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Two State System Described by a Time-Dependent Hamiltonian

  1. Oct 23, 2013 #1
    1. The problem statement, all variables and given/known data
    A two state system is described the time dependent Hamiltonian
    [tex]\hat{H}=|\psi\rangle E\langle\psi|+|\phi\rangle E \langle\phi|+|\psi\rangle V(t)\langle\phi|+|\phi\rangle V(t)\langle\psi|[/tex]
    Where
    [tex]\langle \psi|\psi \rangle = 1=\langle \phi|\phi\rangle, \langle \phi|\psi \rangle=0=\langle \psi|\phi \rangle
    \\V(t)→0, t→±∞[/tex]
    Given that as t→-∞ the system was in the state $$|\psi\rangle$$ find the probability that it will end up in state $$|\phi\rangle$$ as t→+∞.



    2. Relevant equations
    $$\hat{H}_\psi|\psi\rangle = E|\psi\rangle$$
    $$\hat{H}_\phi|\phi\rangle = E|\phi\rangle$$
    $$\hat{H}_0= \hat{H}_\psi+\hat{H}_\phi$$
    $$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$

    3. The attempt at a solution
    To be completely honest, I don't even really know were to begin. I can somewhat grasp the fact that this time dependent Hamiltonian can be written as the sum of the time independent Hamiltonian and some perturbation V(t), but beyond that, I'm at a loss.
     
    Last edited: Oct 23, 2013
  2. jcsd
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