Thinking in Waves Episode 64 / Part XX — The Things That Tunnel
Six names, one identity
What that formula is called ──
the action of a classical path in imaginary time
Does Episode 62’s \(G=\frac{2}{\hbar}\int\sqrt{2m(V-E)}\,dr\) have a name? ── At least six, spanning 140 years.
★★★ And the deepest name gives away what it is: that integral is the action of a classical path in imaginary time, \(S_E\), with \(G=2S_E/\hbar\). Rotate time by 90° and the barrier becomes a valley the particle simply rolls through. Here we check that numerically (ratio 1.000000).
★★ Further ── Arrhenius and Gamow are two limits of one formula, meeting at \(T_c=\hbar\omega_b/2\pi k_B\). And that \(2\pi\) is the same \(2\pi\) as in the Unruh temperature.
“Does that formula have a name?” ── It does. Almost too many.
★ And following the names tells you what the integral actually is.
01Six names
| name | who | year | context |
|---|---|---|---|
| Liouville–Green approximation | Liouville, Green | 1837 | mathematics (asymptotics of 2nd-order ODEs) |
| (and Jeffreys) | Jeffreys | 1923 | hence “WKBJ” in some traditions |
| WKB approximation | Wentzel / Kramers / Brillouin | 1926 | quantum mechanics |
| Gamow factor | Gamow | 1928 | the Coulomb barrier (Episode 62) |
| Kemble formula | Kemble | 1935 | the uniformly valid \(1/(1+e^G)\) |
| instanton / bounce | Coleman | 1977 | field theory, imaginary time |
The core of §01
★ One integral collected six names over 140 years. Liouville and Green wrote it as “the asymptotics of a second-order differential equation” 89 years before quantum mechanics.
★★★ The deepest name is the last ── the integral itself is the action of a classical path in imaginary time, \(S_E\), and the tunnelling amplitude is \(e^{-S_E/\hbar}\). That is an instanton.
★★ So tunnelling is classical motion in imaginary time.
02Why \(2/\hbar\)?
★ The 2 in Episode 62’s \(2/\hbar\) is there because the amplitude is squared to get a probability. In decades: the amplitude is suppressed by \(G/(2\ln10)\), the probability by \(G/\ln10\).
★★ Episode 58’s Bogoliubov coefficient \(|\beta|\) is also an amplitude ── which is why its exponent was half. The factor-of-two offset is only about whether you are looking at an amplitude or a probability.
03★★★ In imaginary time, the barrier becomes a valley
The sign flips. In imaginary time the particle moves in \(-V\), the upside-down potential ── where the barrier is a valley and the forbidden region can be crossed classically.
Using energy conservation \(\frac{m}{2}(dx/d\tau)^2=V-E\), that path’s Euclidean action is
★ Let us check that by actually solving the inverted-potential equation of motion (\(V=V_0\,\mathrm{sech}^2(x/a)\), \(\hbar=m=1\), \(V_0=1\), \(a=3\)).
| \(E\) | turning points \(\pm x_t\) | ★ Euclidean action \(S_E\) | WKB integral | ratio |
|---|---|---|---|---|
| 0.2 | 4.3309 | 7.367896 | 7.367896 | 1.000000 |
| 0.4 | 3.0952 | 4.898871 | 4.898871 | 1.000000 |
| 0.6 | 2.2365 | 3.004322 | 3.004322 | 1.000000 |
| 0.8 | 1.4436 | 1.407143 | 1.407143 | 1.000000 |
The core of §03 ── the summit of this episode
★★★ Agreement to six decimal places. This is not algebra rewritten: it is the result of running the particle through the inverted potential with RK4 and accumulating the action.
★★ “Classically forbidden” was a statement about real time. Rotate time by 90° and the barrier is a valley the particle simply rolls through.
★ The path returns to where it started, which is why it is called a bounce. ★★★ Looking for “the moment of crossing” in tunnelling finds nothing because it is not an event in real time.
04Against the exact solution ── the decades are right, the prefactor is not
\(V=V_0\,\mathrm{sech}^2(x/a)\) (the Eckart barrier) can be solved exactly. Compare.
| \(E/V_0\) | \(G\) | \(e^{-G}\) | exact | ★ exact / \(e^{-G}\) |
|---|---|---|---|---|
| 0.10 | 18.2275 | 1.213e−8 | 1.460e−8 | 1.2036 |
| 0.20 | 14.7358 | 3.984e−7 | 4.797e−7 | 1.2041 |
| 0.30 | 12.0565 | 5.807e−6 | 6.992e−6 | 1.2041 |
| 0.50 | 7.8077 | 4.066e−4 | 4.893e−4 | 1.2036 |
| 0.70 | 4.3542 | 1.285e−2 | 1.524e−2 | 1.1858 |
The core of §04
★★★ Across eight decades of transmission the ratio stays at 1.20 (1.5 % relative spread).
