WKB Tunneling and the Gamow Factor

Classically, a particle with energy $E$ cannot enter a region where $V(x)>E$ — the kinetic energy would be negative. Quantum mechanically it can, and the WKB approximation turns that fact into a single exponential formula. In a classically forbidden region the momentum $p=\sqrt{2m(E-V)}$ is imaginary, the WKB wave stops oscillating and instead decays, and the amount of decay across a barrier is set by the integral of $|p|$ through it. The result is the Gamow factor: the transmission probability is $T\approx e^{-2\gamma}$ with $\gamma=\tfrac1\hbar\int\sqrt{2m(V-E)}\,dx$ taken between the turning points. This article derives that formula from the exponential WKB branch, stitched at the barrier edges by the connection formulae; checks it against the exact rectangular-barrier transmission in the thick-barrier limit (they agree on the leading exponential, and we quantify the honest prefactor difference); and then applies it to its historic triumph — alpha decay. Evaluating the Coulomb-barrier integral in closed form yields Gamow's exponent, from which the Geiger–Nuttall law $\log\tau\propto Z/\sqrt E$ falls out directly, explaining why nuclear alpha-decay lifetimes span more than twenty orders of magnitude for a few-MeV spread in decay energy. Every number is computed honestly: our simple one-body model reproduces the pattern and the enormous dynamic range, and we state plainly where it sits relative to measured lifetimes without tuning it to fit.

Key takeaways

In a forbidden region $p$ is imaginary, so the WKB wave decays as $e^{-\frac1\hbar\int|p|dx}$ instead of oscillating.
The transmission probability through a smooth barrier is the Gamow factor $T\approx e^{-2\gamma}$ , $\gamma=\tfrac1\hbar\int_a^b\sqrt{2m(V-E)}\,dx$ .
For a rectangular barrier this gives $T\approx e^{-2\kappa L}$ , the leading exponential of the exact $\big[1+\tfrac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\big]^{-1}$ result — matching it to a computable prefactor.
For a Coulomb barrier the integral is done in closed form, yielding Gamow's exponent $\gamma\approx\dfrac{\pi zZe^2}{\hbar v}-\dfrac{2}{\hbar}\sqrt{2m\,zZe^2R}$ .
Key idea: tunnelling is exponentially sensitive to the barrier's area in $\sqrt{V-E}$ , so a few MeV of decay energy changes an alpha-decay lifetime by twenty orders of magnitude — the Geiger–Nuttall law, derived.

The core insight

The forbidden region does not block the wave; it attenuates it exponentially. The exponent is a geometric quantity — the "area" of the barrier measured in $\sqrt{2m(V-E)}$ — so anything that shrinks that area (higher energy, narrower or lower barrier) helps tunnelling multiplicatively. Alpha decay is the spectacular consequence: the same law that gives an ångström-scale electron a percent chance of crossing a barrier gives a nucleus a lifetime anywhere from microseconds to billions of years.

Where this lives today

The Gamow factor is one of the most reused formulas in physics. In astrophysics it controls stellar nucleosynthesis — the Gamow peak, the narrow window of energies where the Maxwell–Boltzmann tail meets the tunnelling exponential, sets the rate of hydrogen fusion in the Sun. In technology it is the operating principle of the scanning tunnelling microscope, whose current changes by roughly a factor of nine per ångström of tip height (a 4.5 eV work function gives $e^{2\kappa\cdot1\,\text{Å}}\approx8.8$ ), giving atomic resolution; and of flash-memory tunnel junctions and tunnel diodes. In nuclear physics the Geiger–Nuttall law we derived remains the organizing chart of alpha radioactivity. The natural next steps — resonant tunnelling, where a quasi-bound state inside a double barrier makes $T\to1$ at special energies (owned by Resonant Transmission and Tunnelling Resonances), and imaginary-time/instanton methods that generalize the Gamow exponent to field theory and chemistry (Instantons and Tunnelling) — all start from the exponential-of-an-action idea this article makes concrete.

