Quantum physics time evolution of an overlap

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1. Jul 22, 2017

Monci

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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.

3. 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.

2. Jul 23, 2017

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 $\Delta H^2 = \langle H^2\rangle - \langle H\rangle ^2$.

3. Jul 23, 2017

Monci

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