Averaging Over Time and the Uncertainty Principle in Quantum And Classical Statistical Mechanics

It is possible to have an average which exists almost instantaneously in time, but one can also have averages which are computed over a time interval. It was argued in previous notes that average kinetic energy in quantum mechanics is one such average computed over time, as a particle in a potential scatters from one plane wave state to another. The frequency for this process is E from exp(iEt), the time dependent part of the wavefunction. If this is the case, then one only establishes conservation of energy at each point in space on average. As a result, for short time periods the energy may different from E which seems to be related to the uncertainty principle delta E delta time >= hbar. In this note, we also examine the idea of averaging over time in classical statistical mechanics. It seems that averages here are also computed over time. If this is the case, then it seems that an uncertainty principle should also apply in this classical case. In this note, we also consider the idea of collective motion in a dilute gas with a potential at equilibrium