The following is a section "Correlation Amplitude and the Energy-Time Uncertainty Relation" from *Sakurai's Modern Quantum Mechanics book page 79*: 

[![enter image description here][1]][1]
[![enter image description here][2]][2]

**Question:**

 -  What is the importance of the observation that if the interval for which $|E-E_0| \approx \frac{\hbar}{t}$ holds is much narrower than $\Delta E$ then there essentially no contribution to $C(t)$? Lastly, what is essentially the point of these observations?




 


  [1]: https://i.sstatic.net/W3NJz.png
  [2]: https://i.sstatic.net/aOxGL.png