6.1: Time Dependence

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The exact expression for the time dependence of a system with \(N\) states required a set of \(N\) simultaneous differential equations. One case where we can solve this problem exactly is when we have a small number of states. Consider a system which requires only two basis states. Say we prepare it in initial state \(|1 \rangle\) and we want to know how long it will take to go to the other state \(|2 \rangle\). From section 5, we have two coupled equations in the time dependent \(c_1\) and \(c_2\):

\[i\hbar \dot{c}_1 = V_{11}c_1 + V_{12}c_2e^{i\omega_{12}t} \\ i\hbar \dot{c}_2 = V_{22}c_2 + V_{21}c_1e^{i\omega_{21}t} \nonumber\]

where \(c_1(0) = 1\) and \(c_2(0) = 0\).

If the change is slow, we can use first order time-dependent perturbation theory. We thus replace the \(c_n(t)\) by \(c_n(0)\), and integrate whence:

\[c_1 \approx \text{ exp}(iV_{11}t/\hbar ) \\ |c_1|^2 \approx 1 \\ c_2 \approx \frac{−i}{\hbar} \int^t_0 V_{21}e^{i\omega_{21}t} dt \nonumber\]

Including the constant of integration for \(c_1(0) = 1\).