Perturbation which is switched on slowly

4.1.1 Perturbation which is switched on slowly

Let ${\hat{H}}^{(1)} (t) = {\hat{H}}^{(1)} e^{t ∕ τ}$ start acting at time $t = - \infty$ so that it reaches full strength at $t = 0$ . Then

\begin{array}{rcl} d_{n} (0) & = & - \frac{i}{ℏ} ⟨ n | {\hat{H}}^{(1)} | i ⟩ \int_{- \infty}^{0} e^{t ∕ τ} e^{i ω_{n i} t} d t \\ = & - \frac{i}{(1 ∕ τ + i ω_{n i}) ℏ} ⟨ n | {\hat{H}}^{(1)} | i ⟩ \end{array}

In the limit $τ ≫ 1 ∕ ω_{n i}$ we simply recover the expression for the ﬁrst-order shifts of the eigenkets in time-independent perturbation theory. With an adiabatic (exceedingly slow) perturbation the system evolves smoothly from the state $| i^{(0)} ⟩$ to the state $| i ⟩$ .

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