Roll a ball toward a hill that is taller than it can climb. It rolls partway up, stops, and rolls back — every single time. In the everyday world a barrier you don't have the energy to climb is simply a wall. Quantum tunneling is the strange fact that a tiny particle, like an electron, can sometimes turn up on the far side of such a wall — without ever having enough energy to go over the top. It doesn't smash through; it's just that its odds of being found on the other side are small but never exactly zero.
Three things set those odds: the particle's energy $E$, how tall the barrier is ($V_0$), and how wide it is ($L$). The bigger the shortfall $V_0 - E$ and the wider the barrier, the harder it is to get across. Quantum mechanics packs the first two into a single "stubbornness" number, the decay constant $\kappa$ (kappa): the wave that describes the particle doesn't stop dead inside the barrier, it fades away smoothly, and $\kappa$ tells you how fast it fades.
$T$ is the transmission probability — the fraction of tries that get through. Here is a worked number: an electron facing a barrier $1\ \text{eV}$ above its energy gives $\kappa \approx 5.1\ \text{nm}^{-1}$. For a barrier $L = 0.5\ \text{nm}$ wide, $2\kappa L \approx 5.1$, so $T \approx e^{-5.1} \approx 0.6\%$. Now double the width to $1\ \text{nm}$: $2\kappa L \approx 10.2$ and $T \approx e^{-10.2} \approx 0.004\%$ — about $160$ times smaller. Tunneling is exponential in width, which is the whole game.
The precise version
Solving the Schrödinger equation gives the exact decay constant $\kappa = \sqrt{2m(V_0-E)}/\hbar$ and the full transmission coefficient $T = \big[\,1 + \tfrac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\,\big]^{-1}$, which collapses to the $e^{-2\kappa L}$ rule for a thick barrier. The mass $m$ sits right inside $\kappa$: heavier particles fade faster, so electrons tunnel easily while protons almost never do. In the sim, the $E$, $V_0$, $L$ and $m$ sliders move exactly these quantities, and the readout shows $\kappa$, $\kappa L$ and $T$ live.
Try this in the sim above: (1) Slide $L$ up and watch $T$ in the readout plunge while the $|\psi|^2$ curve on the far side shrinks — exponential decay you can see. (2) Push $E$ up toward $V_0$: $\kappa$ shrinks toward zero and the barrier becomes almost transparent. (3) Crank the mass $m$ up to $10\,m_e$ and watch tunneling all but vanish — the reason a heavy object never tunnels through a real wall.
Section 03
Equations & Derivation
Quantum tunneling is a purely quantum-mechanical phenomenon in which a particle penetrates a potential barrier even when its kinetic energy $E$ is less than the barrier height $V_0$. Classically forbidden, it follows directly from solving the time-independent Schrödinger equation across the three regions of a rectangular barrier.
For a rectangular barrier $V(x) = V_0$ for $0 < x < L$ and zero elsewhere, write:
$$\psi_I = A e^{ikx} + B e^{-ikx}, \quad \psi_{II} = C e^{\kappa x} + D e^{-\kappa x}, \quad \psi_{III} = F e^{ikx}$$
where $k = \sqrt{2mE}/\hbar$ and $\kappa = \sqrt{2m(V_0-E)}/\hbar$.
Step 2 — Match boundary conditions
Continuity of $\psi$ and $d\psi/dx$ at $x=0$ and $x=L$ gives four equations. Solving for $F/A$:
This explains why tunneling is extraordinarily sensitive to barrier width — a few picometers change in an STM tip-sample gap changes the current by orders of magnitude.
Mapping to the simulation
The slider $E$ sets the incoming energy; $V_0$ and $L$ define the barrier; $m$ scales $\kappa$. The "Probability" view shows $|\psi|^2$ — a standing-wave pattern in region I (interference), exponential decay in II, and a plane wave in III with reduced amplitude $|F/A|^2 = T$.
The animation plots the steady-state wavefunction $\psi(x)$ (real part) and probability density $|\psi(x)|^2$ across a potential barrier. The incoming wave from the left partially reflects (interference fringes) and partially transmits through the barrier. Inside the barrier, $\psi$ decays exponentially — but it is non-zero on the far side, which means the particle has a finite probability of being found there.
Stellar nuclear fusion (protons tunnel through their Coulomb repulsion in the Sun's core), alpha decay (Gamow's 1928 model), the scanning tunneling microscope (STM), tunnel diodes, flash memory (Fowler-Nordheim tunneling), and — speculatively — enzyme catalysis and some olfactory mechanisms. Without tunneling, the Sun would not shine at its measured rate.
