= Quadratic short-time quantum survival probability
{title2=$|\langle\psi,e^{-iHt/\hbar}\psi\rangle|^2=1-(\Delta H)^2t^2/\hbar^2+O(t^4)$}
For a normalized <quantum state> $\psi$ and a <self-adjoint operator> $H$ on a finite-dimensional <Hilbert space>, expand its <unitary time evolution>:
$$
\langle\psi,e^{-iHt/\hbar}\psi\rangle
=1-\frac{i\langle H\rangle t}{\hbar}-\frac{\langle H^2\rangle t^2}{2\hbar^2}+O(t^3).
$$
Multiplying by its <complex conjugate> gives the survival <probability>
$$
|\langle\psi,e^{-iHt/\hbar}\psi\rangle|^2
=1-\frac{\langle H^2\rangle-\langle H\rangle^2}{\hbar^2}t^2+O(t^4).
$$
The function is even in $t$, so its odd coefficients vanish. The coefficient $(\Delta H)^2=\langle H^2\rangle-\langle H\rangle^2$ is the <variance> of the <energy>. This quadratic departure yields the <quantum Zeno effect> when ideal <projective measurements> repeatedly reset the <quantum state>.
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