2.2: First order energy shifts

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In first order perturbation theory, we assume that the change in the wavefunction is small, i.e. \(|c_{i0}/c_{00}| \ll \forall i\) and neglect the second term in equation 1 which becomes.

\[\Delta E_0 \approx \langle n_0 | \hat{V} | n_0 \rangle \equiv V_{00} \nonumber\]

which is one of the most useful results in quantum mechanics. It tells us how to calculate the change in the nth energy eigenvalue, to first order:

\[\text{The shift in energy induced by a perturbation is given to first order by the expectation value of the perturbation with respect to the unperturbed state.} \nonumber\]

Thus first order time independent perturbation is equivalent to making the approximation that the wavefunction does not change. Loosely, this works because the energy depends on the perturbation to first order, but on wavefunction squared.