W Struyve ◽  
H Westman
Keyword(s):  
Wave Function ◽  
Electromagnetic Field ◽  
Quantum Theory ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Wave Model ◽  
Relativistic Quantum Theory ◽  
Pilot Wave ◽  
Free Electromagnetic Field

We present a way to construct a pilot-wave model for quantum electrodynamics. The idea is to introduce beables corresponding only to the bosonic and not to the fermionic degrees of freedom of the quantum state. We show that this is sufficient to reproduce the quantum predictions. The beables will be field beables corresponding to the electromagnetic field and will be introduced in a way similar to that of Bohm's model for the free electromagnetic field. Our approach is analogous to the situation in non-relativistic quantum theory, where Bell treated spin not as a beable but only as a property of the wave function. After presenting this model, we also discuss a simple way for introducing additional beables that represent the fermionic degrees of freedom.

Quantum Radiation Field: Spontaneous Emission and Entangled Photons

2021 ◽  
pp. 247-273
Author(s):  
Duncan G. Steel
Keyword(s):  
Electromagnetic Field ◽  
Excited State ◽  
Radiation Field ◽  
Quantum Electrodynamics ◽  
Equations Of Motion ◽  
Single Mode ◽  
Vacuum Field ◽  
Level System ◽  
Quantum Radiation ◽  
System P

One of the greatest successes in quantum theory, and certainly one of the more important parts for application to devices and applications is the prediction of the emission of light through the quantization of an electromagnetic field. Broadly, this is the field of quantum electrodynamics. In this chapter, we develop the Hamiltonian for the classical electromagnetic field. It is seen that the Hamiltonian for each mode (identified by the k-vector and polarization of the field) of the plane wave electromagnetic field is identical to that of the harmonic oscillator. One unit of energy, ℏω, in a mode is a called a photon. The eigenkets for the system are number states (Fock states). We then consider a two-level system described by a Hamiltonian which couples the two-level quantum system to the quantized electromagnetic field. Using the Weisskopf–Wigner formalism developed in Chapter 14, we solve the equations of motion for the time dependent Schrödinger equation assuming the system starts in the excited state with no radiation present in the vacuum field. The results show the creation of one unit of energy in an electromagnetic mode corresponding to the emission of a photon. The excited state probability decays exponentially with the emission of this photon. We consider the important and special case of such a two-level system but in a cavity restricting the radiation field to a single mode. The Jaynes–Cummings Hamiltonian shows that this system, if started in the excited state, Rabi oscillates with no radiation incident on the system.

CONSISTENT QUANTIZATION OF NONLINEAR SIGMA MODELS COUPLED TO WEYL FERMIONS

1989 ◽  
Vol 04 (11) ◽  
pp. 2797-2810
Author(s):  
E. MORENO ◽  
C. VON REICHENBACH ◽  
F.A. SCHAPOSNIK
Keyword(s):  
Quantum Theory ◽  
Sigma Models ◽  
Path Integral ◽  
Effective Action ◽  
Degrees Of Freedom ◽  
Integral Approach ◽  
Correct Treatment ◽  
Definition Of ◽  
Weyl Fermions ◽  
Nonlinear Sigma Models

We discuss the quantization of 2-dimensional nonlinear sigma models defined in G/H spaces using the path-integral approach. We show that even when anomalies are present, a careful definition of the quantum effective action leads to a consistent quantum theory. The correct treatment of the H degrees of freedom uncovers the presence of a Wess-Zumino action and the anomaly is absorbed.

scholarly journals Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

2018 ◽  
Vol 28 (4) ◽  
pp. 661-677 ◽  
Author(s):  
Gustavo Cordero ◽  
Víctor Santibáñez ◽  
Alejandro Dzul ◽  
Jesús Sandoval
Keyword(s):  
Differential Equations ◽  
Potential Energy ◽  
Degrees Of Freedom ◽  
Direct Method ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Positive Definite Function ◽  
Open Loop ◽  
Definite Function ◽  
Passivity Based Control

Abstract In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2- DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.

scholarly journals The electromagnetic energy of a point charge

1938 ◽  
Vol 168 (934) ◽  
pp. 389-401 ◽  
Keyword(s):  
Quantum Theory ◽  
Classical Theory ◽  
Quantum Electrodynamics ◽  
Point Charge ◽  
Equations Of Motion ◽  
Empty Space ◽  
Electromagnetic Energy ◽  
Point Model ◽  
Field Equations ◽  
Small Sphere

There are many regions for preferring the point model of the electron, in which the field equations of empty space hold all the way up to the centre of the electron, to the Lorentz model, in which the charge is distributed over a small sphere. The point model is not without difficulties, however, and two have attracted special attention. The first is that the field becomes infinite at the charge, so that the Lorentz equations of motion cannot be applied directly; the second is that the ordinary expression leads to an infinite electromagnetic energy in the neighbourhood of the charge. As these difficulties occur both in classical and in quantum electrodynamics it seems reasonable to look for their solution, first in the classical theory, and then try to translate it into the quantum theory. A recent paper by Dirac (1938) has satisfactorily removed the first difficulty from the classical theory. The present paper shows how the second can be removed also. The translation of these methods to quantum theory has not yet been accomplished. Some papers by Wentzel (1933, 1934) have also dealt with this subject, both from the classical and the quantum standpoint, but they do not seem to be altogether without difficulties, and the method is rather complicated to use in actual problems.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

Investigation of Train Dynamics in Passing Through Curves Using a Full Model

Joint Rail ◽  
2004 ◽  
Author(s):  
Mohammad Durali ◽  
Mohammad Mehdi Jalili Bahabadi
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Full Model ◽  
Bogie Frame ◽  
Nonlinear Characteristics ◽  
Train Derailment ◽  
Train Dynamics

In this article a train model is developed for studying train derailment in passing through bends. The model is three dimensional, nonlinear, and considers 43 degrees of freedom for each wagon. All nonlinear characteristics of suspension elements as well as flexibilities of wagon body and bogie frame, and the effect of coupler forces are included in the model. The equations of motion for the train are solved numerically for different train conditions. A neural network was constructed as an element in solution loop for determination of wheel-rail contact geometry. Derailment factor was calculated for each case. The results are presented and show the major role of coupler forces on possible train derailment.

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