 Solution

Insert position resolutions between short-time evolution factors for $H=p^2/(2M)+V(q)$. Integrating the momenta gives the path integral, with real-time slice factor $[M/(2\pi i\hslash \Delta t){]}^{1/2}$ and weight $e^{iS[q]/\hslash }$. A real classical trajectory cannot pass through a barrier with insufficient energy, but the path integral includes nonclassical paths.

Continue $t=-is$, with s physical imaginary time. The damped kernel isThe Euclidean action saddle obeys $M\ddot{q}=V^{\prime }(q)$: motion in the inverted potential. For degenerate minima at $\pm a$, subtract their common potential energy. A trajectory joining them with zero Euclidean energy satisfiesIt is an instanton; reversing its direction gives an anti-instanton. Thus a crossing amplitude is nonperturbative, proportional to $e^{-S_I/\hslash }$. For the representative quartic well $V=M{\omega }_0^2(q^2-a^2{)}^2/(8a^2)$, direct integration givesThe arbitrary center $s_0$ is an instanton translation zero mode. Expanding $q=q_I+\eta$ gives a Gaussian path integral involving $L_I=-{\partial }_s^2+V^{\prime \prime }(q_I)/M$. Treating the zero mode as a collective coordinate rather than a zero Gaussian denominator produces the instanton fluctuation prefactorCommon endpoint regularization defines the functional determinant ratio; K and kappa have inverse-time units. The prime removes only the zero eigenvalue, while integrating the center supplies the total available time. Large $S_I/\hslash$ and $\kappa \ll {\omega }_0$ justify a dilute instanton gas.

Multiple crossings alternate direction. Integrating n ordered centers gives ${\mathcal{T}}^n/n!$. Return paths contain even n, and transfer paths contain odd n, so their common local-well factor multiplies $\cosh (\kappa \mathcal{T})$ and $\sinh (\kappa \mathcal{T})$ respectively. The two lowest levels are consequentlyThe lower state is the even combination of the two localized states. Local anharmonic corrections modify $E_{\mathrm{well}}\simeq \hslash {\omega }_0/2$ and higher semiclassical corrections modify K. The thermal trace as a closed Euclidean path integral gives $Z\simeq 2e^{-\beta E_{\mathrm{well}}}\cosh (\beta \hslash \kappa )$ when higher local levels are negligible. This describes coherent level splitting in a stable well, not irreversible decay.

For a scattering barrier at energy E, the leading WKB tunnelling exponent gives an amplitude $\exp [-{\int }_{q_1}^{q_2}\sqrt{2M(V-E)}dq/\hslash ]$ and a transmission probability with twice that exponent. Endpoint matching fixes prefactors. Metastable escape instead uses a Euclidean bounce, with a negative fluctuation mode and a decay rate. Neither its negative mode nor the probability/amplitude factor should be transplanted into the stable double-well splitting. ^[ref]