# Classical string as a limit case of elastic quantum states superposition

The usuall way to "half-popularly" show that the quantum mechanics formalism is not unconnected to the classical mechanics is to demonstrate a classical case as a limit of a quantum case.

The main question: Is there a way to show that macroscopic eigenmode is formed of microscopic eigenmodes superposition?

More details: Classical solution for spatial eigenmodes of vertical displacement $y=y(x)$ is:

$$y=\sum_{n=1}^{\infty}A_n\sin \left(\frac{n\pi x}{L}\right)$$

so there must be a $n$ (regarding the string length $L$) for which is the system not sufficiently classical any more.

On the other hand, there is a superposition of elastic arrangement states (eigenmodes) of the lattice oscillations resulting in macroscopic classical motion.

Note 1: I have chosen string because of its general familiarity. I will accept an answer e.g. for longitudinal waves in 1D rod as well.

Note 2: Of course, the classical and quantum description are based on different assumptions, formalism etc. I don't expect the classical description to magically become a quantum at some point.

• This isn't particularly clear. The separation into modes has little to do with QM and it is also present in the classical case. The CM/QM distinction comes into whether each mode is in a classical dynamical state (e.g. a coherent state, or a thermal state, etc.) or in a nonclassical state like a Fock number state. You might be interested in this experiment, though. Nov 20 '15 at 16:43
• Why do you think that the system will not sufficiently classical for some $n$? Nov 21 '15 at 20:47
• @EmilioPisanty: Sorry, that's probably my flawed reading: Did you mean that the problem itself is not particularly clear or my question is unclear couse of my formulation? Nov 23 '15 at 10:48
• @Inmaurer: Because for $n \rightarrow \infty$ the distance between nodes and antinodes $\rightarrow 0$. Therefore at some point (or at some domain), you will need to switch to the quantum description. Nov 23 '15 at 10:50
• Your question is unclear. (The physics is perfectly well understood.) It seems you are conflating the (mainly classical) mode with its (possibly) quantum state. Nov 23 '15 at 21:33