= 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 $[M/(2\pi i\hbar\Delta t)]^{1/2}$ and weight $e^{iS[q]/\hbar}$. 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 is
$$
K_E(q_f,q_i;\mathcal T)=\langle q_f|e^{-H\mathcal T/\hbar}|q_i\rangle
=\int\mathcal Dq\,e^{-S_E[q]/\hbar},\qquad
S_E=\int_0^{\mathcal T}ds\left[\frac M2\dot q^2+V(q)\right].
$$
The <Euclidean action> saddle obeys $M\ddot q=V'(q)$: motion in the inverted potential. For degenerate minima at $\pm a$, subtract their common potential energy. A trajectory joining them with zero Euclidean energy satisfies
$$
\frac M2\dot q^2=V(q),\qquad
\boxed{S_I=\int_{-a}^{a}dq\,\sqrt{2MV(q)}.}
$$
It is an <instanton>; reversing its direction gives an <anti-instanton>. Thus a crossing amplitude is nonperturbative, proportional to $e^{-S_I/\hbar}$. For the representative quartic well $V=M\omega_0^2(q^2-a^2)^2/(8a^2)$, direct integration gives
$$
q_I(s)=a\tanh\frac{\omega_0(s-s_0)}2,\qquad S_I=\frac23M\omega_0a^2.
$$
The arbitrary center $s_0$ is an <instanton translation zero mode>. Expanding $q=q_I+\eta$ gives a <Gaussian path integral> involving $L_I=-\partial_s^2+V''(q_I)/M$. Treating the zero mode as a collective coordinate rather than a zero Gaussian denominator produces the <instanton fluctuation prefactor>
$$
K=\sqrt{\frac{S_I}{2\pi\hbar}}\left[\frac{\det L_0}{\det' L_I}\right]^{1/2},\qquad
L_0=-\partial_s^2+\omega_0^2,\qquad \kappa=K e^{-S_I/\hbar}.
$$
Common 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 $S_I/\hbar$ 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 consequently
$$
\boxed{E_\pm=E_{\rm well}\mp\hbar\kappa,\qquad
\Delta E=2\hbar K e^{-S_I/\hbar}.}
$$
The lower state is the even combination of the two localized states. Local anharmonic corrections modify $E_{\rm well}\simeq\hbar\omega_0/2$ and higher semiclassical corrections modify K. The <thermal trace as a closed Euclidean path integral> gives $Z\simeq2e^{-\beta E_{\rm well}}\cosh(\beta\hbar\kappa)$ 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 $\exp[-\int_{q_1}^{q_2}\sqrt{2M(V-E)}\,dq/\hbar]$ 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. <Gauge Theory by David Tong>[Chapter2, §§2.3.2–2.3.3, printedpp.56–60: quantum-mechanical tunnelling, fluctuation determinants and repeated instanton events]{ref}
\Image[/past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2012/paper-1-qcm2012-instanton.png]
{title=Double-well instanton}
{source=paper-1-qcm2012-instanton.py}
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