Linear and nonlinear physical theories

[Higgsono]

Classical physics is a nonlinear theory, but how is it that? Why is it nonlinear? Also quantum mechanics is a linear theory so that the sum of the solutions of the schrödinger equation is itself a solution.

But I'm not sure I grasp this completely. Why is quantum mechanics linear while classical mechanics is not? Could someone give an example why classical mechanics is not linear.

 

[kith]

The crucial thing to look at are the equations of motion and what they are about.

In classical electrodynamics, the equations of motion are the Maxwell equations. They can be essentially expressed by certain linear operators (differential operators) acting on the fields $\vec E(\vec r,t)$ and $\vec B(\vec r,t)$, so they are linear.

In wave mechanics, we have the Schrödinger equation where the wavefunction $\psi(\vec r_1,\vec r_2,...,t)$ is also acted upon by a linear operator (the Hamiltonian). Different physical systems correspond to different linear operators.

In classical mechanics, the differential operators are essentially fixed by Newton's second axiom. Instead, different physical systems correspond to different functional dependencies on the positions $\vec r_i$ in the equations of motion. (Here is an example of a nonlinear equation of motion: the pendulum. Also, typical forces between interacting bodies lead to nonlinear equations of motion)

So in order to emphasize the difference between classical mechanics and wave mechanics, imagine that we have two particles and add a potential term of the form $$\frac{1}{|\vec r_1 - \vec r_2|}$$ In classical mechanics, this causes the equations of motion to become nonlinear in the $\vec r_i$. In wave mechanics, the change occurs only in the linear operator which acts on the wave function. We never get something like $\frac{1}{\psi}$.

(Ironically, I just found out that the classical Kepler problem can be expressed in linear form. Of course, this doesn't change the main point.)

 

[mpresic]

There are aspects of classical mechanics which are linear (e.g. simple harmonic motion), and aspects which are nonlinear (space motions of a free rigid asymmetric body). Most aspects of non-relativistic quantum mechanics at the graduate level are linear (quantum harmonic oscillator, free particle), but not all of quantum mechanics is linear (quadratic terms are often eliminated in perturbation theory). The non-linear schroedinger equation is generally not presented at this level.