Quadratic short-time quantum survival probability ($|⟨\psi ,e^{-iHt/\hslash }\psi ⟩{|}^2=1-(\Delta H{)}^2{t}^2/{\hslash }^2+O(t^4)$)

For a normalized quantum state $\psi$ and a self-adjoint operator $H$ on a finite-dimensional Hilbert space, expand its unitary time evolution:Multiplying by its complex conjugate gives the survival probabilityThe function is even in $t$, so its odd coefficients vanish. The coefficient $(\Delta H{)}^2=⟨H^2⟩-⟨H{⟩}^2$ is the variance of the energy. This quadratic departure yields the quantum Zeno effect when ideal projective measurements repeatedly reset the quantum state.