Double-well tunneling splitting ($E_{\pm }=E_{\mathrm{well}}\mp \hslash K{e}^{-S_I/\hslash }$)

A symmetric two-well system has nearly localized left/right states coupled by a positive quantum tunnelling magnitude $\Delta =\hslash K{e}^{-S_I/\hslash }$. Its effective Hamiltonian is $E_{\mathrm{well}}I-\Delta {\sigma }_x$, so the superposition with an even spatial wavefunction is lower, and the level splitting is $2\Delta$. The dilute instanton gas reproduces these two exponents through even/odd crossing sums.

For normalized localized states related by reflection and real overlap $s=⟨{\varphi }_L|{\varphi }_R⟩$, the displayed superpositions are normalized even and odd states. The even state is lower for the lowest tunnelling pair; the odd state has one central node. Small overlap explains nearly identical probability densities in the two distant wells, but the relative phase remains different. A localized one-well state is a superposition of the pair rather than an exact stationary parity eigenstate.