De Broglie waves, the Schrödinger equation, and relativity. I. Exclusion of the rest mass energy in the dispersion relation

2020 ◽  
Vol 33 (1) ◽  
pp. 96-98
Author(s):  
Masanori Sato
Keyword(s):  
Dispersion Relation ◽  
Synchrotron Radiation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rest Mass ◽  
Mass Energy ◽  
De Broglie Waves ◽  
Virtual Photons ◽  
The Difference

The difference between de Broglie waves and the Schrödinger equation is the rest mass. The dispersion relation of de Broglie waves includes the rest mass, but the Schrödinger equation does not. Synchrotron radiation is when de Broglie waves shake off virtual photons and emit real photons. It also shows that synchrotron radiation is not compatible with relativity.

The Schrödinger Equation

2020 ◽  
pp. 265-288
Author(s):  
M. Suhail Zubairy
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Quantum Tunneling ◽  
Schrodinger Equation ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Particle Dynamics ◽  
Quantization Condition ◽  
De Broglie Waves ◽  
Macroscopic Objects

In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.

Nonlinear dispersion relation for nonlinear Schrödinger equation

1988 ◽  
Vol 39 (2) ◽  
pp. 297-302 ◽  
Author(s):  
J. C. Bhakta
Keyword(s):  
Dispersion Relation ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Instability Region ◽  
Nonlinear Dispersion ◽  
Nonlinear Schrödinger ◽  
Nonlinear Dispersion Relation ◽  
Cubic Nonlinear Schrödinger Equation

By using the average-Lagrangian method (average variational principle), a nonlinear dispersion relation has been derived for the cubic nonlinear Schrödinger equation. It is found that the size of the instability region in wavenumber space decreases with increasing field amplitude in comparison with the linear theory.

scholarly journals COMPLEXIFIER VERSUS FACTORIZATION AND DEFORMATION METHODS FOR GENERATION OF COHERENT STATES OF A 1D NLHO I: MATHEMATICAL CONSTRUCTION

2013 ◽  
Vol 10 (10) ◽  
pp. 1350056 ◽  
Author(s):  
R. ROKNIZADEH ◽  
H. HEYDARI
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Harmonic Oscillator ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Jacobi Polynomials ◽  
One Dimensional ◽  
The Difference ◽  
Mathematical Construction

Three methods: complexifier, factorization and deformation, for construction of coherent states are presented for one-dimensional nonlinear harmonic oscillator (1D NLHO). Since by exploring the Jacobi polynomials [Formula: see text], bridging the difference between them is possible, we give here also the exact solution of Schrödinger equation of 1D NLHO in terms of Jacobi polynomials.

scholarly journals Determination of a Control Parameter for the Difference Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mesut Urun
Keyword(s):  
Schrödinger Equation ◽  
Difference Scheme ◽  
Schrodinger Equation ◽  
Control Parameter ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
Accuracy Difference ◽  
The Difference ◽  
The Stability

The first order of accuracy difference scheme for the numerical solution of the boundary value problem for the differential equation with parameterp,i(du(t)/dt)+Au(t)+iu(t)=f(t)+p,0<t<T,u(0)=φ,u(T)=ψ, in a Hilbert spaceHwith self-adjoint positive definite operatorAis constructed. The well-posedness of this difference scheme is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained.

scholarly journals About calculation of the ground-state energy of the helium atom

2021 ◽  
Vol 2067 (1) ◽  
pp. 012002
Author(s):  
E V Baklanov ◽  
P V Pokasov ◽  
A V Taichenachev
Keyword(s):  
Perturbation Theory ◽  
Numerical Calculation ◽  
Schrödinger Equation ◽  
Helium Atom ◽  
Ground State Energy ◽  
Ground State ◽  
Schrodinger Equation ◽  
State Energy ◽  
Calculation Results ◽  
The Difference

Abstract Two versions of the numerical calculation of the ground state energy of the helium atom are compared. First, the nonrelativistic Schrödinger equation with a fixed nucleus is solved, and then the perturbation theory is used. Another version solves this problem exactly. Comparison shows that the difference between the calculation results is 94 kHz.

Time-dependent versus time-independent probabilities of transmission and reflection of a quantum particle incident upon potentials with large changes

2015 ◽  
Vol 93 (11) ◽  
pp. 1227-1234 ◽  
Author(s):  
Mark R.A. Shegelski ◽  
Kevin Malmgren
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Quantum Particle ◽  
Time Dependent ◽  
Step Potential ◽  
Localized State ◽  
Square Well ◽  
The Difference ◽  
Transmission And Reflection

We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.

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