Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

146 submissions , 123 unreviewed
3,953 questions , 1,403 unanswered
4,889 answers , 20,761 comments
1,470 users with positive rep
506 active unimported users
More ...

How does non-linear behaviour arise from the inherently linear QM framework?

+ 1 like - 0 dislike
492 views

Quantum mechanics is a linear theory, living in a Hilbert space with built-in linearity. It has even been argued that introducing non-linearity in the quantum theory would allow for superluminal signalling.

As far as I know there is also no experimental evidence showing that QM breaks down at a certain scale.

So why is it then that the world exhibits rich non-linear behaviour? Where does the non-linearity arise from mathematically?

EDIT: On examples of nonlinear behaviour:

  • Properties of materials (electrical resistance, elasticity)
  • Chaotic dynamics
  • Complex systems
This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user miha priimek
asked Jan 18, 2015 in Theoretical Physics by miha priimek (5 points) [ no revision ]
we can't know what examples of non-linearity you have in mind. Can you give some examples?

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user Sofia
There are at least two points where it can enter: (1) A linear PDE can have a close relation to a non-linear ODE. Classical Hamilton-Jacobi theory allows you to formulate classical mechanics in the form of a linear PDE. (2) If you describe a subsystem by a density matrix, the evolution equation for the density matrix can have non-linear terms modeling the interaction of the subsystem with the environment. (I don't want to prevent anybody from writing a proper answer by this comment, even if it should use the same examples. I'm just too lazy to write a detailed answer.)

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user Thomas Klimpel
as for the break-down of QM, see this answer physics.stackexchange.com/questions/159922/…. The scale where such break-down is expected to occur is given by Planck's length.

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user Phoenix87

1 Answer

+ 0 like - 0 dislike

Non-linearity arises when one takes the limit of quantum dynamics in some sense. Two standard examples are:

1) the semiclassical approximation (i.e. Born's correspondence principle) where in the limit of large quantum numbers ("$\hslash\to 0$") quantum linear dynamics becomes the classical (usually non-linear) one;

2) Mean field approximation (i.e. the limit of a very large number of particles), where the dynamics of each component of the system is modelled by an effective non-linear dynamics (i.e. Hartree or Gross-Pitaevskii equations as the mean field limit of many-bosons systems in condensed matter).

The subject of analyzing rigorously this classical or mean field limit is a very active subject in the domain of mathematical physics/analysis of PDEs.

This post imported from StackExchange Physics at 2015-05-13 18:58 (UTC), posted by SE-user yuggib
answered Jan 18, 2015 by yuggib (360 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...