# Two State System Described by a Time-Dependent Hamiltonian

1. Oct 23, 2013

### sharrington3

1. The problem statement, all variables and given/known data
A two state system is described the time dependent Hamiltonian
$$\hat{H}=|\psi\rangle E\langle\psi|+|\phi\rangle E \langle\phi|+|\psi\rangle V(t)\langle\phi|+|\phi\rangle V(t)\langle\psi|$$
Where
$$\langle \psi|\psi \rangle = 1=\langle \phi|\phi\rangle, \langle \phi|\psi \rangle=0=\langle \psi|\phi \rangle \\V(t)→0, t→±∞$$
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