Some arguments for the wave equation in Quantum theory

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

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The first and second order equations of the quantum theory

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1929.0102 ◽

1929 ◽

Vol 124 (793) ◽

pp. 143-150 ◽

Cited By ~ 1

Keyword(s):

Electromagnetic Field ◽

Wave Equation ◽

Quantum Theory ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Electromagnetic Theory ◽

Second Order ◽

Electromagnetic Mass ◽

First Order ◽

Second Order Equations

The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator ( h /2 πi ∂/∂ x α - eϕ α ) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc . Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.

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Fractional Laplacians viscoacoustic wavefield modeling with k-space-based time-stepping error compensating scheme

Geophysics ◽

10.1190/geo2019-0151.1 ◽

2020 ◽

Vol 85 (1) ◽

pp. T1-T13 ◽

Cited By ~ 1

Author(s):

Ning Wang ◽

Tieyuan Zhu ◽

Hui Zhou ◽

Hanming Chen ◽

Xuebin Zhao ◽

...

Keyword(s):

Wave Equation ◽

Wave Equations ◽

Time Derivative ◽

Time Stepping ◽

Long Time ◽

Sampling Intervals ◽

Spatial Derivatives ◽

The Stability ◽

Wavefield Extrapolation ◽

Space Approach

The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.

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Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.24200/squjs.vol17iss2pp245-253 ◽

2012 ◽

Vol 17 (2) ◽

pp. 245

Author(s):

N.H. Sweilam ◽

T.A. Assiri

Keyword(s):

Wave Equation ◽

Finite Difference ◽

Error Analysis ◽

Wave Equations ◽

Finite Difference Approximation ◽

Difference Approximation ◽

Fractional Wave Equation ◽

Fractional Wave Equations ◽

Explicit Finite Difference ◽

The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.

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The Dual Continuity Equations

10.20944/preprints201808.0341.v1 ◽

2018 ◽

Author(s):

Arbab Arbab ◽

Norah N. Alsaawi

Keyword(s):

Electromagnetic Field ◽

Wave Equation ◽

Gauge Invariance ◽

Maxwell Equations ◽

Continuity Equation ◽

Mathematical Structure ◽

Magnetic Monopoles ◽

System Of Equations ◽

Duality Transformation ◽

Continuity Equations

The ordinary continuity equation relating the current and density of a system is extended to incorporate systems with dual (longitudinal and transverse) currents. Such a system of equations is found to have the same mathematical structure as that of Maxwell equations. The horizontal and transverse currents and the densities associated with them are found to be coupled to each other. Each of these quantities are found to obey a wave equation traveling at the velocity of light in vacuum. London's equations of super-conductivity are shown to emerge from some sort of continuity equations. The new London's equations are symmetric and are shown to be dual to each other. It is shown that London's equations are Maxwell's equations with massive electromagnetic field (photon). These equations preserve the gauge invariance that is broken in other massive electrodynamics. The duality invariance may allow magnetic monopoles to be present inside superconductors. The new duality is called the comprehensive duality transformation.

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PARTICLE EQUATIONS FROM NON-ASSOCIATIVE ALGEBRAS

Canadian Journal of Physics ◽

10.1139/p59-024 ◽

1959 ◽

Vol 37 (2) ◽

pp. 183-188

Author(s):

Richard Bourret

Keyword(s):

Wave Equation ◽

Associative Algebra ◽

Continuity Equation ◽

Wave Equations ◽

Relativistic Wave Equations ◽

Lorentz Transformations ◽

Relativistic Wave ◽

Associative Algebras ◽

Mass Terms

Attention is called to the neglect of linear algebras not representable by matrices in the formation and study of possible relativistic wave equations. An eight-unit non-associative algebra of Cayley is used to construct a bilocal wave equation obeying a continuity equation and possessing invariance under bilocal gauge and (proper) Lorentz transformations. Mass terms are extracted from the equations and particle and interaction interpretations are briefly discussed.

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Constraints on the anisotropic parameters for pseudoelastic vertical transverse isotropy wave equations and the applications on imaging carbonate reservoirs

Geophysics ◽

10.1190/geo2017-0778.1 ◽

2019 ◽

Vol 84 (2) ◽

pp. C85-C94 ◽

Cited By ~ 1

Author(s):

Houzhu (James) Zhang ◽

Hongwei Liu ◽

Yang Zhao

Keyword(s):

