Question: Suppose a particle is subjected to a potential and we measure the position of the particle as it goes along. If we let the time interval tends to zero, we would be able to find a path of such a particle as a function of t. It seems that we would be able to obtain the velocity of the particle from such path. Of course we cannot because in doing so we would ascribe properties to the system additional to that contained in the wavefunction. But this reminds me of some strange brownian motion. Brownian motion is continous yet (almost surely) non-differentiable. Is it possible that as the time interval tends to zero, the particle takes on such a path?

James M Monk answered on 15 Jun 2011:

This is a bit like Maxwell’s demon I think. Maxwell suggested a thought experiment where a demon could determine the position of all of the air molecules in two rooms with a door between them. The demon opens and closes the door so that all the fast moving molecules end up in one room, and all the slow moving ones end up in the other. So the demon would have been able, just knowing the random positions and speeds of the molecules, to separate the air into a hot room and a cold room (which is supposed to be impossible under the second law of thermodynamics).

In fact, Maxwell’s demon was recreated recently as a *real* experiment. Experimenters managed to get a particle to climb some small stairs by watching its random motion. Whenever it started fluctuating downwards they switched on an electric field (I think!) to prevent it moving, but if it fluctuated upwards they did nothing. That is pretty amazing, I think.