I need to find the momentum expectation value of the function in the attached picture. It is the function of the harmonic oscillator (first excited state). I know that the expectation value is the value that we measure with the highest probability if we measure the system. But what the hell does the standard deviation d(p)=<p^2>-<p>^2 mean for the harmonic oscillator? Does it mean that if we make 1000 times the same experiment and measure with every experiment the momentum that we result in this d(p) if we always measure p? And another question is can we measure in one experiment <p> and <x> (the two expectation values) exactly at the same time? I think no, because the uncertainty principle would not be sadisfied.