★★ And that 1.2041 matches the analytic \(\exp\!\bigl(\pi[2a\sqrt{2V_0}-\sqrt{8V_0a^2-1}]\bigr)\) ── a next-order correction coming from the “\(-1\)” inside \(\sqrt{\lambda-1}\).
★★★ So WKB’s error lives in the prefactor, not in the exponent. That is exactly why Episode 62 could say “the decades are right”.
★ Conversely, if you want the prefactor, WKB is not enough. Episode 62’s \(\alpha\) decay being off by a few decades in absolute value has the same origin.
05★★ At the top of the barrier \(e^{-G}\) breaks ── the Kemble form
| \(E/V_0\) | \(G\) | \(e^{-G}\) | ★ \(1/(1+e^G)\) | exact |
|---|---|---|---|---|
| 0.90 | 1.3329 | 2.637e−1 | 0.208686 | 0.234657 |
| 1.00 | 0.0000 | 1.000e+0 | 0.500000 | 0.546309 |
| 1.05 | −0.6664 | 1.947e+0 | 0.660704 | 0.699322 |
| 1.50 | −6.6643 | 7.839e+2 | 0.998726 | 0.997927 |
The core of §05 ── what to fix in Episode 62’s calculator
★★★ At \(E=V_0\), \(e^{-G}=1\) (everything passes) is plainly wrong. Exact is 0.546; Kemble gives 0.500.
★★ Above the barrier \(e^{-G}\) reaches 784 ── a probability greater than one.
★ Kemble’s \(1/(1+e^G)\) stays between 0 and 1 everywhere, and for a parabolic barrier it is the exact answer.
★★★ Why it breaks at the top: the two turning points merge. Episode 57’s “edge” ── we have left the range where the expansion applies. There we counted decades lost as \((1/n)\log(1/\varepsilon)\); here, near the edge, the approximation itself collapses.
06★★★ Arrhenius and Gamow are two limits of one formula
The exponent is \(-E/k_BT+2\pi E/(\hbar\omega_b)+\text{const}\). The boundary is where the coefficient of \(E\) changes sign:
What an experiment measures is the apparent activation energy \(E_{\text{eff}}=-d(\ln\Gamma)/d(1/T)\).
| \(T/T_c\) | ★ \(E_{\text{eff}}\) | \(E_{\text{eff}}/V_0\) | high-\(T\) prediction \(V_0+k_BT\) | regime |
|---|---|---|---|---|
| 0.3 | 0.0322 | 3.2 % | 1.0225 | tunnelling |
| 0.5 | 0.0750 | 7.5 % | 1.0375 | tunnelling |
| 0.8 | 0.2614 | 26 % | 1.0600 | tunnelling |
| 1.0 | 0.5091 | 51 % | 1.0750 | ★ crossover |
| 2.0 | 1.0009 | 100 % | 1.1501 | Arrhenius |
| 5.0 | 1.3234 | 132 % | 1.3751 | Arrhenius |
Crossover temperature calculator
The core of §06 ── the experimental signature of tunnelling
★★★ At low temperature \(E_{\text{eff}}\) is only 3 % of the true barrier. Lowering the temperature stops slowing the rate ── that is the sign that tunnelling has taken over.
★★ The Arrhenius plot (\(\ln\Gamma\) versus \(1/T\)) flattens at low temperature. When the apparent activation energy comes out below the true barrier, the difference is the tunnelling.
★ At high temperature \(E_{\text{eff}}\to V_0+k_BT\) ── slightly above the true barrier, by the \(k_BT\) of the prefactor (1.323 against a predicted 1.375 at \(T=5T_c\)).
★★★ Episode 56 discussed activation as \(\tau=\tau_0e^{E/kT}\) ── that \(E\) was an apparent activation energy. Sweep the temperature, find the knee, and you have \(T_c\) and hence \(\omega_b\).
07That \(2\pi\) is the Unruh \(2\pi\)
★ At temperature \(T\), imaginary time \(\tau\) is a circle of circumference \(\hbar/k_BT\). That is the correspondence between temperature and imaginary time — the same framework as Episode 45’s Unruh effect.
★★ A bounce near a parabolic top has period \(2\pi/\omega_b\). It fits on that circle only if \(\hbar/k_BT\ge2\pi/\omega_b\), i.e. \(T\le T_c\).
| setting | formula | where the \(2\pi\) comes from |
|---|---|---|
| crossover (here) | \(T_c=\hbar\omega_b/2\pi k_B\) | circumference of imaginary time |
| Unruh (Episode 45) | \(T=\hbar a/2\pi ck_B\) | circumference of imaginary time |
| Hawking | \(T=\hbar\kappa/2\pi ck_B\) | circumference of imaginary time |
★★★ Episode 45’s “an accelerated observer sees the vacuum as warm” and this episode’s “above a certain temperature you go over classically” are two faces of the same imaginary-time geometry.
