For waves described by the focusing nonlinear schrodinger equation fnls, we present an effective dispersion relation edr that arises dynamically from the interplay between the linear. In this video i show how the dispersion relation and schrodinger equation are important to eachother. There is no rigorous derivation of schrodingers equation from previously. The numerical dispersion relation equation dre for the linear schrodinger equation can be derived as 29. The solutions to this equation can be built up from exponential functions. This is known as the debroglie relation, and it actually predates the schrodinger equation. An example energy k ko consider a conduction energy band with the dispersion. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. This is the socalled dispersion relation for the above wave equation. But as well see, it is somewhat of a trivial dispersion relation, in. For nonrelativistic quantum physics the basic equation to be solved is the schr odinger equation.
Musslimani department of mathematics, florida state university, tallahassee, florida 323064510, usa. The schrodinger equation describes for example how quantum states of a physical system change in time. Which quantum theory must be reconciled with gravity. The dispersion relation, a relation between the wave frequency. Recall that we did not derive the tise, we simple constructed a differential equation that is consistent with the freeparticle wave function. Dispersion relations, stability and linearization 1. The nonlinear schrodinger equation and sources 15 where p is the integration constant, which has been set to zero in the following. Now that we understand the dispersion relation for systems, its easy to understand the dispersion relation for the schrodinger equation.
In reverse, if we accept schrodinger equation as a basic principle, then the classical newtons law of motion can be derived from schrodinger equation. The schrodinger wave equation and probability interpretation. The schrodinger and kleingordon equations are directly derived from classical lagrangians. The usual dispersive estimate for the schrodinger equation in. Deriving time dependent schrodinger equation from wavemechanics, schrodinger time independent nilesh p. Pdf the nonlinear schrodinger equation with white noise. Schroedinger equation einsteins relation between particle energy and frequency eq. A side goal of this homework task is to also determine the dispersion relation that i can get by substituting the supposed solution. We extend a result on dispersion for solutions of the linear schr.
Pdf effective dispersion in the focusing nonlinear. Thus the single nonlinear ordinary differential equation we have to solve is 2. Deriving time dependent schrodinger equation from wave. Derivation of the schrodinger equation and the kleingordon equation from first principles gerhard grossing austrian institute for nonlinear studies parkgasse 9, a1030 vienna, austria abstract.
Because the energy eis the eigenvalue of the hamitonian, we act hagain on the dirac wave function and. Section 3 introduces the idea of operators in quantum mechanics and develops differential operators for momentum and kinetic energy in onedimensional motion. The aim of this work is to prove that this latter equation is globally well posed in l 2 or h 1. This search for an equation describing matter waves was carried out by erwin schroedinger.
On dispersive equations and their importance in mathematics. But as well see, it is somewhat of a trivial dispersion relation, in the sense that there is no dispersion. Dispersion of discontinuous periodic waves math user home pages. For nondispersive systems, like most of what weve covered so far. Handout 24 schrodinger equation cornell university. After including the nonlinearity, we resort to the multiple time scales method to reduce the dispersion relation to the three dimensional nonlinear schrodinger equation. Schrodinger equation reading french and taylor, chapter 3 quantum mechanics sets probabilities outline wave equations from. On dispersive equations and their importance in mathematics gigliola sta. Apr 06, 2020 the schrodinger equation also known as schrodingers wave equation is a partial differential equation that describes the dynamics of quantum mechanical systems via the wave function. Quartic normal forms for the periodic nonlinear schrodinger. By combining maxwells equations, the wave equation for electromagnetic waves can be obtained. In addition to the geometrydependent and materialdependent dispersion relations, the overarching kramerskronig relations describe the frequency dependence of wave propagation and attenuation.
Vlasov equation and poissons equation, the dispersion relation can be derived for example by following steps in gurnett and bhattacharjee 2005. The kleingordon equation is a relativistic equation, the schr. Walter gordon published the equation that now bears there name the kleingordon equation that was exactly the same as schr odinger s rst equation. In the previous derivation, we assumed that the waves were. We start with the classical wave equation, as derived from maxwells equations governing classical electrodynamics see the appendix. Lecture 1 position representation of quantum state function. Wave mechanics and the schrodinger equation theory of. Dispersive estimates in rn n2 for the schrodinger and the. Nonetheless, the derivation of the newtonian potential from the weak.
Newtons laws, the schrodinger equation does not give the trajectory of a particle, but rather the wave function of the quantum system, which carries information about the wave nature of the particle, which allows us to only discuss the probability of nding the particle in di erent regions of space. Nonlinear dispersion relation for nonlinear schrodinger equation. The completely integrable variant of semidiscrete nonlinear schrodinger equation, known as the ablowitzladik equation was introduced. Dispersion for the schrodinger equation on discrete trees. Oct 03, 2011 in this video i show how the dispersion relation and schrodinger equation are important to eachother.
Mesoscopic systems are those that have some dimensions which are too small to be treated using classical physics while they are too large to be considered on the quantum level only. Our aim is to partially complete the discussion on waveguide solutions in 11, section. Now we will consider how to solve the schrodinger equation in general, when there is also a nonzero potential energy term, vx say, due to an electric. The wavefunction of a free particle is discussed along with the corresponding probability density function and its relation to the heisenberg uncertainty principle. May 28, 2014 now that we understand the dispersion relation for systems, its easy to understand the dispersion relation for the schrodinger equation. Recall the dispersion relation is defined as the relationship between the. In theoretical physics, the onedimensional nonlinear schrodinger equation nlse is a nonlinear variation of the schrodinger equation. Bardapurkar 32 introduction quantum mechanics is an essential part of undergraduate syllabus in physics as well as in chemistry. The dispersion relation of nonlinear equations is that of their linearization. Thus, we see that if the amplitude function of our wave packet is a solution of the nls equation, all terms of order o 3 or lower that are proportional to the carrier wave will cancel. Nonlinear schrodinger equation dispersion relation. The momentum region n p a equat ion exce p t if the p oten tial is tim e dep end en t, a d evelopmen t w e wil l n ot b e d iscu ssing h ere.
In other words, we have a solution of the nonlinear maxwells equations, up to terms of o 4. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. The trajectory, the positioning, and the energy of these systems can be retrieved by solving the schrodinger equation. Dispersion relation and schrodinger equation youtube. Quartic normal forms for the periodic nonlinear schrodinger equation with dispersion management panayotis panayotaros. Dispersion plays a very important role in the analysis of certain pdes, and in particular schr odinger s equation. In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. Bloch managed to prove a very important theorem, which states that the solution to this equation must take the following form. With periodic potential in 1d scipio cuccagna abstract. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to boseeinstein condensates confined to highly anisotropic cigarshaped traps, in the meanfield regime. For a conservative system, this relation can be expressed through \ \rm e h\vec p,\vec r, \ where h is the hamiltonian function. It is found that the size of the instability region in wavenumber space decreases with increasing. Waveguide solutions for a nonlinear schrodinger equation with mixed dispersion denis bonheure and robson nascimento ao nosso amigo djairo com admira. In ecis03 the dispersion relation as described by c.
Phys a528 1991 253, is added to the real potencial, it is a generalisation of the dispersion relation as described in mahaux, c. From the dispersion relation follows by multiplication with the wave function. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines. Department of applied mathematics university of colorado at boulder boulder, co 803090526 abstract. Heuristically, one sometimes imagines that a \typical solution to a dispersive pde is. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. Dispersion for the schrodinger equation on discrete trees the motivation for studying thin structures comes from mesoscopic physics and nanotechnology. Chapter 4 schroedinger equation mit opencourseware. Dispersion relation equation preserving fdtd method for.
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