Quantum Tunneling Probability Calculator
Compute the WKB transmission probability T ≈ exp(−2·w·κ) for a particle tunnelling through a rectangular potential barrier, given barrier height, width, and particle energy. Returns a dimensionless probability between 0 and 1.
Last updated: May 2026
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About this calculator
For a particle with energy E less than barrier height V₀, the WKB (Wentzel-Kramers-Brillouin) approximation gives the transmission probability T ≈ exp(−2·w·κ), where w is the barrier width and κ = √(2m(V₀ − E))/ℏ is the wavevector inside the classically forbidden region. The calculator hardcodes m = 9.109 × 10⁻³¹ kg (electron mass) and ℏ = 1.055 × 10⁻³⁴ J·s. Inputs: barrier_height (V₀) and particle_energy (E) in joules, barrier_width (w) in metres. T is strongly suppressed exponentially with both w and √(V₀ − E), so thicker or higher barriers exponentially reduce tunnelling probability. Edge cases: if E ≥ V₀, the formula gives imaginary κ (NaN) — the particle has enough energy to cross classically and tunnelling is replaced by classical transmission near 1 (the WKB formula doesn't apply). At very small w or V₀ − E, T approaches 1 (high tunnelling). The formula is the leading-order WKB result, accurate for moderately opaque barriers but missing pre-exponential factors (~order 1) that more careful matched-asymptotic treatments include. It also assumes a rectangular barrier; for smoothly varying potentials, the WKB exponent integrates κ(x) over the classically forbidden region: T ≈ exp(−2·∫κ(x)dx). Tunnelling has many manifestations: α-decay of radioactive nuclei, scanning tunnelling microscopy (STM), Esaki diodes, Josephson junctions, biochemical proton tunnelling in enzymes, and quantum simulations.
How to use
Example 1 — STM-style barrier. barrier_height = 2 × 10⁻¹⁹ J (~1.25 eV, typical work function), particle_energy = 1 × 10⁻¹⁹ J, barrier_width = 1 × 10⁻⁹ m (1 nm). Step 1: V₀ − E = 2e−19 − 1e−19 = 1e−19 J. Step 2: 2m(V₀ − E) = 2 × 9.109e−31 × 1e−19 = 1.822e−49. Step 3: √(...) = √1.822e−49 ≈ 4.27e−25 kg·m/s. Step 4: κ = 4.27e−25 / 1.055e−34 ≈ 4.046e9 1/m. Step 5: 2·w·κ = 2 × 1e−9 × 4.046e9 = 8.09. Step 6: T = exp(−8.09) ≈ 3.06e−4 (about 0.03%). Verify: STM tip-sample distances of 1 nm with ~1 eV barriers typically give T in the 10⁻⁴–10⁻³ range, consistent ✓. STMs work because even tiny T produces measurable current at high atom densities. Example 2 — much thicker barrier. Same V₀ and E, but w = 5 × 10⁻⁹ m (5 nm). Step 1–4: same κ = 4.046e9 1/m. Step 5: 2·w·κ = 2 × 5e−9 × 4.046e9 = 40.46. Step 6: T = exp(−40.46) ≈ 2.66e−18 (effectively zero). Verify: increasing barrier width by 5× drops T by ~10¹⁴ — the exponential dependence on thickness is the reason STM uses sub-nanometre tip-sample distances and the reason α-decay half-lives vary by 24+ orders of magnitude across different isotopes despite barriers differing only by factors of a few ✓.
Frequently asked questions
Why is tunnelling exponentially suppressed with thickness?
Inside the classically forbidden region (E < V₀), the wavefunction's spatial dependence is exp(−κx) rather than oscillating like exp(ikx), where κ = √(2m(V₀−E))/ℏ is the decay constant. The wavefunction's amplitude drops by a factor of exp(−κw) across a barrier of width w, and probability (|ψ|²) drops by exp(−2κw). This exponential decay is fundamental: in just a few decay lengths the wavefunction becomes negligible. For an electron with V₀ − E = 1 eV, κ ≈ 5 × 10⁹ /m, so the decay length 1/κ is about 0.2 nm — a barrier of 1 nm reduces probability by exp(−10) ≈ 5 × 10⁻⁵, of 2 nm by exp(−20) ≈ 2 × 10⁻⁹. Doubling the barrier doesn't just cut probability in half; it squares the (already tiny) probability. This is why STM is exquisitely sensitive to surface topography — sub-Å changes in tip-sample distance change current by orders of magnitude.
