In the book on QM by D.J. Griffiths, the time-energy uncertainty relation is in general proved for any observable $Q$ whose operator is not a function of time.
Now it has been derived that if uncertainty in energy is $\Delta E$ and the time required for an observable's expectation value to change by its standard deviation be $\Delta T$, then $$\Delta E \Delta T\geq h/4\pi$$
Now, my question is,say, for the time evolution of a free particle wavefunction, when the standard deviation itself is a function of time, there how can we interpret this result? Because then we cannot define a time in which the expectation changed by an s.d as the s.d is also a function of time.