5.2: Time-dependent Perturbation Theory

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Consider the Hamiltonian

\[\hat{H} = \hat{H}_0 + \hat{V} (t) \nonumber\]

where the time dependent part is small. We can write the time dependent coefficients \(c_n\)

\[c_n (t) = c_n (0) + \Delta c_n (t) \nonumber\]

Where \(c_n (0)\) is the value of \(c_n\) at t=0. We substitute in the equation for \(\dot{c}_m\) derived above to give

\[\dot{c}_m(t) = (i\hbar )^{−1} \sum_n [c_n (0) + \Delta c_n (t)]V_{mn} \text{ exp}(i \omega_{mn} t) \nonumber\]

We can assume that for a perturbation \(c_n (0) >> \Delta c_n (t)\), and ignore the second term. This allows us to obtain the coefficients \(c_m(t)\) by integrating the first–order differential equation to give:

\[c_m(t) = (i\hbar )^{−1} \sum_n c_n (0) \int^t_0 V_{mn} \text{ exp}(i \omega_{mn}t) dt \nonumber\]

In the special case where the system is known to be in an eigenstate of \(\hat{H}_0\), say \(|k \rangle\), at \(t = 0\), then \(c_k(0) = 1\) and all other \(c_m(0) = 0\), \(m \neq k\), giving

\[c_m(t) = (i\hbar )^{−1} \int^t_0 V_{mk} \text{ exp}(i \omega_{mk}t) dt \nonumber\]

Thus a system starting in a known eigenstate of the unperturbed system may transform to a different eigenstate through the action of the perturbing potential. Notice that \(c_m(t)\) is an integral over time, if we wait a long time, the transition may become more likely.

The probability of finding the system at a later time, \(t\), in the state \(|m \rangle\) where \(m \neq k\) is given by

\[p_m(t) = |c_m(t)|^2 \nonumber\]

Since we have assumed a small perturbation, this result is only reliable if \(p_m(t) \ll 1\). “Small” here applies to both \(V_{mk}\) and its integral over time.