★ But only the origin of the \(2\pi\) is shared; the physics is different ── the same caution as Episode 63’s “the same statistic is not the same physics”.
08★★★ Perturbation theory cannot see \(e^{-S/\hbar}\) ── but its divergence knows it
The anharmonic oscillator \(H=\frac{p^2}{2}+\frac{x^2}{2}+gx^4\). For \(g<0\) it becomes metastable, with bounce action \(S=1/(3|g|)\) (checked numerically: \(S=6.666667\) at \(g=0.05\) against the analytic 6.666667).
Now compute the coefficients of \(E(g)=\sum a_ng^n\) out to order 40.
| \(n\) | \(a_n\) | \(a_n/a_{n-1}\) | ★ ratio \(/(-3)\) | predicted \(n-\frac12\) |
|---|---|---|---|---|
| 2 | −2.6250e+0 | −3.5000 | 1.1667 | 1.5 |
| 5 | 3.5810e+3 | −14.8410 | 4.9470 | 4.5 |
| 10 | −2.4479e+10 | −29.3674 | 9.7891 | 9.5 |
| 20 | −7.6940e+26 | −58.7709 | 19.5903 | 19.5 |
| 40 | −6.6419e+65 | −118.6129 | 39.5376 | 39.5 |
The core of §08 ── the deepest point here
★ The series does not converge. \(a_n\sim3^n\,\Gamma(n+\tfrac12)\): factorial divergence.
★★ So perturbation theory can never produce \(e^{-1/(3g)}\) ── its Taylor expansion vanishes at every order. A function that is zero to all orders. ★ That is what “non-perturbative” means.
★★★ And yet the divergence rate \(3^n\) has a “3” that is exactly \(1/S\) for the bounce action \(S=1/(3g)\).
★★★ “Perturbation theory cannot see tunnelling — but the way it fails knows about it.” The quantitative form of Dyson’s argument (1952), computed by Bender–Wu (1969).
★★ This is the main reason that integral collected six names ── it is very nearly the only tool for counting what lies outside perturbation theory.
Exercises
- §03 says tunnelling is classical motion in imaginary time. So how long does the crossing take?
Show answer
The question is framed in real time, and as posed it has no answer.
★ The “duration” \(\tau\) in imaginary time is computable (this episode’s code produces it), but it is not a number of seconds.
★★ Real-time tunnelling time still has several competing definitions and no settled answer (Wigner time, Larmor time, Büttiker–Landauer time, …). “How long was the particle inside the barrier” is hard to translate into an observable in the first place.
★★★ In this series’ language: it is the same type of problem as Episode 60’s “trend or signal is decided by the record, not the system”. “Time spent inside the barrier” is undefined until you declare how you measure it.
★ All this episode claims is that the actions agree ── it says nothing about time. - §05 introduced the Kemble form. Should Episode 62’s calculator be fixed?
Show answer
It depends on the use.
★ For Episode 62’s purposes, no. Fusion, cold fusion, \(\alpha\) decay and STM all sat at \(G\gg1\). At \(G=10\) the difference between \(e^{-G}\) and \(1/(1+e^G)\) is below \(10^{-4}\).
★★ It matters near the barrier top: in Episode 62’s table the riskiest row is d+t at 20 keV (\(G=7.7\)), where the difference is about 0.05 % ── still fine.
★★★ Where it really bites is chemistry ── at room temperature \(G\sim1\) is common, and there \(e^{-G}\) is wrong by more than a factor of two. ★ The handbook entry: use Kemble when \(G<5\). Not a blanket rewrite, but counting where the difference appears first ── Episode 55’s discipline. - §08 says the divergence knows the action. Is that a coincidence?
Show answer
Not a coincidence: two views of the same integral.
★ Dyson’s argument (1952): for \(g<0\) the system is unstable, so \(E(g)\) cannot be analytic at \(g=0\); the radius of convergence is zero and the series must diverge.
★★ The degree of that instability — the decay rate \(\sim e^{-S/g}\) — is what sets the divergence rate. Under Borel transformation, \(e^{-S/g}\) corresponds to a singularity in the complex Borel plane, located at \(S\). The \(n!/S^n\) growth is precisely the distance to that singularity.
★★★ So “how the perturbation series breaks” and “how big the non-perturbative effect is” are the same information ── organised into the theory of resurgence.
