Overview#

The Wick rotation of §3.4.1 revealed that imaginary-time motion obeys Newton’s law in the inverted potential \(-V(x)\) — barriers become hills. §3.4.2 put this geometric fact to dramatic use: demanding that the Euclidean black-hole geometry close smoothly into a thermal circle delivered the Hawking temperature without quantizing a single field. Here we exploit the inverted-potential trick for a third miracle, this time inside ordinary quantum mechanics: classically forbidden tunneling through a barrier becomes a classically allowed trajectory in imaginary time. The trajectory that crosses the barrier — the instanton — has finite Euclidean action, and summing over instanton paths (the dilute instanton gas) produces the ground-state energy splitting \(\Delta \propto \mathrm{e}^{-S_{\mathrm{inst}}/\hbar}\). This is the paradigm calculation of non-perturbative quantum mechanics.

The Inverted Potential#

Two ingredients from §3.4.1 set up everything that follows. First, the Wick rotation \(t \to -\mathrm{i}\tau\) replaces the real-time action \(S = \int(\tfrac{1}{2}m\dot{x}^2 - V)\,\mathrm{d}t\) with the Euclidean action

in which kinetic and potential terms add. Second, varying \(S_E\) gives a second-order equation of motion with the opposite sign from real-time Newton’s law:

In imaginary time the particle moves as if it were rolling in the inverted potential \(-V(x)\). Barriers in \(V\) become hills in \(-V\) that the particle can climb over with finite Euclidean action; trajectories that are classically forbidden in real time become classically allowed in imaginary time. The rest of this lesson exploits this single sign flip in the case of a double-well potential.

The Double-Well Potential#

The symmetric double-well potential is

with two degenerate minima at \(x = \pm a\) (where \(V(\pm a) = 0\)) and a barrier of height \(\lambda a^4\) at the origin.

Classical vs. quantum. Classically, a particle with energy \(E < \lambda a^4\) is trapped in one well. Quantum mechanically, tunneling splits the ground state into a symmetric combination \(\vert +\rangle\) (energy \(E_0 - \Delta\)) and an antisymmetric combination \(\vert -\rangle\) (energy \(E_0 + \Delta\)). The splitting \(2\Delta\) is non-perturbative in \(\hbar\): it is \(O(\mathrm{e}^{-1/\hbar})\) and invisible at any finite order in perturbation theory.

The Instanton Solution.

A first-order reduction (Bogomolny trick) simplifies the second-order equation. Any path satisfying

automatically solves the Euclidean equation of motion (verify by differentiating). For the double-well \(V = \lambda(x^2 - a^2)^2\):

Separating variables and integrating gives the instanton (kink) solution:

The anti-instanton \(\bar{x}_{\mathrm{inst}}(\tau) = -a\tanh(\omega(\tau - \tau_0)/2)\) traverses the barrier in the opposite direction, from \(+a\) to \(-a\).

The Instanton Action.

The Euclidean action of the instanton is

Energy Level Splitting#

The key is to compute the transition amplitude \(\langle{-a}\vert\mathrm{e}^{-\hat{H}T/\hbar}\vert{a}\rangle\) in two independent ways, then match.

Step 1 — Amplitude from Eigenstate Expansion.

Insert a complete set of energy eigenstates \(\{\vert n\rangle\}\):

For large \(T\), the sum is dominated by the two lowest states. Since \(V\) is symmetric, \(\psi_+\) is even and \(\psi_-\) is odd, so \(\psi_+(-a) = \psi_+(a)\) and \(\psi_-(-a) = -\psi_-(a)\). With \(E_\pm = E_0 \mp \Delta\):

The \(\sinh\) arises from the sign difference between \(\psi_+\) and \(\psi_-\) at \(x = \pm a\).

Step 2 — Amplitude from the Dilute Instanton Gas.

The same amplitude equals the Euclidean path integral \(\int_{x(0)=a}^{x(T)=-a}\mathcal{D}[x]\,\mathrm{e}^{-S_E/\hbar}\).

Single instanton centered at \(\tau_0 \in [0,T]\): the dominant saddle. Including Gaussian fluctuations around the saddle (functional determinant), one instanton contributes

where \(K\) is the pre-exponential factor from the functional determinant (it involves the ratio of the fluctuation operators with and without the instanton).

Many instantons. Over a large interval \(T\), the particle can tunnel back and forth \(n\) times. The \(n\)-instanton path has action \(\approx nS_{\mathrm{inst}}\) (dilute gas: widely separated instantons don’t interact), and contributes \((K\,\mathrm{e}^{-S_{\mathrm{inst}}/\hbar}\,T)^n/n!\).

Going from \(x = +a\) to \(x = -a\) requires an odd number of crossings (\(n = 1, 3, 5, \ldots\)):

Including the zero-point factor \(\mathrm{e}^{-E_0 T/\hbar}\):

Step 3 — Matching.

Both expressions have the same \(\mathrm{e}^{-E_0 T/\hbar}\sinh(\cdots T)\) structure. Identifying the arguments:

Summary#

• Double-well potential and tunneling: \(V(x) = \lambda(x^2-a^2)^2\) has degenerate minima; quantum tunneling splits them by \(2\Delta\), a non-perturbative effect invisible at any finite order in \(\hbar\).

• Euclidean equation of motion: \(m\ddot{x} = +\mathrm{d}V/\mathrm{d}x\) inverts the potential; the kink solution \(x_{\mathrm{inst}}(\tau) = a\tanh(\omega\tau/2)\) tunnels the barrier classically in imaginary time.

• Instanton calculation: The instanton action \(S_{\mathrm{inst}} = 4\sqrt{2\lambda m}a^3/3\) enters the splitting via instanton gas: matching eigenstate expansion (\(\sinh(\Delta T/\hbar)\)) to instanton gas (\(\sinh(K\mathrm{e}^{-S/\hbar}T)\)) yields \(\Delta = \hbar K\mathrm{e}^{-S_{\mathrm{inst}}/\hbar}\).

• Non-perturbative suppression: The splitting is \(O(\mathrm{e}^{-1/\hbar})\)—exponentially small and inaccessible to any finite perturbation series, exemplifying tunneling as a purely quantum phenomenon.

• 3.4.1 Wick Rotation — time becomes temperature; Euclidean path integral and the inverted-potential equation of motion.

• 3.4.2 Black Hole Temperature — Wick rotation applied to spacetime geometry; another payoff of “imaginary time is periodic”.

• 3.3.2 WKB Approximation — real-time semi-classical tunneling, for comparison with the instanton exponent.