Sudden perturbation

4.1.2 Sudden perturbation

Consider a constant perturbation ${\hat{H}}^{(1)}$ which is switched on at $t = - ε ∕ 2$ and oﬀ at $t = ε ∕ 2$ .

\begin{array}{rcl} d_{n} (\infty) & = & - \frac{i}{ℏ} ⟨ n | {\hat{H}}^{(1)} | i ⟩ \int_{- ε ∕ 2}^{ε ∕ 2} e^{i ω_{n i} t} d t \\ = & - ⟨ n | {\hat{H}}^{(1)} | i ⟩ \frac{2 i}{ℏ ω_{n i}} sin (ε ω_{n i} ∕ 2) \overset{ε \to 0}{\to} - i \frac{ε}{ℏ} ⟨ n | {\hat{H}}^{(1)} | i ⟩ \to 0 . \end{array}

So sudden changes to a system leave the state unchanged.

Actually we didn’t need to switch oﬀ at $t = ε ∕ 2$ , we can instead regard the expression above as $d_{n} (ε ∕ 2)$ , ie the coeﬃcient immediately after turning on the perturbation. The conclusion is the same: a change in the system which is sudden (compared with $1 ∕ ω_{n i}$ ) doesn’t change the state of the system. We can use this to conclude, for instance, that the electronic conﬁguration is unchanged when a tritium nucleus beta-decays to helium-3. Of course this conﬁguration is not an eigenstate of the new Hamiltonian, but we can take it as an initial state for the subsequent time evolution.

[prev] [up] [top]