We live in a world where almost all macroscopic physical phenomena are non-linear, while the description of microscopic phenomena is based on quantum mechanics which is linear by definition. What are the physics points of connection between the two descriptions?
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4$\begingroup$ I'm not sure this question makes any real sense. It what way are you calling quantum mechanics linear? Sure, wavefunctions superpose linearly, but what's the problem? Voting to close, I'm afraid. $\endgroup$– NoldorinCommented Nov 22, 2010 at 14:46
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32$\begingroup$ What's wrong with this question? (Aside from being really difficult to answer properly.) $\endgroup$– Peter ShorCommented Nov 22, 2010 at 21:21
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7$\begingroup$ Excellent question, I should ask something similar but perhaps better defined. $\endgroup$– Carl BrannenCommented Feb 5, 2011 at 0:26
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6$\begingroup$ Why is this question closed, it is legitimate to ask about the connection between linear macroscopic laws and macroscopically observed nonlinearities. $\endgroup$– DilatonCommented Aug 3, 2013 at 16:25
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9$\begingroup$ Voting to reopen because closing this made absolutely zero sense. It's a good question - I can't even begin to imagine what people thought was wrong with it. $\endgroup$– N. VirgoCommented Aug 3, 2013 at 17:07
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5 Answers
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There is an all too common misconception that because the Schrodinger equation is linear, non-linear phenomena (like chaos) are only classical. The wavefunction does obey a linear equation, the Schrodinger equation, but it is not directly related to observable physics. Observables quantities, like expectation values of operators, obey non-linear equations. In fact, many times the same equations as their classical counterparts, with small corrections.
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Assuming you mean "linear" in the mathematical sense of "the sum of two solutions to the relevant equation is also a solution," there's no particular reason why macroscopic objects are inherently non-linear. In fact, there is a large body of work in the quantum foundations community on ways to have macroscopic objects behave in a linear manner but look non-linear. That's the whole point of things like the Many-Worlds interpretation of quantum mechanics, and the research into decoherence by people like W. Zurek. There may be a scale above which it is impractical to see superposition states, but that doesn't mean that they can't exist.
If that's not what you mean, then I don't know how to answer you.
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Mean field dynamics, describing the effective evolution of a particle in a system with a very large number of particles, is non-linear, even if the quantum dynamics is linear. Convergence towards mean field dynamics has been rigorously proved for quantum systems of many particles (and even quantum fields), and is nowadays a thoroughly studied subject in mathematical physics. In this sense there are solid foundations on the connection between the linear quantum dynamics and the nonlinear effective evolution of macroscopic systems.
The idea is that time-evolved reduced density matrices of the quantum system converge, in the limit $N\to\infty$, to the projector on the solution of mean field nonlinear equations (at least for some particular quantum states, e.g. coherent states, with general states the picture gets more complicated, but the nonlinear dynamics governs the evolution in the limit).
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1$\begingroup$ This is a good answer, but an enterprising young researcher might want some sources for an introductory investigation of the new topic being introduced. Do you have any recommendations? $\endgroup$– psitaeCommented Sep 13, 2018 at 11:21
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Linear in the quantum mechanics has nothing to do with its complexity. A two-state spin can be described by a simple 2-by-2 matrix; however, 30 interacting spin, in general, must be described by a 1 billion by 1 billion matrix. It grows exponentially as the number of spins increases, for $10^{23}$ spin, you may need a matrix of size $2^{10^{23}}$. It is not easy to understand and not simple in most sense. If you learn some statistical mechanics, you will know that this number is large enough to have new emergent phenomenon.
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$\begingroup$ You start with a huge mistake. Newton's equation of motion is in general non-linear. Only for special cases such as the harmonic oscillator is the equation linear. Take for instance Newton's equation for the Kepler problem (two gravitating masses) and see if you can combine two solutions linearly to obtain a new one. It is however correct that linear equations will never lead to chaos, but that doesn't mean that linear equations can't be difficult. As you correctly point out, quantum systems have exponentially more variables as compared to their classical counterparts. $\endgroup$ Commented Nov 22, 2010 at 16:22
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$\begingroup$ Thanks for the correction. I mixed the deterministic and non-linear when I just started typing. It is clear that Newton's equation is non-linear cos we can set any force, say $F(x)=x^3$, to make it non-linear. Let's remove that part of answer. $\endgroup$– unsymCommented Nov 22, 2010 at 17:20
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$\begingroup$ I agree it is not a good question though. $\endgroup$– unsymCommented Nov 22, 2010 at 17:27
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There is other "domain of linearity"; $\ddot{x}=-x$ is a linear equation with solutions nonlinear in time.
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3$\begingroup$ That's true but this is never meant by linearity of equations/theory. Linearity always has to do with superposition. $\endgroup$– MarekCommented Nov 22, 2010 at 15:56
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1$\begingroup$ @Marek Yes, but I can't see why it is a problem. One can superpose nonlinear solutions to get a nonlinear solution. $\endgroup$– user68Commented Nov 22, 2010 at 16:03
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1$\begingroup$ @mbq: first of all, I wouldn't call a solution of linear equation, which is nonlinear in time, nonlinear. The solutions are almost never linear in time, so it is just plain confusing. Second, OP's question is not a real question (I voted to close) so I don't think there is any reasonable answer. Third, even if there were a good answer, yours is more like a comment about quite irrelevant piece of terminology. $\endgroup$– MarekCommented Nov 22, 2010 at 16:12
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$\begingroup$ @Marek If so, ok; indeed I also voted to close. $\endgroup$– user68Commented Nov 22, 2010 at 16:14
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1$\begingroup$ @KennyTM Quite obvious while this is a nonlinear equation. My point was just that "linearity" of equations does not imply that the solutions are linear functions (though it implies superposition). $\endgroup$– user68Commented Nov 22, 2010 at 16:21