Instantons and tunnelling

With the equivalence proven last section, you may well ask: if the path integral only re-derives known results, why learn it? This section is the answer. There is a class of problems where the wavefunction language gets the numbers but not the picture, while the path integral delivers a geometric image beautiful enough to be unforgettable — barrier tunnelling.

The phenomenon: double wells and ammonia

Consider a double-well potential: two wells of equal depth separated by a barrier of height $V_0$ (picture $V(x)=\lambda(x^2-a^2)^2$, with the two minima at $x=\pm a$).

This is no toy. The ammonia molecule NH₃ is a genuine double well: the nitrogen atom can sit above or below the plane of the three hydrogens, the two configurations are symmetric and equivalent, and between them stands the barrier of “nitrogen pushing through the hydrogen plane”.

Classical mechanics makes a clean prediction: a particle with energy below $V_0$ is trapped in one well, forever. Experiment says otherwise — the nitrogen atom of ammonia oscillates back and forth between the two configurations at about 24 GHz. In spectral terms: if the two wells were truly isolated, the ground state would be doubly degenerate (one state per well); what is actually measured is a pair of split levels, with splitting $\Delta E=h\times23.87\ \text{GHz}\approx10^{-4}$ eV — the bound-state face of the tunnelling we met in section 2.11. The first maser, in 1954, ran on exactly this pair of levels.

Why the real-time path integral gets stuck

Try to compute the tunnelling amplitude with last chapter’s stationary-phase method and you hit a wall immediately. Stationary phase lives on finding classical paths — but there is no real classical path from the bottom of the left well to the bottom of the right: inside the barrier $E<V$, the classical kinetic energy $\frac{m\dot x^2}2=E-V<0$, so the velocity would have to be imaginary.

With no stationary point, the real-time path integral is a heap of wildly oscillating phases, none more important than any other — impossible to compute and impossible to read. Tunnelling is an “exponentially small” effect ($T\sim\ee^{-2\gamma}$, recall section 2.11), and the one thing oscillatory integrals are worst at is fishing an exponentially small remainder out of a page full of cancellations.

“The velocity would have to be imaginary” — it sounds like a death sentence, but it is actually a signpost. Velocity is displacement divided by time; the most elegant way to make the velocity imaginary is to make the time imaginary.

The Wick rotation: turning time into the imaginary axis

Turn the double well $V(x)$ upside down: the two well bottoms become two hilltops, and the barrier between them becomes a valley. Rolling from the left hilltop to the right one — in the inverted potential that is a perfectly legitimate classical path! The predicament of “no classical path under the barrier” evaporates in imaginary time.

The instanton solution and the splitting formulaadvanced~10 minexpand

Step 1: energy conservation in imaginary time. The equation $m x''=+V'(x)$ has the conserved quantity $\frac m2x'^2-V(x)=E_E$ (the “energy” in the inverted potential). The path we want starts from the left hilltop (at rest at $x=-a$ as $\tau\to-\infty$, where $V=0$), so $E_E=0$, giving

$$\frac{\dd x}{\dd\tau}=\sqrt{\frac{2V(x)}{m}}\tag{11.4.4}$$

Step 2: solve for the instanton. For $V=\lambda(x^2-a^2)^2$, separating variables and integrating gives

$$x(\tau)=a\tanh\!\left[\frac{\omega(\tau-\tau_0)}{2}\right], \qquad \omega\equiv\sqrt{\frac{8\lambda a^2}{m}}\tag{11.4.5}$$

($\omega$ is exactly the small-oscillation frequency near the bottom of a single well.) The solution completes its crossing from $-a$ to $+a$ within a window of width about $1/\omega$ around $\tau_0$, sitting nearly still before and after — the whole “event” occupies but an instant of imaginary time, hence the name instanton. $\tau_0$ is arbitrary: the crossing can happen at any moment, and this freedom contributes a factor in the exact calculation.

Step 3: the instanton’s action. Use $E_E=0$ to merge the two terms in $S_E$ ($\frac m2x'^2=V$, so the integrand $=mx'^2=x'\cdot m x'$) and change to an integral over $x$:

$$S_E=\int_{-\infty}^{\infty}m\,x'^2\,\dd\tau =\int_{-a}^{a}\sqrt{2mV(x)}\,\dd x\tag{11.4.6}$$

Substituting the quartic double well: $\sqrt{2mV}=\sqrt{2m\lambda}\,(a^2-x^2)$, and $\int_{-a}^{a}(a^2-x^2)\dd x=\frac{4a^3}{3}$; using $\sqrt{2m\lambda}=\frac{m\omega}{2a}$ then gives

$$S_E=\frac{4a^3}{3}\cdot\frac{m\omega}{2a}=\frac{2}{3}\,m\omega a^2\tag{11.4.7}$$

Recognise an old friend: $\int\sqrt{2mV}\,\dd x$ is precisely the numerator of the WKB tunnelling exponent $\gamma=\frac1\hbar\int\sqrt{2m(V-E)}\,\dd x$ of section 7.4 at $E\approx0$. The two methods must agree in the exponent — they are keeping the same books.

