A proof of the indeterminary relation of lifetime and energy
type: Item
instance of: scholarly article
title: A proof of the indeterminary relation of lifetime and energy
zbMATH Open document ID: 0585.46065
DOI: 10.1007/BF00400974
author: Marian Grabowski
published in: Letters in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: 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 \(\psi_ t=e^{-iHt}\psi\).
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 3937847
zbMATH Keywords: Lavine's uncertainty relation
zbMATH Keywords: life time of a normalized state vector
zbMATH Keywords: energy dispersion
zbMATH Keywords: Mandelstam-Tamm inequality
zbMATH Keywords: bounded selfadjoint operator
reviewed by: Huzihiro Araki
MaRDI profile type: Publication
cites work: Exponential decay in the Stark effect
cites work: Estimation of shift parameters of a quantum state
cites work: Q5526485
cites work: Q4185363
cites work: Q3874904
description: scientific article; zbMATH DE number 3937847