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.

WKB in Classically Forbidden Regions owns the local growing and decaying basis, its boundary-condition logic, and the current carried by a complex branch mixture. This page owns the global barrier action and leading transmission probability after both turning points are connected.

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

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

This is the canonical WKB tunneling formula. It captures the leading exponential dependence on barrier height, width, particle mass, and energy.

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

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

The exponent is often the most reliable part of the calculation. If $K/\hbar\gg1$, even a modest uncertainty in the prefactor is much less important than the exponential suppression.

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.

Prefactor Caveats

The leading WKB formula suppresses prefactors. More accurate transmission formulas require Airy connection formulas at both turning points and careful matching to incoming and outgoing waves. Prefactors can depend on:

• 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 on the two sides.

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

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

This exponential structure appears in field-emission estimates, where an applied electric field turns a binding barrier into an approximately triangular barrier.

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.

Common Mistakes

• Forgetting that $T$ contains twice the under-barrier amplitude exponent.
• Mixing momentum $\kappa=\sqrt{2m(V-E)}$ with decay constant $\kappa/\hbar$.
• Applying the smooth WKB barrier formula to abrupt barriers without checking matching.
• Trusting only the exponent near the top of the barrier, where the forbidden region shrinks and connection details matter.
• Interpreting tunneling as a classical trajectory moving under the barrier in real time.

Exercises

1. Show that the WKB formula reduces to the leading rectangular-barrier exponential.

Solution

For a rectangular barrier, $\kappa(x)=\sqrt{2m(V_0-E)}$ is constant as a momentum, so

$$K = \int_0^a \kappa\,dx = \kappa a.$$

Therefore

$$T\approx e^{-2K/\hbar} = \exp\left( - \frac{2a}{\hbar} \sqrt{2m(V_0-E)} \right).$$

If $\kappa_{\mathrm{decay}}=\sqrt{2m(V_0-E)}/\hbar$, this is $T\approx e^{-2\kappa_{\mathrm{decay}}a}$.

2. Evaluate the triangular-barrier action

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

Solution

Let $u=U-Fx$, so $du=-Fdx$. The limits are $u=U$ at $x=0$ and $u=0$ at $x=U/F$. Then

$$K = \frac{\sqrt{2m}}{F} \int_0^U u^{1/2}\,du = \frac{\sqrt{2m}}{F} \frac{2}{3}U^{3/2}.$$

Thus

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

3. Why is the exponent often more robust than the prefactor in WKB tunneling?

Solution

When $K/\hbar\gg1$, the probability contains the very small factor $e^{-2K/\hbar}$. Multiplicative prefactors change the result algebraically, while small changes in $K/\hbar$ change it exponentially. Therefore the leading exponent usually controls the order of magnitude, even when the prefactor is only approximate.

References

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