Can anyone help me develop some intuition for the idea of avoided crossing in QM. To explain what confuses me consider a two-level system given by
$ M= \begin{bmatrix} E1 & 0 \\ 0 & E2 \\ \end{bmatrix} $
Let us denote its eigenvectors as $\lvert 1 \rangle$ and $\lvert 2 \rangle$ and assume we set $E1=1$ and want to plot $E2$ along with the eigenstate that follows this energy as a function of $E1$. What is now the correct way to make this plot? Is it that the state $\lvert 2 \rangle$ follows $E2$ as it crosses $E1$ or should I switch around my states when passing the degeneracy point. The reason why I ask is that when you have a small perturbation on the off-diagonal terms you will have an avoided crossing where the lower level avoids the upper one at the degeneracy point and somehow it makes sense that the states should then be switched around at the degeneracy point. Hope it is clear what I mean.