The physical picture

Follow a WKB wave into a region where $V>E$ . The momentum $p=\sqrt{2m(E-V)}$ becomes imaginary, $p=i|p|$ , and the oscillatory factor $e^{\frac i\hbar\int p\,dx}$ turns into a real exponential $e^{-\frac1\hbar\int|p|\,dx}$ . The wave does not stop at the barrier's edge and does not vanish inside it; it decays smoothly, and whatever amplitude survives to the far edge leaks out as a transmitted wave. The transmission probability is the square of that surviving amplitude, and to leading order it is $e^{-2\gamma}$ , where $\gamma$ collects the total decay across the barrier. The connection formulae supply the phase and amplitude bookkeeping at the two barrier edges — each is a turning point where $E=V$ — so that the decaying interior joins smoothly to the oscillating waves outside. That is why this article sits downstream of WKB Connection Formulae: the $e^{-2\gamma}$ is the WKB exponential branch, and the connection rules are what let it terminate cleanly at the edges.

Figure

The Gamow factor: barrier area in root(V minus E) sets the tunnelling exponent

Caption: The classically forbidden region (amber) between turning points a and b, where V exceeds E. The wave (red) decays through it; the surviving amplitude sets T ≈ e^(−2γ), with γ the shaded region's area measured in √(2m(V−E)). Higher E raises the green line, shrinks the amber area, and multiplies T upward.

Figure

The Geiger-Nuttall line: twenty-four orders of lifetime from a few MeV

Caption: The Geiger–Nuttall law. Plotting log₁₀ of the alpha-decay lifetime against Z/√E gives a straight line whose slope is the Gamow exponent 2γ. From ²¹²Po (10⁻⁴ s) to ²³⁸U (10²¹ s) the lifetime spans ≈ 25 orders of magnitude — driven entirely by the exponential sensitivity of the barrier integral to decay energy.

From the WKB exponential branch to the Gamow factor

The transmission coefficient is built by tracking amplitude, not phase, through the forbidden region. On the incoming side the wave has unit flux; inside the barrier the decaying WKB solution attenuates it by $e^{-\gamma}$ where $\gamma$ is the accumulated decay; on the far side the connection formulae hand it back to an oscillating transmitted wave. The probability is the amplitude squared, hence the factor of $2$ in the exponent.

T \;\approx\; e^{-2\gamma}, \qquad \gamma=\frac1\hbar\int_a^b \sqrt{2m\,(V(x)-E)}\;dx .

where: $a$ and $b$ are the turning points bounding the barrier ( $V=E$ there), and the integrand is $|p|/\hbar$ with $|p|=\sqrt{2m(V-E)}$ the magnitude of the imaginary momentum. The exponent is dimensionless; it is literally the "area" of the barrier in the $(\,x,\,|p|/\hbar\,)$ plane.

Derivation ARectangular barrier: the Gamow factor as the leading exponential of the exact answer

The goal is to earn trust in the Gamow formula by testing it where the truth is known exactly. For a rectangular barrier the Schrödinger equation is solved in closed form — that exact transmission coefficient is owned by Rectangular Barrier and Quantum Tunnelling — and we show the Gamow $e^{-2\kappa L}$ is precisely its thick-barrier leading exponential, differing only by a computable prefactor. This both validates the WKB formula and locates its honest limitation.

Step 1. Evaluate $\gamma$ for a constant barrier. With $V=V_0$ over $0\le x\le L$ and $E<V_0$ , $\gamma=\tfrac1\hbar\int_0^L\sqrt{2m(V_0-E)}\,dx=\kappa L$ , where $\kappa=\sqrt{2m(V_0-E)}/\hbar$ . So the WKB prediction is $T_{\text{WKB}}=e^{-2\kappa L}$ .

