Inequality for the time evolution of an overlap

[Dyatlov]

Hello.
I am trying to prove that the uncertainty in energy for a normalized state limits the speed at which the state can become orthogonal to itself.
The problem is number 2 on https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps6.pdf
Having issues with point a).
Any input is greatly appreciated.

 

[hilbert2]

You should post this in the Homework forum, using the correct question template.

The function $\left<\psi (0)|\psi(t)\right>$ is called an autocorrelation function. You could start by playing with a two-state system where the state space is spanned by only two vectors, and finding out how the autocorrelation function of $a|\psi_1 > + b|\psi_2 >$ depends on how similar in absolute value the coefficients $a,b$ are. $|\psi_1 >$ and $|\psi_2 >$ are the energy eigenstates here.