★ In this series’ terms: Episode 57 said “an edge is set by its leading correction term”. Here the edge of a series (\(n\to\infty\)) reports the action of a barrier in real space. The edge of the series and the edge of the potential point at the same number.
Summary — six names, one identity
“Does that formula have a name?” ── At least six, over 140 years. Liouville–Green (1837), Jeffreys (1923), WKB (1926), the Gamow factor (1928), Kemble (1935), instanton / bounce (1977).
★★★ And its identity ──
the action of a classical path in imaginary time, \(S_E\), with \(G=2S_E/\hbar\).
★ Rotating time by 90° inverts the potential and turns the barrier into a valley. Running the particle through that inverted potential with RK4 and accumulating the action reproduces the WKB integral to six decimal places. ★★ “Classically forbidden” was a statement about real time.
★★ Against the exact Eckart solution, WKB’s error lives in the prefactor, not the exponent ── the ratio holds at 1.20 across eight decades of transmission (1.5 % spread), matching the analytic \(\exp(\pi[2a\sqrt{2V_0}-\sqrt{8V_0a^2-1}])=1.2041\). That is why Episode 62 could claim the decades.
★ At the barrier top \(e^{-G}\) breaks (1 at \(E=V_0\), 784 at \(1.5V_0\)). Kemble’s \(1/(1+e^G)\) is uniformly valid and exact for a parabolic barrier. It breaks because the two turning points merge ── Episode 57’s edge.
★★★ Arrhenius and Gamow are two limits of one formula, meeting at \(T_c=\hbar\omega_b/2\pi k_B\). The apparent activation energy runs from \(V_0+k_BT\) at \(T=5T_c\) down to 3 % of the true barrier at \(0.3T_c\). ★★ A flattening Arrhenius plot is the experimental signature of tunnelling.
★★ And that \(2\pi\) is the Unruh \(2\pi\) ── both are the circumference of imaginary time. Episode 45 and this episode are two faces of one geometry. ★ Though only the \(2\pi\)’s origin is shared; the physics differs (the same caution as Episode 63).
★★★ And the deepest point: the series \(E(g)=\sum a_ng^n\) diverges as \(a_n\sim3^n\Gamma(n+\frac12)\) and never converges. Perturbation theory can never produce \(e^{-1/(3g)}\) — it is zero to all orders. Yet the “3” in that divergence is exactly \(1/S\) for the bounce action \(S=1/(3g)\) (ratio \(/(-3)=39.5376\) at \(n=40\) against a predicted 39.5). Perturbation theory cannot see tunnelling; the way it fails knows about it.
★ Episode 57 counted how an edge is set by its leading correction; this episode found that a series’ edge knows a non-perturbative effect.
★★ Next we join the two ── where is the best place to truncate an asymptotic series? ★ Optimal truncation, and the error left behind being exactly \(e^{-S/\hbar}\) (superasymptotics). It may be the final form of this series’ question: how many decades can you buy?
This document is Episode 64 of the “Thinking in Waves” series. The computations are in kenshou/namae.js (zero dependencies, all 14 checks passing). Not discoveries of this episode: the WKB / Liouville–Green approximation, the Gamow factor, Kemble’s uniform formula, the exact solution of the Eckart (Pöschl–Teller) barrier, the imaginary-time formalism and the instanton / bounce (Coleman 1977), the crossover temperature \(T_c=\hbar\omega_b/2\pi k_B\) (Affleck 1981), the periodicity of imaginary time at finite temperature, the Unruh and Hawking temperatures, Dyson’s argument (1952), the Bender–Wu large-order asymptotics (1969), and resurgence ── all well-known; this episode is a numerical confirmation. What it did was ① integrate the classical motion in the inverted potential and verify to six figures that \(S_E\) equals the WKB integral, ② compare against the exact Eckart solution to localise WKB’s error in the prefactor and match that prefactor’s analytic form, ③ compute the apparent activation energy \(E_{\text{eff}}(T)\) and exhibit the knee at \(T_c\) numerically, and ④ compute the perturbation coefficients to order 40, confirm \(a_n/a_{n-1}\to-3(n-\frac12)\), and match it against the bounce action. Limits: §03 uses \(d\tau=10^{-5}\) starting \(10^{-7}\) inside the turning point, so agreement is to six figures only. The “1.20” in §04 is specific to this barrier (\(V_0=1\), \(a=3\)); other barriers give other constants. §06 is a model calculation using the parabolic transmission, and for real systems \(\omega_b\) is itself an approximation. The §08 coefficients use a 260-state harmonic basis, and beyond order 40 the truncation begins to matter. This episode too uses only analytic formulas and numerics; no measured data were used. ── To print, use your browser’s Print → Save as PDF.