2.3: 2.3 Mixing of the eigenstates of Hˆ 0

Turning to equation 2, we make the approximation \(c_{i0} \ll c_{00} \approx 1 \forall i \neq 0\) so that the only significant term in the sum comes from \(i = 0\), and also that \(\Delta E_0\) is negligible compared to the energy difference between states 0 and \(k\):

\[c_{k0} \approx \langle n_k|\hat{V} |n_0 \rangle / (E_0 − E_k) \]

Using these coefficients, we see that the perturbation causes a first-order correction to the energy eigenvector \(|n_0 \rangle \):

\[|\phi_0 \rangle = |n_0\rangle + \sum_{k \neq 0} \frac{\langle n_k |\hat{V} |n_0\rangle }{(E_0 − E_k)} |nk\rangle \equiv |n_0\rangle + \sum_{k \neq 0} \frac{V_{k0}}{(E_0 − E_k)} |n_k\rangle \nonumber\]

Which defines the matrix element \(V_{ij}\) for \(i = k\), \(j = 0\). We speak of the perturbation mixing the unperturbed eigenfunctions since the effect is to add to the unperturbed eigenfunction, \(|n_0 \rangle \), a small amount of each of the other unperturbed eigenfunctions. The denominator suggests that states with similar energies are more strongly mixed, and the “matrix element” determines how the perturbation mixes the states.

Unlike the formula for the energy shift, we are faced in general with evaluating an infinite sum to find the correction to the eigenfunctions.