# Solitons and Heisenberg uncertainty

• zetafunction
In summary, the conversation discusses the relationship between soliton solutions for the Nonlinear Schroedinger equation and the uncertainty principle. It is noted that although a soliton is a localized wave packet, it still carries some uncertainty. The conversation also addresses the use of the Nonlinear Schroedinger equation, specifically the Gross-Pitaevskii equation, in describing weakly interacting bosons and boson condensates.

#### zetafunction

if we can find a 'soliton' solution for Nonlinear Schroedinguer equation , then does this imply that Uncertainty principle is false ??

since a soliton is a localized wave packet then i can find the position of the soliton and its momentum so apparently i have violated uncertainty.

Well, a soliton doesn't have to be localized that 'well'. It's only localized within a certain region - it would still 'carry some uncertainty'. The same goes for its momentum distribution. But then again, I'm not too familiar with the nonlinear schroedinger equation ;)

Solitons are localized, but they are still spread over some displacement, I don't get how you can violate the HUP.

A wave packet certainly does not violate the HUP, and non-dispersive wave packets certainly do not either.

Also, the Schrodinger equation is linear (otherwise superposition principle would not work), what's the nonlinear Schrodinger equation?

Matterwave said:
what's the nonlinear Schrodinger equation?

There are a family of equations called NSE, one of the most common is the http://en.wikipedia.org/wiki/Gross-Pitaevskii_equation" [Broken]. The unknown function in these equations is not a single particle wave function but rather a quantum field, so things like superposition, probability interpretation, uncertainty principle, etc must be abandoned or modified for this case.

To OP's question is just as false for solitons as it is for ordinary quantum wave packets, so my advice to the OP would be to further study single particle QM.

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If the position and velocity of the whole wave packet means
the position and velocity of one particle, this violates the uncertainty principle.

Because we can determine the position and momentum of the particle at the same time.

And the wave packet is spreading in all space as the particle is moving around and rebounds from the wall.

So the wave packet does not mean a particle.

ExactlySolved said:
There are a family of equations called NSE, one of the most common is the http://en.wikipedia.org/wiki/Gross-Pitaevskii_equation" [Broken]. The unknown function in these equations is not a single particle wave function but rather a quantum field, so things like superposition, probability interpretation, uncertainty principle, etc must be abandoned or modified for this case.

To OP's question is just as false for solitons as it is for ordinary quantum wave packets, so my advice to the OP would be to further study single particle QM.

That is, i have to present a paper about it in my Master , the question is what are these Nonlinear Schroedinguer equation used for ?? could someone give me an example

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zetafunction said:
if we can find a 'soliton' solution for Nonlinear Schroedinguer equation , then does this imply that Uncertainty principle is false ??

since a soliton is a localized wave packet then i can find the position of the soliton and its momentum so apparently i have violated uncertainty.

I just wanted to mention the Schroedinger's coherent states: they represent wave packets of a constant shape; thay are similar to solitons but are linear combinations of eigenstates (principle of superposition holds). Such states minimize the HU relationship. There applets on internet demonstrating that.

A certain average position R(t) does not mean there is no position spread in a wave packet.

Bob_for_short.

zetafunction said:
That is, i have to present a paper about it in my Master , the question is what are these Nonlinear Schroedinguer equation used for ?? could someone give me an example

The Gross-Pitaevski equation is used to describe weakly interacting bosons in an external potential. It can be used to describe bose condensates.

## 1. What are solitons and how do they relate to Heisenberg uncertainty?

Solitons are solitary waves that maintain their shape and velocity as they propagate through a medium. They arise from nonlinear interactions between waves. Heisenberg uncertainty is a fundamental principle in quantum mechanics that states that the position and momentum of a particle cannot both be precisely known at the same time. Solitons are interesting in the context of Heisenberg uncertainty because, unlike traditional waves, they have well-defined positions and momenta, which challenges the principle.

## 2. How do solitons behave differently from traditional waves?

Solitons have a unique ability to maintain their shape and speed as they travel through a medium. This is due to their nonlinear interactions, which allow them to overcome dispersion effects that cause other waves to spread out and lose energy. Solitons also have well-defined positions and momenta, unlike traditional waves which can only be described by probability distributions.

## 3. What applications do solitons have in science and technology?

Solitons have a wide range of applications in fields such as optics, fluid dynamics, and quantum physics. In optics, solitons can be used for long-distance data transmission in fiber optics and for creating ultra-short pulses of light in lasers. In fluid dynamics, solitons can be found in tsunamis, where they can travel great distances without losing energy. In quantum physics, solitons have potential applications in quantum computing and information storage.

## 4. Can solitons violate the Heisenberg uncertainty principle?

No, solitons do not violate the Heisenberg uncertainty principle. While they have well-defined positions and momenta, this does not mean that both quantities can be measured simultaneously with perfect accuracy. The uncertainty principle still holds, but it is not as straightforward to apply to solitons as it is to traditional waves.

## 5. How do scientists study solitons and Heisenberg uncertainty?

Scientists use a variety of experimental and theoretical techniques to study solitons and their relationship to Heisenberg uncertainty. These can include numerical simulations, laboratory experiments, and mathematical models. By studying the behavior of solitons in different systems, scientists can gain a better understanding of the fundamental principles of quantum mechanics and explore potential applications for solitons in various fields.