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Linear superposition 
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#1
Nov2706, 08:04 AM

P: 140

I have some difficulty in understanding linear superposition's domain of validity and reading Jackson's 1st chapter.I found the following lines but to confess it out, could not grasp the whole meaning...Please help.
"The Maxwell equations in vacuum are linear in the fields E and B.This linearity is exploited so often, for example,...,that it is taken for granted.There are of course, circumstances where nonlinear effects occur..." "What evidence do we have to support the idea of linear superposition?At the macroscopic level, all sorts of experiments test linear superposition at the level of 0.1% accuracy...At the macroscopic and even at the atomic level, linear superposition is remarkably valid." "It is in the subatomic domain that departures from linear superpossition is legitimately sought.As charged particles approach each other very closely, electric field stengths become enormous.If we think of a chargd particle as a localized disstribution of charge, we see that its electromagnetic energy grows larger an larger as the charge is localized more and more.It is natural,to avoid infinite selfenergies of point particles,to speculate that some sort of saturation occurs,that field strengths have some upper bound.such classical nonlinear theories have been studied in the past...(BornInfld equation)...It suffices to illustrate the general idea.Fields are obviously modified at short diastances;all electromagnetic energies are finite.But for such theories suffer from arbitrariness in the the manner of how nonlinearity occrs and also from from grave problems with a trnsition to quantum theory.Furthermore, there is no evidence of this classical nonlinearity...(the next part contains examples that quantum theory correctly predicts using linear superposition)." So, where we are?Once he says at subatomic level linearity does not hold good; after that he says quantum mechanics correctly predicts many things using linear superpostion at the subatomic level.I think what he says is true...But, I am going wrong somewhere...Please help. 


#2
Nov2706, 08:13 AM

Sci Advisor
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P: 11,896

There are plenty of nonlinear effects in macroscopic physics and classical electromagnetism makes no exception. For instance the magnetic media. There no longer is a linear relationship between the magnetic induction B and the magnetic field intensity H. The same happens with dielectric media as well.
As for the linearity in the quantum regime, the discussion is vast. For starters, it suffices to say that a linear theory of QM works pretty well. Daniel. 


#3
Nov2706, 12:03 PM

P: 140

So, we have
1.Plenty of examples where linear superposition is valid in macroscopic world. 2.Some nonlinear behaviour in macroscopic world, as idicated by you and overlooked by me. As it appears, you supplied some information but did not clear the confusionin the subatomic domain Linear superposition does not hold good but a linear theory in QM works pretty well...These two are contradicting...atleast apparently.Is it that according to classical theory the nonlinearity exists, without proof, but in QM it may be ignored in the first approx.?But as we believe the reality should be unique.Classical theory and QM should predict the same thing... 


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