## Nonlinear Schrodinger Equation

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Suffusion of Yellow
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### Nonlinear Schrodinger Equation

Hey all. I'm reading about the derivation of the NLSE (http://aflb.ensmp.fr/AFLB-303/aflb303m376.pdf) and while I understand the math, the physical meaning of the equation is hard for me to grasp.

As I understand it: a particle is represented/formulated as a complex-valued function. If you take the value of this function at any point and multiply it by it's conjugate, you get the probability that your particle is there (I guess the integral of the function times its conjugate over all R has to be one?). Is this the case? And why does Wikipedia say that the NLSE never describes the evolution a quantum system (this seems pretty quantum to me?) Are all particles represented like this?

Any help would be appreciated! (Disclaimer: I know nearly nothing about physics.)

MarvinM
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### Re: Nonlinear Schrodinger Equation

It sounds like you haven't learned the ordinary Schroedinger equation yet, I can manage that, but not this one. N.B. this may involve numbers that exceed 4. You are right in thinking the integral of PsiPsi* over all space needs to be 1, this does not happen automatically, 'normalising' it is usually a separate step in solving. The usual form of the SE, like the NL one is time independent, so it cannot tell you how things are changing/evolving.

Korrente
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### Re: Nonlinear Schrodinger Equation

Is this the one where the line is both linear and nonlinear at the same time?

Xezlec
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### Re: Nonlinear Schrodinger Equation

Suffusion of Yellow wrote:Hey all. I'm reading about the derivation of the NLSE (http://aflb.ensmp.fr/AFLB-303/aflb303m376.pdf) and while I understand the math, the physical meaning of the equation is hard for me to grasp.

As I understand it: a particle is represented/formulated as a complex-valued function. If you take the value of this function at any point and multiply it by it's conjugate, you get the probability that your particle is there (I guess the integral of the function times its conjugate over all R has to be one?). Is this the case? And why does Wikipedia say that the NLSE never describes the evolution a quantum system (this seems pretty quantum to me?) Are all particles represented like this?

I don't know all that much physics either, but this guy says some things on the first page that set off some alarm bells for me, so I looked it up and apparently this journal is not well-respected by the only person whose opinions came up right away on a Google search. The author has apparently published in serious journals before, but I find no evidence that he's a professor or anything and his niece is apparently insane. This is all very circumstantial and I'm not really trying to claim that this paper is bogus, just that I don't have any immediate reason to trust it and that there are all kinds of weird people in the world, so be careful what you believe.

In any case, since this NLSE thing is apparently advanced advanced quantum mechanics,it's probably a little ambitious to expect to comprehend it without the basics. From Wikipedia's description, it sounds like, yes, it is "pretty quantum" in the sense that it is an equation that has to do with some part of the very broad subject of "quantum physics" (specifically, certain kinds of problems in optics), but no, it does not describe the evolution of a quantum system. That's what the Schrodinger equation is for, so maybe you would want to go read about that instead?

doogly
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### Re: Nonlinear Schrodinger Equation

Steer clear of NLSE. It is not like SE is the linear approximation to NLSE. SE is the actually important thing. The hugely important thing, even!
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BlackSails
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### Re: Nonlinear Schrodinger Equation

doogly wrote:Steer clear of NLSE. It is not like SE is the linear approximation to NLSE. SE is the actually important thing. The hugely important thing, even!

I dont know, it looks sort of not lorentz invariant to me.....

doogly
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### Re: Nonlinear Schrodinger Equation

Oh yeah, but we can wait to fix that. I suppose.
Then again, Klein Gordon did come first historically, so starting with it might not be a terrible anyway.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

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### Re: Nonlinear Schrodinger Equation

Suffusion of Yellow wrote:Hey all. I'm reading about the derivation of the NLSE (http://aflb.ensmp.fr/AFLB-303/aflb303m376.pdf) and while I understand the math, the physical meaning of the equation is hard for me to grasp.

As I understand it: a particle is represented/formulated as a complex-valued function. If you take the value of this function at any point and multiply it by it's conjugate, you get the probability that your particle is there (I guess the integral of the function times its conjugate over all R has to be one?). Is this the case? And why does Wikipedia say that the NLSE never describes the evolution a quantum system (this seems pretty quantum to me?) Are all particles represented like this?

Any help would be appreciated! (Disclaimer: I know nearly nothing about physics.)

Like other people have said the SE is not a linear approximation of the NSLE. The NSLE is (in the context I know) used to model macroscopic nonlinear waves. Do you know what a dispersion relation is? For many kinds of waves, the dispersion relation goes [imath]\omega=Dk^2[/imath]. The NLSE is used when it happens that the frequency of a wave depends on its amplitude. Then you have a dispersion relation [imath]\omega=Dk^2+NA^2[/imath] if A is the amplitude. If the wave is a pulse, rather than a plane wave, this interaction between the amplitude and the frequency creates interesting effects.
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