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.)