Are equivalent classical systems equivalent also quantum mechanically?

We introduced a model where the electric attraction of a nucleus and an electron is replaced with a classical zero-mass elastic rod on which the nucleus and the electron grip. The rod model suggests that the mediating virtual photon "in reality" is a real phonon of the elastic rod.

This raises a general question:

Question 1. If we are studying two classical systems which, in some sense, have equivalent classical behavior, is then their quantum mechanical behavior equivalent, or even identical?


In the rod example, we replaced the classical electric attraction with a classical rod. Is the quantum mechanical behavior of the systems identical?

We can calculate the quantum mechanical behavior of a classical system by using a path integral and a lagrangian on the classical system.

S1 <--- P mapping of paths ---> S2

Conjecture 2. Suppose that we have classical systems S1 and S2. If we have

i) a one-to-one mapping P from the paths of S1 to the paths of S2, and

ii) P preserves the value of the integral of the lagrangian (= action) on each path, and

iii) P also gives some natural mapping from the observables of S1 to observables of S1,

then the quantum mechanical behavior of S1 and S2 is equivalent.