Quantum physics time evolution of an overlap

[Monci]

Homework Statement

[/B]
I'm trying to solve the following problem. (a) was easy but I am stuck at (b).

Homework Equations

[/B]
Since we are told that the Hamiltonian is conserved, and the answer is in terms of the uncertainty of H, I assume I have to use the conservation of uncertainty. Maybe I could use the Schrödinger equation to see how time affects the wave function.

The Attempt at a Solution

Using the Schrödinger equation I have $$\psi (t) = \psi (0) + \frac{1}{i\hbar}H\psi(0)t + O(t^2)$$
However I don't find this particularly useful since I can't get from here to the uncertainty of H easily. I have tried the case with just two states but didn't accomplish anything. Dimensional analysis suggests something like $$ 1 - \frac{\Delta H^2}{\hbar^2}dt^2 + O(t^3) $$
I have no idea how to proceed.

 

[blue_leaf77]

You also need the 2nd order term
$$
-\frac{H^2t^2}{\hbar}\psi(0)
$$
May be the problem asks you to make use of the uncertainty formula for energy ##\Delta H^2 = \langle H^2\rangle - \langle H\rangle ^2##.

 

[Monci]

You also need the 2nd order term
$$
-\frac{H^2t^2}{\hbar}\psi(0)
$$
May be the problem asks you to make use of the uncertainty formula for energy ##\Delta H^2 = \langle H^2\rangle - \langle H\rangle ^2##.

Thank you. Once I added the second term it was very clear how I should proceed.