A proof of the indeterminary relation of lifetime and energy
A proof of Lavine's uncertainty relation \[ \tau_ H(\psi)\sqrt{\Delta_{\psi}H}\geq 1/2 \] (for a selfadjoint operator H without singular spectrum) between the life time of a normalized state vector \(\psi\) \(\tau_ H(\psi)=\int^{\infty}_{0}| (\psi,e^{- itH}\psi)|^ 2dt\) and the energy dispersion \(\Delta_{\psi}H=\| H\psi \|^ 2-(\psi,H\psi)^ 2\), with equality only for an eigenvector of H, is given on the basis of the Mandelstam-Tamm inequality \[ (\Delta_{\psi_ t}A)(\Delta_{\psi}H)\geq 1/4| d/dt(\psi_ t,A\psi_ t)|^ 2 \] for any bounded selfadjoint operator A, where \(\psi_ t=e^{-iHt}\psi\).
scientific article; zbMATH DE number 3937847