Word Count : 501

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We are asked to solve a two-level system under a time-dependent perturbation. Let’s solve it manually, step by step. We have: * Unperturbed Hamiltonian: [ \hat{H}_0 |1\rangle = E_1 |1\rangle, \quad \hat{H}_0 |2\rangle = E_2 |2\rangle, \quad E_2 > E_1 ] * Time-dependent perturbation: [ \hat{V}(t) = V_0 \cos(\omega t) \big( |1\rangle\langle 2| + |2\rangle\langle 1| \big) ] * Initial condition: at (t=0), system is in (|1\rangle). We need probability (P_{1\to 2}(t)). --- ### Step 1: Time-dependent Schrödinger equation Let the state at time (t) be: [ |\psi(t)\rangle = c_1(t) e^{-i E_1 t / \hbar} |1\rangle + c_2(t) e^{-i E_2 t / \hbar} |2\rangle ] Here (c_1(0) = 1, c_2(0) = 0). Time-dependent perturbation theory gives: [ i \hbar \frac{d c_2(t)}{dt} = \langle 2 | ________ ______ _______ _______ ________ _____ __________ _____ _______.
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