The Dispersion Electrons in the One-Electron Problem

1929 ◽  
Vol 25 (3) ◽  
pp. 323-330
Author(s):  
J. Hargreaves
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Dipole Moment ◽  
Optical Spectrum ◽  
Wave Equations ◽  
Simple Wave ◽  
Incoherent Scattering ◽  
Central Field ◽  
Dispersion Formula ◽  
The One

In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.

scholarly journals Schrodinger Wave Equation

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Aaron C.H. Davey
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Quantum Theory ◽  
System Change ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
One Dimensional ◽  
Erwin Schrödinger ◽  
The One ◽  
Over Time

The father of quantum mechanics, Erwin Schrodinger, was one of the most important figures in the development of quantum theory. He is perhaps best known for his contribution of the wave equation, which would later result in his winning of the Nobel Prize for Physics in 1933. The Schrodinger wave equation describes the quantum mechanical behaviour of particles and explores how the Schrodinger wave functions of a system change over time. This project is concerned about exploring the one-dimensional case of the Schrodinger wave equation in a harmonic oscillator system. We will give the solutions, called eigenfunctions, of the equation that satisfy certain conditions. Furthermore, we will show that this happens only for particular values called eigenvalues.

A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. A1-A6 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Numerical Modeling ◽  
Wave Equation ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Computational Domain ◽  
Perfect Absorption ◽  
Transition Area ◽  
Efficient Scheme ◽  
Grid Points ◽  
The One

We propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations. This scheme divides the computational domain into boundary, transition, and inner areas. The wavefields within the inner and boundary areas are computed by the wave equation and the one-way wave equation, respectively. The wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one-way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone. The results from our finite-difference numerical modeling tests of the 2D acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area. We obtain equally good performance using pseudospectral and finite-element modeling with the same scheme. Our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption.

The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part IV. Further Results relating to Terms of the Optical Spectrum

1929 ◽  
Vol 25 (3) ◽  
pp. 310-314 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Equation ◽  
Optical Spectrum ◽  
Coulomb Field ◽  
Central Field ◽  
Wave Mechanics

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.

The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III. Term Values and Intensities in Series in Optical Spectra

1928 ◽  
Vol 24 (3) ◽  
pp. 426-437 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Coulomb Field ◽  
Practical Method ◽  
Optical Spectra ◽  
Central Field ◽  
Wave Mechanics ◽  
In Series ◽  
Characteristic Values ◽  
Image Position

In two recent papers the writer has given an account of a practical method of finding the characteristic values and functions of Schrödinger's wave equations for a given non-Coulomb central field. For terms of optical spectra the method is effectively the following. We take the wave equation in the formand require the values of ɛ for the solutions which are zero at the origin and at r = ∞. We consider the result of integrating this equation outwards from P = 0 at r = 0 to a radius r0 at which the deviation from a Coulomb field is negligible, and inwards from P = 0 at r = ∞ to the same radius, with a given value of ɛ; the characteristic values are those values for which these two solutions join smoothly on to one another, i.e. for which they have the same value of η = −P′/P at this radius. For a given ɛ, the solution zero at the origin depends on the particular atom; the solution zero at infinity can be expressed in a form independent of any particular atom.

scholarly journals The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods

1928 ◽  
Vol 24 (1) ◽  
pp. 89-110 ◽  
Author(s):  
D. R. Hartree
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Initial Conditions ◽  
Energy Parameter ◽  
Coulomb Field ◽  
Characteristic Functions ◽  
Central Field ◽  
Wave Mechanics ◽  
Numerical Work ◽  
Characteristic Values

