N.11 Better description of two-state systems

An approximate definition of the states $\psi_1$ and $\psi_2$ would make the states $\psi_{\rm {L}}$ and $\psi_{\rm {H}}$ only approximate energy eigenstates. But they can be made exact energy eigenfunctions by defining $(\psi_1+\psi_2)$$\raisebox{.5pt}{$/$}$$\sqrt2$ and $(\psi_1-\psi_2)$$\raisebox{.5pt}{$/$}$$\sqrt2$ to be the exact symmetric ground state and the exact antisymmetric state of second lowest energy. The precise basic wave function $\psi_1$ and $\psi_2$ can then be reconstructed from that.

Note that $\psi_1$ and $\psi_2$ themselves are not energy eigenstates, though they might be so by approximation. The errors in this approximation, even if small, will produce the wrong result for the time evolution. (The small differences in energy drive the nontrivial part of the unsteady evolution.)