How does tunnelling differ from classical 'barely getting over' a barrier?
Classical physics requires E > V₀ to cross a barrier; below that, the particle is reflected with 100% probability. Quantum mechanics allows nonzero transmission even when E < V₀, the hallmark of tunnelling. Equally, when E > V₀, classical physics predicts 100% transmission while quantum mechanics predicts some reflection — a particle scattering off a 'cliff' even when it has enough energy. These wave-like reflections at boundaries are real and observable in optics (light from glass-air interface reflects partially despite the photon having far more than enough energy) and in quantum mechanics (resonant transmission, antireflection coatings designed to suppress these reflections). Below the barrier (E < V₀) the wavefunction is evanescent (exponentially decaying); above the barrier (E > V₀) it's oscillatory inside the barrier too but with reduced wavelength. Both regimes have rich behaviour including resonance phenomena (perfect transmission at certain energies even with barriers), bound states in wells, and quasi-bound (metastable) states with finite lifetimes.
Where in nature is quantum tunnelling crucial?
Tunnelling underlies many fundamental and applied phenomena. Alpha decay: α particles tunnel out of the strong-force binding potential of heavy nuclei despite having less energy than the barrier height, with the WKB formula explaining the enormous range of half-lives (uranium-238: 4.5 billion years; polonium-212: 0.3 µs — both alphas tunnel through similar-shaped barriers, but small differences in barrier dimensions give huge differences in T). Nuclear fusion in stars: protons must tunnel through their mutual Coulomb repulsion to fuse — the Sun couldn't burn at observed luminosity without tunnelling. STMs: image surfaces at atomic resolution by measuring tunnelling current. Flash memory: stores data by tunnelling electrons into and out of floating gates. Tunnel diodes (Esaki): exploit negative differential resistance from sharply doped semiconductor junctions. Josephson junctions: superconducting Cooper-pair tunnelling, underpinning SQUID magnetometers and superconducting qubits. Enzyme catalysis: many enzymes accelerate proton-transfer reactions partly via quantum tunnelling at room temperature.
What are the common mistakes when computing tunnelling probability?
The biggest mistake is using non-SI units — barrier height and particle energy in eV without converting to joules (1 eV = 1.602e−19 J), or width in nm without converting to m. The second is applying the formula to E > V₀, where κ becomes imaginary and the formula returns NaN; in that regime use the above-barrier transmission formula. The third is forgetting that the WKB approximation is a leading-order result that omits pre-exponential factors of order 1; for the exact rectangular-barrier T, use the full quantum scattering solution. People also use electron mass m_e for non-electron tunnelling problems (proton tunnelling in enzymes uses m_p ≈ 1.673e−27 kg, ~1836× the electron mass, making tunnelling much harder at the same barrier dimensions). For varying-height barriers (not rectangular), the WKB exponent is the integral ∫κ(x)dx across the forbidden region, not simply κ·w with constant κ. Finally, treating T > 1 (which can occur from numerical or input errors) as physical: T must lie in [0, 1]; T > 1 indicates the formula has been pushed outside its domain or there are numerical issues.
When should I not use this calculator?
Do not use it when the particle energy E exceeds the barrier height V₀ — the formula gives NaN (imaginary κ); for above-barrier transmission use the appropriate quantum-scattering formula or 1 minus reflection probability. It is not appropriate for non-rectangular barriers; integrate κ(x) over the forbidden region instead. Do not use it for resonant tunnelling structures (double barriers, quantum wells in series), where coherent transmission can be much higher than the single-barrier WKB prediction. It is unsuitable for very thin or low barriers where pre-exponential factors of WKB become important and the simple exp(−2κw) understates T by significant factors. The calculator hardcodes electron mass; for proton or other-particle tunnelling, replace m manually. For relativistic particles (high-energy γ-rays, fast electrons in heavy materials), the non-relativistic WKB formula breaks down — use Dirac-equation tunnelling. For multi-particle correlated tunnelling (Cooper pairs in Josephson junctions, alpha particles in heavy nuclei), the simple formula needs significant modification. Finally, the formula assumes time-independent barrier; for AC bias or photo-assisted tunnelling, time-dependent perturbation theory is required.