The paper is concerned with the practical determination of the characteristic values and functions of the wave equation of Schrodinger for a non-Coulomb central field, for which the potential is given as a function of the distance r from the nucleus.The method used is to integrate a modification of the equation outwards from initial conditions corresponding to a solution finite at r = 0, and inwards from initial conditions corresponding to a solution zero at r = ∞, with a trial value of the parameter (the energy) whose characteristic values are to be determined; the values of this parameter for which the two solutions fit at some convenient intermediate radius are the characteristic values required, and the solutions which so fit are the characteristic functions (§§ 2, 10).Modifications of the wave equation suitable for numerical work in different parts of the range of r are given (§§ 2, 3, 5), also exact equations for the variation of a solution with a variation in the potential or of the trial value of the energy (§ 4); the use of these variation equations in preference to a complete new integration of the equation for every trial change of field or of the energy parameter avoids a great deal of numerical work.For the range of r where the deviation from a Coulomb field is inappreciable, recurrence relations between different solutions of the wave equations which are zero at r = ∞, and correspond to terms with different values of the effective and subsidiary quantum numbers, are given and can be used to avoid carrying out the integration in each particular case (§§ 6, 7).Formulae for the calculation of first order perturbations due to the relativity variation of mass and to the spinning electron are given (§ 8).The method used for integrating the equations numerically is outlined (§ 9).

scholarly journals SCHRÖDINGER EQUATION OF GENERAL POTENTIAL

2011 ◽  
Vol 25 (15) ◽  
pp. 2009-2017
Author(s):  
XIANG-YAO WU ◽  
BO-JUN ZHANG ◽  
HAI-BO LI ◽  
XIAO-JING LIU ◽  
NUO BA ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Equations ◽  
General Potential ◽  
Quantum Wave ◽  
Quantum Wave Equation

A generalization of quantum mechanics is proposed, where the Lagrangian is the general form. The new quantum wave equation can describe the particle which is in general potential [Formula: see text], and the Schrödinger equation is only suited for the particle in common potential V(r, t). We think these new quantum wave equations can be used in some fields.

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Wave Equations ◽  
Propagation Speed ◽  
Fractional Diffusion ◽  
Maximum Point ◽  
Diffusion Wave ◽  
Constant Coefficients ◽  
The One ◽  
Signalling Problem

AbstractIn this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

scholarly journals A Symmetry Hidden at The Center of Quantum Mathematics Causes a Disconnect Between Quantum Math and Quantum Mechanics

2017 ◽  
Vol 13 (4) ◽  
pp. 7379-7386
Author(s):  
Jeffrey Boyd
Keyword(s):  
Quantum Mechanics ◽  
Time Reversal ◽  
Wave Equations ◽  
Wave Direction ◽  
The One ◽  
Richard Feynman ◽  
The Universe ◽  
Direction Reversal ◽  
Successful Science ◽  
Quantum Mathematics

Although quantum mathematics is the most successful science ever, that does not mean we live in the universe described by quantum mechanics. This article is entirely based on symmetry. Two symmetrical universes could have exactly the same mathematics, but differ in other respects. The motivation for seeking symmetry inside quantum mathematics is that the QM picture of nature is bizarre. Richard Feynman says no one can understand it. We propose that the quantum world is not bizarre. QM portrays the wrong universe: the symmetrical one, not the one we inhabit. If quantum waves travel in the opposite direction as what is expected, then we would have the same math but a different universe, one that is recognizable and familiar. Wave equations are symmetrical with respect to time reversal. This means they are symmetrical with respect to wave direction reversal (with time going forwards). This wave equation symmetry is the basis of the symmetry of two models of the universe, only one of which is congruent with the universe we inhabit.

A Bremmer series for radial wave equations

1977 ◽  
Vol 55 (24) ◽  
pp. 2150-2157 ◽  
Author(s):  
W. E. Couch ◽  
R. J. Torrence
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Wkb Approximation ◽  
Spherically Symmetric ◽  
Radial Wave ◽  
One Dimensional ◽  
Similar Series ◽  
Leading Term ◽  
The One ◽  
Radial Wave Equation

The Bremmer series solution of the one-dimensional Helmholtz equation with variable velocity is generalized to obtain a similar series for the radial wave equation with a spherically symmetric velocity function. Since the leading term of Bremmer's series is the one-dimensional WKB approximation, we obtain an approximation for the radial wave equation analogous to the WKB approximation.

Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind

2014 ◽  
Vol 24 (12) ◽  
pp. 1450160 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Traveling Wave ◽  
Wave Equations ◽  
Solitary Wave Solution ◽  
Nonlinear Wave Equations ◽  
Straight Line ◽  
The One

In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe-α|x-ct|. In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon).

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