Step 4: the splitting formula (result, stated qualitatively). The full calculation also requires the Gaussian fluctuations around the instanton (beyond the scope of this book); the result is

$$\Delta E=2\hbar\omega\,\sqrt{\frac{2S_E}{\pi\hbar}}\;\ee^{-S_E/\hbar}\tag{11.4.8}$$

The skeleton is what really matters:

$$\Delta E\;\sim\;\hbar\omega\,\ee^{-S_E/\hbar}\tag{11.4.9}$$

An attempt frequency $\omega$ (how many times per second the particle “hits the wall” in its well) times the per-attempt tunnelling suppression $\ee^{-S_E/\hbar}$. Make the barrier taller and wider → $S_E$ grows → the splitting shrinks exponentially.

The instanton picture reaches far beyond the double well: tunnelling corrections to low-temperature reaction rates in chemistry, quantum phase slips in superconducting junctions, even the rate at which a “false vacuum decays by bubble nucleation” in cosmology — all use the same template: find the Euclidean classical solution, compute $S_E$, rate $\propto\ee^{-S_E/\hbar}$.

∎Key formulas

• Wick rotation

  $$t=-\ii\tau,\qquad \ee^{\ii S/\hbar}\to\ee^{-S_E/\hbar}$$

  Oscillation becomes decay; stationary phase becomes least action

• Euclidean action

  $$S_E=\int\Bigl[\frac m2\Bigl(\frac{\dd x}{\dd\tau}\Bigr)^2+V(x)\Bigr]\dd\tau$$

  A plus sign! Equivalent to ordinary mechanics in the inverted potential −V

• Instanton solution (quartic double well)

  $$x(\tau)=a\tanh\frac{\omega(\tau-\tau_0)}{2}$$

  Rolls from left hilltop to right hilltop in the inverted potential; width ∼1/ω

• Instanton action and splitting

  $$S_E=\int_{-a}^{a}\sqrt{2mV}\,\dd x,\qquad \Delta E\sim\hbar\omega\,\ee^{-S_E/\hbar}$$

  Exponent matches the WKB tunnelling exponent; ΔE sets the left-right beat frequency

?Self-check4 questions

1. 1.

   The fundamental difficulty of the real-time path integral with barrier tunnelling is:

   • No real classical path exists inside the barrier, stationary phase loses its foothold, and the oscillatory integral cannot extract the exponentially small net contribution
   • The path integral only applies to free particles
   • The wavefunction vanishes inside the barrier, so there are no paths to take
   • Tunnelling violates energy conservation, which the path integral cannot describe

   Submit
2. 2.

   Which statements about the world after the Wick rotation t=−iτ are correct? (Select all that apply.)

   Select all that apply

   • The weight changes from the oscillating e^{iS/ħ} to the decaying e^{−S_E/ħ}, and the dominant path is the one minimising S_E
   • The imaginary-time equation of motion is equivalent to classical mechanics in the inverted potential −V
   • The particle really does cross the barrier along the instanton trajectory in real time
   • e^{−S_E/ħ} has the same structure as the Boltzmann factor of statistical mechanics

   Submit
3. 3.

   Raise the double well’s barrier (S_E goes from 5ħ to 10ħ, ω unchanged). The ground-state splitting ΔE becomes roughly:

   • e⁻⁵ ≈ 1/150 of its old value
   • Half
   • e⁻¹⁰ of its old value
   • Unchanged — the splitting depends only on ω

   Submit
4. 4.

   The inversion oscillation frequency of ammonia is ν=23.87 GHz. What is the corresponding ground-state splitting ΔE=hν in eV? (h=6.626×10⁻³⁴ J·s, 1 eV=1.602×10⁻¹⁹ J)

   eV0% relative tolerance

   Submit

What comes next

The path integral has rewritten quantum mechanics and built a bridge to statistical mechanics and field theory. But the whole edifice still rests on an unspoken foundation: time is Newton’s absolute time, energy is $p^2/2m$ — all non-relativistic. What happens when an electron moves near the speed of light? When energies are high enough to create particles out of nothing? The next chapter generalises in a different direction: making quantum mechanics compatible with special relativity. At the end of that road wait two of quantum mechanics’ greatest gifts to physics — the origin of spin, and the prediction of antimatter.