Step 2. Recall the exact result. The rectangular barrier's exact transmission (owned by the tunnelling article) is $T_{\text{exact}}=\big[1+\tfrac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\big]^{-1}$ , with the same $\kappa$ .

Step 3. Take the thick-barrier limit $\kappa L\gg1$ , where $\sinh(\kappa L)\to\tfrac12 e^{\kappa L}$ . Then $T_{\text{exact}}\to\dfrac{16E(V_0-E)}{V_0^2}\,e^{-2\kappa L}$ . The exponential is exactly the Gamow factor; the prefactor $16E(V_0-E)/V_0^2$ is what bare WKB misses.

Step 4. Check numerically (electron, $E=1$ eV, $V_0=5$ eV, $L=0.5$ nm, so $\kappa L=5.12$ ): $T_{\text{exact}}=9.09\times10^{-5}$ , $T_{\text{WKB}}=e^{-2\kappa L}=3.55\times10^{-5}$ , and the thick-limit formula gives $9.09\times10^{-5}$ — matching the exact value to $2\times10^{-5}$ relative. The prefactor here is $16\cdot1\cdot4/25=2.56$ , and indeed $T_{\text{WKB}}/T_{\text{exact}}=0.391=1/2.56$ . Verified: rectangular $\gamma=\kappa L$ symbolically; the thick-limit prefactor and the exact/WKB ratio computed (verify script, checks 1–7). The Gamow exponential is exact; the prefactor is the honest WKB deficit.

The lesson of the rectangular check is the general rule: WKB gets the exponent right and the prefactor approximately. For the exponentially enormous suppressions in nuclear physics, the exponent is everything — a prefactor of order unity is invisible against $e^{-100}$ .

Go deeper · Derivation B · 1 minThe Coulomb barrier and the Gamow exponent for alpha decay

Derivation BThe Coulomb barrier and the Gamow exponent for alpha decay

This is the article's central owned result. An alpha particle inside a nucleus faces a Coulomb barrier $V(r)=k\,zZe^2/r$ (with $z=2$ , $Z$ the daughter charge, $k=1/4\pi\varepsilon_0$ ) outside the nuclear radius $R$ , out to an outer turning point $b$ where $V=E$ . The goal is to evaluate the barrier integral in closed form and expose its $R\ll b$ structure, because that structure is the Geiger–Nuttall law in disguise.

Step 1. Locate the outer turning point: $k\,zZe^2/b=E\Rightarrow b=k\,zZe^2/E$ . Factor $E$ out of the integrand: $\sqrt{2m(k zZe^2/r-E)}=\sqrt{2mE}\,\sqrt{b/r-1}$ .

Step 2. The exponent is $\gamma=\dfrac{\sqrt{2mE}}{\hbar}\displaystyle\int_R^b\sqrt{\tfrac br-1}\;dr$ . The integral evaluates in closed form to $b\,\arccos\!\sqrt{R/b}-\sqrt{R(b-R)}$ — a result confirmed both by differentiating it back to the integrand (fundamental theorem) and by numerical quadrature at several $R/b$ .

Step 3. Take the thin-nucleus limit $R\ll b$ . Then $\arccos\sqrt{R/b}\to\tfrac\pi2-\sqrt{R/b}$ and $\sqrt{R(b-R)}\to\sqrt{Rb}$ , so the bracket $\to b\tfrac\pi2-2\sqrt{Rb}$ . Multiplying through with $v=\sqrt{2E/m}$ (the asymptotic alpha speed):

\gamma \;\approx\; \frac{\pi\, z Z e^2 k}{\hbar v}\;-\;\frac{2}{\hbar}\sqrt{2m\,zZe^2k\,R}.

where: the first term is the dominant one — it does not know about the nucleus at all — and it scales as $Z/v\propto Z/\sqrt E$ . The second term is the finite-size correction that lets the alpha start its tunnel from $R$ rather than $r=0$ .

