5.2: Time-dependent Perturbation Theory

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.