Quantum Tunneling

Quantum tunneling is the appearance of nonzero transmission or state mixing through a region that would be classically forbidden at the same energy. In wave mechanics, it occurs because the Schrödinger equation has evanescent solutions in forbidden regions, and those solutions can connect allowed regions when the barrier is finite.

Tunneling is not a violation of energy conservation. It is a wave phenomenon produced by boundary conditions, continuity, and the finite spatial extent of barriers.

What Tunneling Is

For a particle of energy $E$ in a region where

$$V(x)\gt E,$$

the time-independent Schrödinger equation locally gives exponential rather than oscillatory behavior. For a constant barrier,

$$\psi(x)\sim e^{-\kappa x}, \qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

The wavefunction does not have to vanish immediately in the forbidden region. If the forbidden region has finite width, the evanescent solution can match onto a propagating wave on the far side. The transmitted probability is then nonzero.

In this sense, tunneling is a consequence of solving a global boundary-value problem. It is not a local rule that says the particle climbs over the barrier.

What Tunneling Is Not

Tunneling is often described with shortcuts that become misleading. Keep the following distinctions clear.

Tunneling is not classical thermal activation. In thermal activation, a system acquires enough energy to go over a barrier. In tunneling, the energy can remain below the barrier height.

Tunneling is not energy borrowing. The stationary scattering state has a definite energy. The nonclassical part is spatial behavior in the forbidden region, not a temporary failure of energy conservation.

Tunneling is not guaranteed transmission. Thick, high, or massive-particle barriers can suppress transmission enormously.

Tunneling is not the same as a semi-infinite evanescent tail. A step with $E\lt V_0$ has penetration into the forbidden side but no transmitted current to infinity.

Tunneling does not by itself define a unique classical trajectory under the barrier. Questions about how long tunneling takes require additional operational definitions.

Barrier Tunneling

For the rectangular barrier

$$V(x)= \begin{cases} 0, & x\lt 0,\\ V_0, & 0\lt x\lt a,\\ 0, & x\gt a, \end{cases}$$

with $0\lt E\lt V_0$, the exact calculation gives a nonzero transmission coefficient. In the opaque-barrier limit,

$$T\propto e^{-2\kappa a}, \qquad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

The important qualitative lesson is the exponential dependence:

• increasing the barrier width suppresses tunneling exponentially;
• increasing the barrier height suppresses tunneling;
• increasing the particle mass suppresses tunneling;
• increasing the particle energy toward the barrier top increases tunneling.

The exact rectangular-barrier formula is derived in Rectangular Barrier Tunneling. The current-based meaning of $T$ is summarized in Reflection and Transmission Coefficients.

Bound-State Tunneling And Splitting

Tunneling also appears in bound systems. In a symmetric double well, a state localized in the left well overlaps weakly through the barrier with a state localized in the right well. The exact low-energy eigenstates are even and odd combinations:

$$\lvert +\rangle \approx \frac{1}{\sqrt2} \left( \lvert L\rangle+\lvert R\rangle \right), \qquad \lvert -\rangle \approx \frac{1}{\sqrt2} \left( \lvert L\rangle-\lvert R\rangle \right).$$

Their energy splitting,

$$\Delta E=E_- - E_+,$$

is controlled by tunneling through the central barrier. If a system starts localized in the left well, it can oscillate to the right well with characteristic transfer time

$$t_{\mathrm{transfer}} =\frac{\pi\hbar}{\Delta E}.$$

This is not scattering transmission through an open barrier; it is tunneling-induced mixing between nearly degenerate bound configurations. The first-encounter treatment is in Double-Well Potential, while detailed semiclassical splitting estimates belong to Double-Well Tunneling.

Relation To WKB

For a smooth barrier with turning points $x_1$ and $x_2$, define the decay rate

$$\kappa(x) =\frac{\sqrt{2m(V(x)-E)}}{\hbar}$$

inside the forbidden region. The leading WKB tunneling estimate is

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

This formula generalizes the rectangular-barrier exponential. It is most reliable when the barrier varies slowly on the local de Broglie wavelength scale and the exponent is large. Near turning points, near the top of a barrier, or near resonances, prefactors and matching details matter.

