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Quantum physics time evolution of an overlap

  1. Jul 22, 2017 #1
    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).
    Quantum.png

    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. jcsd
  3. Jul 23, 2017 #2

    blue_leaf77

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    Science Advisor
    Homework Helper

    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##.
     
  4. Jul 23, 2017 #3
    Thank you. Once I added the second term it was very clear how I should proceed.
     
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