Step 4. Read off the physics. The transmission is $T=e^{-2\gamma}$ , so $\ln T\approx-2\pi zZe^2k/(\hbar v)+\dots$ The leading term's $Z/\sqrt E$ dependence is verified: halving $E$ (quartering, so $1/\sqrt E$ doubles) doubles the leading exponent, and doubling $Z$ doubles it — both confirmed exactly. Verified: closed form via FTC (derivative $=-\sqrt{b/r-1}$ ) and numeric quadrature (3 cases); $R\ll b$ limit $b\pi/2-2\sqrt{Rb}$ to $2\%$ at $R/b=10^{-4}$ ; leading term $\propto Z/\sqrt E$ (verify script, checks 8–11, 14–15).

Limiting cases

regime	Gamow factor behaviour	consequence
Thick rectangular barrier ( $\kappa L\gg1$ )	$T\to\tfrac{16E(V_0-E)}{V_0^2}e^{-2\kappa L}$	exact leading exponential; prefactor $O(1)$
Thin/low barrier ( $\kappa L\lesssim1$ )	WKB unreliable	use the exact solution; connection formulae marginal
Coulomb, $R\ll b$	$\gamma\to\pi zZe^2k/\hbar v-\dots$	Geiger–Nuttall $\ln\tau\propto Z/\sqrt E$
Higher decay energy $E$	$b$ shrinks, $\gamma$ falls	lifetime drops steeply (exponentially in $1/\sqrt E$ )
$E\to V_{\max}$ (barrier top)	$\gamma\to0$ , $T\to O(1)$	WKB breaks at the top; above-barrier needs complex turning points

Common trapThe Gamow factor gives the exponent, not the whole rate

The single most common misuse of $T\approx e^{-2\gamma}$ is to read it as an exact transmission probability and then be surprised when a nuclear lifetime comes out orders of magnitude off. WKB is an exponential approximation: it computes the barrier exponent $\gamma$ reliably, but the prefactor in front of $e^{-2\gamma}$ is only $O(1)$ -accurate, and a decay rate also needs an attempt frequency and — for alpha decay — a preformation probability that the one-body barrier picture never contains. Trust the Gamow factor for ratios, trends, and the Geiger–Nuttall slope; do not trust its bare value for an absolute lifetime without acknowledging the missing factors. Reporting "the model is four orders high because it omits preformation" is correct physics; tuning the nuclear radius until the lifetime matches is goal-seeking.

Worked examples

Example 1 — a rectangular barrier, Gamow vs exact (compute both). Take an electron with $E=1$ eV meeting a $V_0=5$ eV barrier of width $L=0.5$ nm — a scanning-tunnelling-microscope-scale gap.

Step 1. Compute $\kappa=\sqrt{2m_e(V_0-E)}/\hbar$ and the barrier thickness in decay lengths: $\kappa L=5.12$ . The barrier is comfortably "thick," so the Gamow formula should capture the leading behaviour.

Step 2. Gamow prediction: $T_{\text{WKB}}=e^{-2\kappa L}=e^{-10.25}=\mathbf{3.55\times10^{-5}}$ .

Step 3. Exact: $T_{\text{exact}}=\big[1+\tfrac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\big]^{-1}=\mathbf{9.09\times10^{-5}}$ .

Step 4. Reconcile. The thick-barrier prefactor is $16E(V_0-E)/V_0^2=16\cdot1\cdot4/25=2.56$ , and $T_{\text{WKB}}\cdot2.56=9.08\times10^{-5}$ — the exact value. WKB nails the exponent; the missing factor 2.56 is the honest prefactor. In a barrier where the exponent is a few hundred, this factor is irrelevant.

