I've noticed that when papers in mathematical physics concern themselves with the rate at which a wavepacket spreads, they almost always try to bound the moments of the position operator (the operator whose eigenvalue gives the position of the particle). 


What precisely is the connection between those moments and wavepacket spreading?



Moments of the position operator and wavepacket spreading
Well, the absolute value squared of a wave packet $\Psi\_t(x)$ has the interpretation of a time-dependent probability distribution $P\_t(x)=|\Psi\_t(x)|^2$ for the stochastic variable $x$ (position on a line, or in two- or three-dimensional space). As for any probability distribution, it makes sense to try to characterize it by its moments $\mu\_n(t)=\int x^n P\_t(x) dx$. These are by definition moments of the position operator $\hat{x}$, $\mu\_n(t)=\langle\Psi\_t|\hat{x}^n|\Psi\_t\rangle$.


One then speaks of wavepacket spreading if the variance $\mu\_2(t)-\mu\_1^2(t)$ increases with time, and so bounding the second moment of the position operator is a sensible thing to do.


