5.3: Time–independent Perturbations

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The results obtained in the last section can also be applied to the case where the perturbation, \(\hat{V}\), is actually independent of time (strictly, ‘switched on’ at t=0).

Again, starting the system in eigenstate \(|k \rangle\) of \(\hat{H}_0\) we obtain,

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

for \(m \neq k\), giving for the transition probability

\[p_m(t) = |\Delta c_m(t)|^2 = \frac{|V_{mk}|^2}{\hbar^2}\frac{\sin^2 (\omega_{mk}t/2)}{(\omega_{mk}/2)^2}. \nonumber\]

For sufficiently large values of \(t\), the function

\[f(t, \omega_{mk}) \equiv \frac{\sin^2 (\omega_{mk}t/2)}{(\omega_{mk}/2)^2} \nonumber\]

consists essentially of a large peak, centerd on \(\omega_{mk} = 0\), of height \(t^2\) and width \(\approx 4\pi /t\), as indicated in Fig. 4. Thus there is only a significant transition probability if \(E_m \approx E_k\). That is, if \(|\omega_{mk}| < 2\pi /t\).

Note that we are assuming that the system was prepared in some eigenstate of \(\hat{H}_0\) which is not an eigenstate of \(\hat{V}\): if it were, then the matrix element \(V_{nm}\) would be zero and \(p_m(t) = 0\). Thus although the analysis treats the perturbation as time independent, it is applied to cases where the perturbation is switched on at \(t = 0\). Moreover only perturbations which are incompatible with the Hamiltonian can induce transitions.