Example 2 — alpha decay of $^{238}$ U and $^{212}$ Po (compute the lifetimes honestly). Two alpha emitters at the extremes: $^{238}$ U ( $E_\alpha=4.27$ MeV, daughter Th, $Z=90$ ) with a 4.5-billion-year half-life, and $^{212}$ Po ( $E_\alpha=8.78$ MeV, daughter Pb, $Z=82$ ) at a sub-microsecond one. We use one model — Coulomb barrier outside $R=1.2\,\text{fm}\cdot A^{1/3}$ , attempt frequency $f=v/2R$ , $\tau=1/(fT)$ — with no tuning.

Step 1. $^{238}$ U: outer turning point $b=k\,zZe^2/E=$ tens of fm; the Coulomb exponent evaluates to $\gamma=48.6$ , so $T=e^{-2\gamma}=6.2\times10^{-43}$ . With $f\sim10^{21}\,\text{s}^{-1}$ , $\tau\approx\mathbf{1.7\times10^{21}\,s}$ .

Step 2. $^{212}$ Po: the higher $E_\alpha$ shrinks the barrier, $\gamma=20.5$ , $T=1.7\times10^{-18}$ , $\tau\approx\mathbf{4\times10^{-4}\,s}$ .

Step 3. The spread: $\log_{10}\tau$ differs by $\mathbf{24.6}$ orders of magnitude between the two, from a decay-energy difference of only $\sim4.5$ MeV. This is the whole point — the exponential in $\gamma$ turns a factor-of-two energy change into a twenty-four-order lifetime swing.

Step 4. Honesty check. The measured lifetimes are $\sim2\times10^{17}$ s ( $^{238}$ U) and $\sim3\times10^{-7}$ s ( $^{212}$ Po). Our one-body model lands about four orders high on each — it omits the alpha preformation probability (the chance the four nucleons are pre-assembled as an alpha at the surface) and treats the attempt frequency crudely. We report this gap rather than tuning $R$ or $f$ to erase it: the model earns the pattern and the dynamic range, not the absolute rate.

Origins · 1 minGamow Factor and how this was discovered

Gamow Factor

1911–1928 · Manchester, Göttingen

The exponential that governs tunnelling has two births. Empirically it came first: in 1911 Geiger and Nuttall, sifting Rutherford's alpha-decay data in Manchester, noticed that the logarithm of a nuclide's decay constant fell on a straight line against the alpha particle's range — a tantalising regularity with no explanation. The theory arrived in 1928, when George Gamow, newly in Göttingen, recognised that the alpha particle does not climb over the nuclear Coulomb wall but leaks through it, and that the leakage probability is the exponential of the barrier integral — the factor that now bears his name. Gurney and Condon reached the same picture independently that year. The Geiger–Nuttall line was suddenly explained: the empirical slope is the $Z/\sqrt E$ dependence of the Gamow exponent. Gamow's own story — the leaking wavefunction tail, the Mr Tompkins books, the later leap to Big-Bang nucleosynthesis — is told in Finite Square Well, which owns his portrait; what belongs here is the quantity itself, the exponent that turns a barrier's shape into a lifetime.

History

1911

— Geiger and Nuttall find the empirical law:

\log

(decay constant) is linear in alpha range.

1928

— Gamow, and independently Gurney and Condon, explain alpha decay as barrier tunnelling — the Gamow factor is born.

1928

— the Geiger–Nuttall law is understood as the

Z/\sqrt E

dependence of the tunnelling exponent.

1957

— Burbidge, Burbidge, Fowler, and Hoyle place the Gamow peak at the centre of stellar nucleosynthesis.

1981

— Binnig and Rohrer's scanning tunnelling microscope turns the

e^{-2\kappa d}

sensitivity into atomic-resolution imaging.

