The difference between Schrödinger equation derived from Schrödinger map and Landau–Lifshitz equation

2012 ◽  
Vol 376 (4) ◽  
pp. 231-235 ◽  
Author(s):  
Ganshan Yang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Lifshitz Equation ◽  
The Difference ◽  
Schrödinger Map

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.

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.

scholarly journals Mean-field dynamics for mixture condensates via Fock space methods

2019 ◽  
Vol 31 (08) ◽  
pp. 1950027
Author(s):  
Gustavo de Oliveira ◽  
Alessandro Michelangeli
Keyword(s):  
Schrödinger Equation ◽  
Density Operator ◽  
Schrodinger Equation ◽  
Field Model ◽  
Fock Space ◽  
Mean Field ◽  
Mean Field Model ◽  
Two Component ◽  
The Difference ◽  
Reduced Density

We consider a mean-field model to describe the dynamics of [Formula: see text] bosons of species one and [Formula: see text] bosons of species two in the limit as [Formula: see text] and [Formula: see text] go to infinity. We embed this model into Fock space and use it to describe the time evolution of coherent states which represent two-component condensates. Following this approach, we obtain a microscopic quantum description for the dynamics of such systems, determined by the Schrödinger equation. Associated to the solution to the Schrödinger equation, we have a reduced density operator for one particle in the first component of the condensate and one particle in the second component. In this paper, we estimate the difference between this operator and the projection onto the tensor product of two functions that are solutions of a system of equations of Hartree type. Our results show that this difference goes to zero as [Formula: see text] and [Formula: see text] go to infinity.

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