Understand Linear Superposition:Maxwell, Quantum Mechanics & Beyond

In summary, Daniel is discussing the domain of validity of linear superposition and how it fares in different situations. Daniel points out that there are many nonlinear effects in the macroscopic world, which classical electromagnetism does not account for. However, Daniel argues that a linear theory of QM works well even in the subatomic realm.
  • #1
Kolahal Bhattacharya
135
1
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 self-energies of point particles,to speculate that some sort of saturation occurs,that field strengths have some upper bound.such classical non-linear theories have been studied in the past...(Born-Infld 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 non-linearity occrs and also from from grave problems with a trnsition to quantum theory.Furthermore, there is no evidence of this classical non-linearity...(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.
 
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  • #2
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
So, we have---
1.Plenty of examples where linear superposition is valid in macroscopic world.
2.Some non-linear behaviour in macroscopic world, as idicated by you and overlooked by me.
As it appears, you supplied some information but did not clear the confusion---in 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 non-linearity 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...
 

What is linear superposition?

Linear superposition is a principle in physics that states that the total response of a system to multiple stimuli is the sum of the responses to each individual stimulus. In other words, the overall effect is a combination of the individual effects.

How does Maxwell's theory of electromagnetism relate to linear superposition?

Maxwell's theory of electromagnetism, also known as Maxwell's equations, describes the behavior of electric and magnetic fields. It is based on the principle of linear superposition, meaning that the combined effect of multiple electric and magnetic fields is the sum of the individual fields.

What is the role of quantum mechanics in understanding linear superposition?

Quantum mechanics is a branch of physics that explains the behavior of particles at the atomic and subatomic level. It is essential in understanding linear superposition because it explains how particles can exist in multiple states simultaneously, and how these states can be combined to create new states through superposition.

Can linear superposition be observed in real-world phenomena?

Yes, linear superposition can be observed in various real-world phenomena, such as the interference patterns of light waves, the behavior of quantum particles, and the combination of multiple sound waves to create a complex sound.

Are there any applications of linear superposition in technology?

Yes, linear superposition has various applications in technology, including in quantum computing, where multiple qubits (quantum bits) can exist in a superposition of states, allowing for more complex calculations. It is also used in signal processing, where multiple signals can be combined to create a more accurate signal. Additionally, it is used in medical imaging techniques, such as MRI, to create 3D images by combining signals from multiple sources.

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