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.)