I have got a quantum conservative system whose Hamiltonian is $H$. I consider an selfandjoint operator $O$ whose eigenvalues and eigenvectors are: $$O|\psi _{n}\rangle = \lambda _{n}|\psi _{n}\rangle$$
At initial time, the system is at the state $|\psi_{i}\rangle$. At time $t= \delta t$, I measure with the operator $O$.
The approximation of the measured state to second order in $\delta t$ is
$$|\psi(\delta t)\rangle=U(\delta t, 0)|\psi(0)\rangle \;,$$
with
$$U(\delta t, 0) = e^{-\frac{i}{\hbar}\delta t H} \approx 1 - \frac{i}{\hbar}\delta t H - \frac{1}{2}\left ( \frac{1}{\hbar^2}(\delta t)^2 H^2 \right ) \;;$$
the probability of obtaining $ \lambda _{i}$ in the second order on $\delta t$ is
$$P(\lambda_{i}) \approx 1-\left ( \frac{\delta t}{\hbar}\Delta H_{\psi_{i}} \right )^2 \;,$$ where I denote $$\Bigl(\Delta H_{\psi_{i}}\Bigr)^2= \langle\psi_{i}|H^2|\psi_{i}\rangle-\langle\psi_{i}|H|\psi_{i}\rangle^2$$
Question. In what regime the quadratic approximation on delta $\delta t$ is good?