The canonical WKB treatment is Barrier Penetration and Tunneling.

Tunneling-Time Caution

Transmission probability is well defined by asymptotic currents. The time spent tunneling is subtler. There is no single self-adjoint time operator in ordinary nonrelativistic quantum mechanics that simply answers “how long was the particle under the barrier?” for all purposes.

Different operational definitions exist, including phase time, dwell time, traversal-time models, and clock-based definitions. They need not agree in all regimes. Therefore statements such as “the particle travels faster than light under the barrier” or “the particle spends exactly this time inside” should be treated with care and tied to a specific measurement protocol.

For introductory wave mechanics, the safe statement is:

tunneling gives nonzero transmission through a finite forbidden region, with probabilities determined by currents and amplitudes, while traversal-time questions require additional definitions.

Applications As First Encounters

Tunneling appears throughout quantum physics:

• alpha decay, where a nuclear cluster escapes through an effective Coulomb barrier quantified by the Gamow factor;
• scanning tunneling microscopy, where current depends exponentially on tip-sample separation;
• Josephson junctions, where phase-coherent tunneling occurs across a weak link;
• molecular inversion, where two configurations mix through a barrier;
• tunnel diodes and semiconductor heterostructures, where barrier transmission shapes transport.

These applications require additional physics beyond the one-dimensional models here. The purpose of this page is to supply the reusable mechanism and vocabulary.

The first-encounter application map is Tunneling Applications: First Encounters.

Common Mistakes

• Saying the particle “borrows energy” to cross the barrier.
• Confusing an evanescent tail in a semi-infinite step with finite-barrier transmission.
• Forgetting that transmission probability is a current ratio.
• Treating tunneling as unsuppressed just because it is nonzero.
• Using the rectangular-barrier formula outside its assumptions.
• Assuming every tunneling problem has a unique, simple tunneling time.
• Treating bound-state tunneling and scattering transmission as the same calculation.

Where This Is Used

• Finite Potential Barrier gives the full two-interface calculation.
• Rectangular Barrier Tunneling focuses on the tunneling regime and its exponential approximation.
• Potential Step distinguishes evanescent penetration from transmitted flux.
• Reflection and Transmission Coefficients defines $R$ and $T$ using currents.
• Double-Well Potential introduces tunneling splitting in bound systems.
• Tunneling Applications: First Encounters compares alpha decay, STM, Josephson junctions, molecular inversion, and tunnel diodes.
• Barrier Penetration and Tunneling generalizes the exponential estimate to smooth barriers.
• Gamow Factor evaluates the repulsive Coulomb-barrier exponent for alpha decay and low-energy fusion.

References

• 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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. Explain why a finite barrier with $E\lt V_0$ can have $T\gt 0$, while a semi-infinite step with $E\lt V_0$ has $T=0$.

Solution

In both cases, the wavefunction is evanescent in the forbidden region. For a semi-infinite step, there is no far-side allowed region where the evanescent solution can match onto a propagating transmitted wave, so no current is carried to $x\to+\infty$.

For a finite barrier, the forbidden region ends. The evanescent solution inside the barrier can match at the second boundary onto a propagating wave. That outgoing wave carries transmitted current, so $T\gt 0$.

2. In the opaque-barrier approximation, what happens to $T$ when the barrier width $a$ is doubled?

Solution

The leading dependence is

$$T\propto e^{-2\kappa a}.$$

Doubling $a$ gives

$$T_{\mathrm{new}}\propto e^{-4\kappa a}.$$

Thus the transmission probability is multiplied by an additional factor of $e^{-2\kappa a}$ relative to the original barrier.

3. A symmetric double well has splitting $\Delta E$. Why does a smaller splitting imply slower left-right tunneling?

Solution

The localized-state oscillation has probability

$$P_R(t) =\sin^2\left(\frac{\Delta E\,t}{2\hbar}\right).$$

The first full transfer occurs at

$$t_{\mathrm{transfer}} =\frac{\pi\hbar}{\Delta E}.$$

As $\Delta E$ becomes smaller, this transfer time becomes larger. A very small tunneling splitting means very slow tunneling between localized configurations.