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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.

5.2: Time-dependent Perturbation Theory is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.