# Can quantum fluctuations affect the double pendulum?

The double pendulum is a simple example of a chaotic system which is extremely sensitive to tiny perturbations in its initial conditions.

If we set off two identical double pendulum systems from identical starting positions, and ignored external forces such as air resistance, friction, and vibrations, would internal quantum randomness cause these two pendulums to eventually deviate in their trajectories? And if so, on what time scale (relevant to length of pendulums and mass) would we be waiting before the difference is easily visible?

However, you can find microscopical systems for which classical equations of motion would exhibit chaos but they are actually quantized. The field studying the properties of such systems is called Quantum chaos. The quantum equations are linear which means that they cannot exhibit chaos and the chaos we see is only a property of the averaged "classical" picture of the dynamics. It actually turns out that the delocalization or "smearing out" of tiny deviations in initial condition typical for chaos gets often supressed in a classically-chaotic dynamical system under quantization. (But on time-scales of order $1/\hbar^2$ where $\hbar$ is the Planck constant.)