No. Energy is conserved overall: the particle has the same kinetic energy on both sides. Inside the barrier the wavefunction is evanescent (exponential decay), not a propagating wave with definite kinetic energy. There is no measurable moment when $KE < 0$ — the question 'what is its energy inside the barrier?' is itself ill-posed in standard quantum mechanics.
Inside the barrier $\psi \propto e^{-\kappa x}$, so $|\psi|^2 \propto e^{-2\kappa L}$. Doubling $L$ does not halve the transmission — it squares the (already small) suppression factor. This is why an STM achieves atomic resolution: a 0.1 nm change in tip height changes the tunneling current by roughly an order of magnitude.
This is the famous tunneling-time problem, debated for nearly a century. Different operational definitions (phase time, Larmor time, dwell time, attoclock) give different answers. Recent attoclock experiments suggest tunneling is essentially instantaneous, but the question depends on what you mean by 'time spent' for an evanescent wave.
No — although phase-velocity arguments and certain experiments (Nimtz) appear to show superluminal traversal, no information or energy travels faster than $c$. The leading edge of any signal is unchanged; only the peak appears to advance, which is a pulse-reshaping artifact, not actual superluminal transport.
Because $\kappa = \sqrt{2m(V_0-E)}/\hbar$ — heavier particles have shorter quantum wavelengths and steeper exponential decay. This is why electrons tunnel readily but protons rarely do, and why Earth-mass objects effectively never tunnel through walls.
Resources: Khan Academy; HyperPhysics; MIT OpenCourseWare; Paul's Physics Notes.
Section 05
Common Misconceptions
❌ Misconception: The particle 'borrows' energy from the vacuum to overcome the barrier.
✅ Correction: This is a popular but incorrect picture. The Heisenberg energy-time uncertainty $\Delta E\,\Delta t \gtrsim \hbar/2$ is not a license for energy non-conservation. The wavefunction simply has support in classically forbidden regions; total energy remains $E$ throughout.
📖 Reference: Griffiths — Introduction to Quantum Mechanics, 3rd Ed., §2.5 'The Delta-Function Potential' & §2.6 'The Finite Square Well'.
❌ Misconception: Tunneling is a relativistic effect.
✅ Correction: Tunneling is fundamentally non-relativistic — it follows directly from the Schrödinger equation. The first quantitative theory (Gamow, 1928) used non-relativistic QM. Relativistic corrections only matter for very high-energy or very small barriers.
📖 Reference: Merzbacher — Quantum Mechanics, 3rd Ed., §6.5 'Penetration of a Potential Barrier'.
❌ Misconception: If $E > V_0$, transmission is 100%.
✅ Correction: Even when $E > V_0$ a particle can be reflected — a purely quantum effect with no classical analog. Transmission becomes 100% only at resonance energies where $kL = n\pi$, similar to thin-film optical anti-reflection.
📖 Reference: Cohen-Tannoudji — Quantum Mechanics, Vol. I, Complement HI 'Tunnel Effect: The Square Barrier'.
❌ Misconception: The wavefunction inside the barrier is zero.
✅ Correction: It is exponentially decaying, not zero. The solution $\psi \propto e^{-\kappa x}$ is real and non-zero throughout the barrier; only the probability current of the evanescent wave is zero (no net flow), but $|\psi|^2$ is finite.
📖 Reference: Sakurai — Modern Quantum Mechanics, 2nd Ed., §2.5 'Elementary Solutions to Schrödinger's Wave Equation'.
❌ Misconception: Tunneling occurs because particles are 'really' waves and waves go everywhere.
✅ Correction: Wave-particle duality is a starting point, but tunneling is a specific quantitative prediction. Classical waves (light, sound) also tunnel — frustrated total internal reflection is the optical analog. The phenomenon is more general than 'particles' vs 'waves'.
📖 Reference: Ohanian — Principles of Quantum Mechanics, §4.5; also see Hecht — Optics §4.7 on evanescent waves.
❌ Misconception: Higher temperature increases tunneling probability.
✅ Correction: Tunneling itself is independent of temperature — it depends only on $E$, $V_0$, $L$, $m$. What temperature changes is the population of energetic states attempting to tunnel; this is thermally activated tunneling, not enhanced tunneling per se.