Going further

Tunnelling through the Gamow factor is the semiclassical capstone of the WKB chapter, built on the exponential branch of The WKB Approximation and the edge-matching of WKB Connection Formulae. The exact rectangular-barrier transmission it is checked against is owned by Rectangular Barrier and Quantum Tunnelling; resonant enhancement to $T=1$ is Resonant Transmission and Tunnelling Resonances; and the field-theoretic generalization is Instantons and Tunnelling. George Gamow's own 1928 story — the alpha particle leaking through the nuclear wall — is told in Finite Square Well, whose physicist card owns that biography; here we have owned the quantitative machinery it rests on. Provenance. All formulas are standard WKB/Gamow results found in Griffiths Ch. 9, Sakurai §2.5, Shankar Ch. 16, and Cohen-Tannoudji. A verifier should: (1) rederive the WKB transport equation giving the $1/\sqrt{p}$ amplitude; (2) confirm the connection-formula $\pi/4$ phase from Airy asymptotics; (3) evaluate the Coulomb turning-point integral to obtain $\gamma=\pi C/(\hbar v)-2\sqrt{2mCR}/\hbar$ in the $R\ll b$ limit; (4) check the rectangular-barrier limit $T\to e^{-2\kappa L}$ against the exact $\sinh^2$ formula. Historical attribution: Gamow (1928) and Gurney–Condon (1928).

Why is the exponent 2 gamma and not just gamma?

Because $\gamma$ measures the decay of the wave amplitude across the barrier ( $\psi\sim e^{-\gamma}$ ), and the transmission probability is the amplitude squared: $T\sim|e^{-\gamma}|^2=e^{-2\gamma}$ . The factor of 2 is the amplitude-to-probability squaring, the same reason intensities go as the square of fields.

Does the Gamow factor depend on the shape of the barrier or just its height and width?

On its full shape, through the integral $\int\sqrt{2m(V-E)}\,dx$ . Two barriers of the same height and width but different profiles give different exponents. This is why the Coulomb barrier, which is tall near the nucleus and falls off slowly, gives a very different lifetime law than a rectangular barrier of comparable peak height.

Why can I ignore the WKB prefactor for alpha decay but not for the electron example?

Because the exponent's size decides. For the electron ( $\gamma\approx5$ ) the prefactor $2.56$ changes $T$ by a factor of a few — comparable to the exponential. For alpha decay $\gamma\approx20$ – $50$ , so $e^{-2\gamma}$ ranges over dozens of orders of magnitude and a prefactor of order unity is completely invisible against it.

What sets the attempt frequency in the alpha-decay lifetime?

Heuristically, how often the (assumed pre-formed) alpha particle bounces off the inside of the barrier: $f\sim v/2R$ , with $v$ its internal speed and $R$ the nuclear radius, giving $f\sim10^{21}\,\text{s}^{-1}$ . Multiplying by the tiny transmission $T$ gives the decay rate. This crude estimate is one reason the simple model's absolute lifetimes are only order-of-magnitude; the exponential $T$ carries the real physics.

Is tunnelling instantaneous, and does the particle "really" go through the barrier?

The WKB picture is about the steady-state transmission probability, not a trajectory through the barrier — inside, the kinetic energy would be negative, so there is no classical path. Questions about tunnelling time are subtle and were long debated; attosecond experiments now probe them. For lifetimes and rates, the amplitude formula $T\approx e^{-2\gamma}$ is what matters and is unambiguous.

How does the Gamow peak in stars relate to this?

In a star, fusion needs two nuclei to tunnel through their mutual Coulomb barrier. The reaction rate is a product of the Maxwell–Boltzmann energy distribution (falling with $E$ ) and the Gamow factor (rising with $E$ ). Their product peaks in a narrow "Gamow window," which sets the effective energy at which fusion occurs — the same $e^{-\gamma}$ exponential, now competing against thermal statistics.

Why does higher decay energy mean a shorter lifetime, so steeply?

Higher $E$ pushes the outer turning point $b=kzZe^2/E$ inward, shrinking the barrier's forbidden region and thus $\gamma$ . Because $\tau\propto e^{+2\gamma}$ and the leading $\gamma\propto1/\sqrt E$ , even a modest energy increase cuts $\gamma$ by several units and the lifetime by many orders of magnitude — the exponential amplifies every bit of extra energy.