Barrier Penetration and Tunneling

WKB tunneling estimates the transmission probability through a smooth classically forbidden barrier. The central result is that the probability is exponentially suppressed by the action accumulated under the barrier.

Required background. Forbidden-region WKB supplies the exponential branches and their current; turning-point matching connects them to incoming and outgoing waves. Here we assume a single smooth, opaque barrier, two well-separated simple turning points, and no additional reflecting structure.

Let $x_1$ and $x_2$ be the two turning points of a barrier at energy $E$:

$$V(x_1)=V(x_2)=E, \qquad V(x)>E \quad \text{for} \quad x_1<x<x_2.$$

Define

$$\kappa(x) = \sqrt{2m\left(V(x)-E\right)}.$$

The leading WKB transmission probability is

$$T \approx \exp\left[ - \frac{2}{\hbar} \int_{x_1}^{x_2}\kappa(x)\,dx \right].$$

Here $T$ is the ratio of transmitted to incident probability currents. The result applies when the barrier action is large compared with $\hbar$ and the local branches and turning-point overlap regions are controlled.

Barrier Turning Points

The turning points mark the boundaries between oscillatory and evanescent behavior. Outside the barrier, where $E>V(x)$, the wavefunction is locally oscillatory. Inside the barrier, where $V(x)>E$, it is a combination of growing and decaying exponentials:

$$\psi(x) \approx \frac{D_+}{\sqrt{\kappa(x)}} \exp\left( \frac{1}{\hbar} \int^x\kappa(x')\,dx' \right) + \frac{D_-}{\sqrt{\kappa(x)}} \exp\left( - \frac{1}{\hbar} \int^x\kappa(x')\,dx' \right).$$

The transmitted amplitude is controlled by the net decay across the forbidden region. Squaring the amplitude gives the factor of $2$ in the probability exponent.

Transmission Exponent

It is useful to define the barrier action

$$K(E) = \int_{x_1}^{x_2} \sqrt{2m\left(V(x)-E\right)} \,dx.$$

Then

$$T\approx e^{-2K(E)/\hbar}.$$

For an isolated smooth barrier, leading Airy matching of unit-flux branches gives this expression, including its leading coefficient of one. The action controls the dominant logarithm: $\log T\simeq-2K/\hbar$. This does not guarantee accurate relative transmission: an action error $\delta K$ changes $T$ by approximately $e^{-2\delta K/\hbar}$.

Rectangular-Barrier Check

For a rectangular barrier of height $V_0$ and width $a$ with $0<E<V_0$,

$$\kappa = \sqrt{2m(V_0-E)}$$

is constant in the barrier. The WKB exponent gives

$$T \sim \exp\left( - \frac{2\kappa a}{\hbar} \right)$$

if $\kappa$ is defined as a momentum. In the exact rectangular-barrier page, the decay constant is instead defined as

$$\kappa_{\mathrm{decay}} = \frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

With that convention,

$$T\sim e^{-2\kappa_{\mathrm{decay}}a},$$

matching the leading exponential in Rectangular Barrier Tunneling. Exact matching instead gives, for the same zero potential on both sides,

$$T\sim\frac{16E(V_0-E)}{V_0^2}e^{-2\kappa a/\hbar}, \qquad \kappa a/\hbar\gg1.$$

The different coefficient records the abrupt interfaces. This checks the decay exponent but does not validate smooth Airy matching at a step.

Prefactor Caveats

The unit leading coefficient above is not universal across other geometries or boundary conditions. Matching must account for:

• the shape of the barrier near the turning points,
• reflection outside the forbidden region,
• whether the barrier is smooth or abrupt,
• resonances from nearby wells,
• different asymptotic wave numbers if plane-wave amplitudes, rather than unit-flux amplitudes, are used ($T=(k_R/k_L)|t|^2$).

When the exponent is large, the leading estimate is usually the main physical effect. When the barrier is thin, near the top, or part of a resonant structure, the prefactor and matching details can be decisive.

One-Dimensional Scattering Revisited compares this leading probability with exact transfer and scattering matrices and explains why multibarrier resonances require complex phases rather than multiplied WKB probabilities.

Triangular-Barrier Example

For a linearly decreasing barrier

$$V(x)-E = U-Fx, \qquad 0<x<\frac{U}{F},$$

with $U>0$ and $F>0$, the barrier action is

$$K = \int_0^{U/F} \sqrt{2m(U-Fx)} \,dx.$$

Evaluating the integral,

$$K = \frac{2\sqrt{2m}}{3F} U^{3/2}.$$

Thus the leading exponential factor is

$$T \propto \exp\left[ - \frac{4\sqrt{2m}}{3\hbar F} U^{3/2} \right].$$

The left edge $x=0$ is not a turning point since $V(0)-E=U>0$; its boundary matching is an additional input. This exponential appears in field-emission models. Image forces and the surface matching alter the simple triangular model.

Physical Examples

Alpha decay is a classic tunneling problem: an alpha particle has energy below the effective Coulomb barrier and escapes with a probability controlled primarily by a WKB exponent. Gamow Factor evaluates that Coulomb action, including the finite nuclear radius and its relation to the Geiger–Nuttall trend. Field emission from a metal surface uses the same idea with an electric-field-distorted barrier. Molecular inversion and double-well splittings also rely on exponentially small penetration through forbidden regions.

These examples require additional model-specific ingredients. The shared WKB content is the under-barrier action $K(E)$.

Relation to Instantons

In path-integral language, tunneling can often be estimated by a classical solution in imaginary time. The Euclidean action of that solution plays the role of the tunneling exponent. WKB and instanton methods are different descriptions of the same semiclassical suppression in many one-dimensional problems, though the instanton method generalizes more naturally to field-theory settings. Tunneling Splittings explains the matching needed to convert an under-barrier amplitude into a bound-state level splitting. False Vacuum Decay in Quantum Mechanics instead converts the transmission probability per encounter into a metastable escape rate and resonance width. The chapter overview Instantons, Tunneling, and Nonperturbative Effects distinguishes transmission from level splitting and metastable decay before selecting a saddle calculation.

The Path Integrals bridge page gives the broader action-phase context.

Exercises

1. A calculation finds $K/\hbar=20$ with a 1% uncertainty in $K$. Estimate the corresponding multiplicative uncertainty in $T$.

Solution

Here $|\delta K|/\hbar=0.2$, giving factors $e^{\pm0.4}$, approximately $0.67$ to $1.49$. The prediction is reliable as an exponentially small scale but does not have 1% relative accuracy.

2. Near a smooth maximum, take $V(x)=V_b-m\Omega^2x^2/2$. Evaluate the sub-barrier action and explain why $e^{-2K/\hbar}$ fails at $E=V_b$.

Solution

Writing $\Delta=V_b-E>0$ and $a^2=2\Delta/(m\Omega^2)$,

$$K=m\Omega\int_{-a}^{a}\sqrt{a^2-x^2}\,dx =\frac{\pi\Delta}{\Omega}.$$

The exponential tends to one as $\Delta\to0$, although the exact inverted-parabola transmission is

$$T=\frac{1}{1+e^{2\pi(V_b-E)/(\hbar\Omega)}},$$

which equals $1/2$ at the maximum. The two turning-point layers merge there. A uniform parabolic-cylinder approximation, rather than two separate Airy matches, describes that regime [Berry and Mount].

References

• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315-397, 1972.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.