Quantum Tunneling

Quantum passage through classically forbidden barriers. Theory, formulas, worked examples and interactive graphs.

Conceptual context

In classical mechanics, a particle with $E$ below a $V_0$ cannot cross the barrier. It is strictly forbidden. It turns back at the classical turning point, just like a ball rolling up a hill turns back when its kinetic energy runs out. In quantum mechanics, however, the particle is represented by a wavefunction. Inside the classically forbidden region, that wave does not vanish instantly; it decays. If the $a$ is not too large and the $\Delta U$ is not too high, some probability amplitude remains on the far side of the obstacle.

The $T$ measures that chance. Quantum tunneling is not energy theft, a temporary violation of energy conservation, or a secret path through matter; it is the measurable consequence of evanescent wave behavior. The particle does not gain energy during the process; its energy remains constant, but its wave nature allows it to penetrate regions that classical particles could never access.

🟢 Essential level

Quantum tunneling occurs when a microscopic particle meets a barrier that it should not cross according to classical energy reasoning.

The right physical picture is not a tiny classical ball drilling a hole through a hard wall, but a wave that partly enters a region where ordinary propagation is forbidden. A larger $a$ gives the evanescent amplitude more distance over which to fade, so the $T$ becomes exponentially smaller. A larger $V_0$ compared with $E$ creates a larger $\Delta U$, which makes the wave decay more rapidly inside the barrier, meaning the wave amplitude decreases to a fraction of its initial value in a shorter distance.

The $m$ also matters because heavier particles display less visible wave behavior; a macroscopic object has such a short wavelength that its tunneling probability is effectively zero. The effect may be tiny, yet at microscopic scales it can control measurable electric currents, radioactive nuclear decay, and complex molecular transitions. It is like dampness seeping through a wall: a thick, dry wall leaves almost no stain on the far side, but a thin, slightly porous wall can let a noticeable dampness through without anyone drilling a hole.

🔵 Formal level

Basic formulas

The rectangular-barrier model treats the obstacle as a region with fixed $V_0$ and well-defined $a$. The first step in any formal analysis is to compare the $E$ of the incoming particle with the barrier height and define the $\Delta U$, the energetic gap that prevents classical crossing.

Barrier excess over energy

When $E$ is below $V_0$, the wavefunction inside the barrier becomes evanescent. Its spatial decay rate is described by the $\kappa$, which depends on $m$, $\Delta U$, and $\hbar$.

Barrier decay coefficient

Real student questions

Does the particle pass through like a ghost? No. The wavefunction penetrates the forbidden region and may leave amplitude on the far side.

Is energy conserved? Yes. $E$ does not become extra energy for climbing. Tunneling is about amplitude, not an energy violation.

Why is $a$ so important? Because the evanescent amplitude decays while crossing the barrier.

Can an infinite $V_0$ be tunneled through? In the ideal infinite barrier, no. The wavefunction is forced to vanish at the wall.

Final synthesis

Quantum tunneling shows that a barrier is not judged only by classical height. The $T$ depends sharply on $a$, $V_0$, $E$, $m$, and the quantum scale set by $\hbar$.