Wave Equation ◽

Elastic Wave ◽

Wave Velocity ◽

Wave Equations ◽

Transverse Isotropy ◽

S Wave ◽

Elastic Wave Equation ◽

Computation Efficiency ◽

The Stability ◽

Vertical Transverse Isotropy

Seismic anisotropy is an intrinsic elastic property. Appropriate accounting of anisotropy is critical for correct and accurate positioning seismic events in reverse time migration. Although the full elastic wave equation may serve as the ultimate solution for modeling and imaging, pseudoelastic and pseudoacoustic wave equations are more preferable due to their computation efficiency and simplicity in practice. The anisotropic parameters and their relations are not arbitrary because they are constrained by the energy principle. Based on the investigation of the stability condition of the pseudoelastic wave equations, we have developed a set of explicit formulations for determining the S-wave velocity from given Thomsen’s parameters [Formula: see text] and [Formula: see text] for vertical transverse isotropy and tilted transverse isotropy media. The estimated S-wave velocity ensures that the wave equations are stable and well-posed in the cases of [Formula: see text] and [Formula: see text]. In the case of [Formula: see text], a common situation in carbonate, a positive value of S-wave velocity is needed to avoid the wavefield instability. Comparing the stability constraints of the pseudoelastic- with the full-elastic wave equation, we conclude that the feasible range of [Formula: see text] and [Formula: see text] was slightly larger for the pseudoelastic assumption. The success of achieving high-accuracy images and high-quality angle gathers using the proposed constraints is demonstrated in a synthetic example and a field example from Saudi Arabia.

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The quantum theory of dispersion in metallic conductors

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1929.0125 ◽

1929 ◽

Vol 124 (794) ◽

pp. 409-422 ◽

Cited By ~ 37

Keyword(s):

Electrical Conductivity ◽

Dielectric Constant ◽

Electromagnetic Field ◽

Quantum Theory ◽

Isotropic Medium ◽

Electromagnetic Waves ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Classical Electrodynamics ◽

Metallic Conductivity

1. Formulation of the problem. - The propagation of electromagnetic waves in a homogeneous isotropic medium showing metallic conductivity has been treated phenomenologically on the basis of classical electrodynamics. If in Maxwell's equations for the electromagnetic field curl E = - 1/ c ∂B/∂ t , curl H = 1/ c (∂D/∂ t + 4πI), div D = 4πρ, div B = 0, we assume that D = εE, B = μH, I = σE, (1) where e is the dielectric constant, u the permeability and q the electrical conductivity, we get curl E = - μ/c ∂H/∂ t , curl H = 1/ c (ε ∂E/∂ t 4πσE), div E = 4πρ/ε. div H =0.

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Green’s Function Method for Electromagnetic and Acoustic Fields in Arbitrarily Inhomogeneous Media

10.5772/intechopen.94852 ◽

2020 ◽

Author(s):

Vladimir P. Dzyuba ◽

Roman Romashko

Keyword(s):

Electromagnetic Field ◽

Wave Equation ◽

Inhomogeneous Medium ◽

Acoustic Field ◽

Particle Velocity ◽

Velocity Vector ◽

Function Method ◽

Wave Equations ◽

Inhomogeneous Media ◽

Electric Vector

An analytical method based on the Green\'s function for describing the electromagnetic field, scalar-vector and phase characteristics of the acoustic field in a stationary isotropic and arbitrarily inhomogeneous medium is proposed. The method uses, in the case of an electromagnetic field, the wave equation proposed by the author for the electric vector of the electromagnetic field, which is valid for dielectric and magnetic inhomogeneous media with conductivity. In the case of an acoustic field, the author uses the wave equation proposed by the author for the particle velocity vector and the well-known equation for acoustic pressure in an inhomogeneous stationary medium. The approach used allows one to reduce the problem of solving differential wave equations in an arbitrarily inhomogeneous medium to the problem of taking an integral.

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A relativistic quantum theory of dyons wave propagation

Canadian Journal of Physics ◽

10.1139/cjp-2017-0080 ◽

2017 ◽

Vol 95 (12) ◽

pp. 1200-1207 ◽

Cited By ~ 10

Author(s):

B.C. Chanyal

Keyword(s):

Wave Propagation ◽

Wave Equation ◽

Quantum Theory ◽

Quantum Electrodynamics ◽

Wave Equations ◽

Relativistic Quantum ◽

Relativistic Wave Equation ◽

Field Equations ◽

Continuity Equations ◽

Quantum Wave

Beginning with the quaternionic generalization of the quantum wave equation, we construct a simple model of relativistic quantum electrodynamics for massive dyons. A new quaternionic form of unified relativistic wave equation consisting of vector and scalar functions is obtained, and also satisfy the quaternionic momentum eigenvalue equation. Keeping in mind the importance of quantum field theory, we investigate the relativistic quantum structure of electromagnetic wave propagation of dyons. The present quantum theory of electromagnetism leads to generalized Lorentz gauge conditions for the electric and magnetic charge of dyons. We also demonstrate the universal quantum wave equations for two four-potentials as well as two four-currents of dyons. The generalized continuity equations for massive dyons in case of quantum fields are expressed. Furthermore, we concluded that the quantum generalization of electromagnetic field equations of dyons can be related to analogous London field equations (i.e., current to electromagnetic fields in and around a superconductor).

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