Quantum Tunneling Calculator
Enter the particle mass, barrier height and width, and energy to compute the transmission (tunneling) coefficient. Electron V=1e-18 J, E=5e-19 J, L=1 Å → T≈0.48.
Input Data
Results
At a glance:Quantum tunneling (George Gamow, 1928, used to explain α decay): a particle can cross a barrier higher than its own energy with non-zero probability. Classical mechanics strictly forbids this (100% reflection when E<V), but in quantum mechanics the wave function decays exponentially inside the barrier and remains non-zero on both sides, so the transmission coefficient T>0. Derivation: the Schrödinger equation for a rectangular barrier V(x)=V (0<x<L) with energy E<V: (1) in regions I and III the wave is free, ψ=Ae^(ikx)+Be^(−ikx) and ψ=Fe^(ikx); (2) in region II it is evanescent, ψ=De^(κx)+Ee^(−κx), with κ=√(2m(V−E))/ℏ. Matching ψ and ψ' at the boundaries gives T=|F/A|²=1/(1+V²sinh²(κL)/(4E(V−E))). For κL≫1 the simplified approximation is T≈16E(V−E)/V²·e^(−2κL) (exponential decay). Physical meaning: (1) wave–particle duality makes tunneling possible; (2) the tunneling probability decays exponentially with barrier width — a wide barrier is essentially impenetrable; (3) heavier particles decay faster, so macroscopic objects have negligible tunneling probability. History: Gamow and Gurney–Condon (1928) used tunneling to explain the enormous range of α-decay half-lives (10⁵–10¹⁷ years), validating quantum mechanics in nuclear physics. Applications: (1) α decay and nuclear fusion (proton tunneling inside the Sun); (2) scanning tunneling microscopy (STM, 1986 Nobel Prize); (3) flash-memory write/erase (Fowler–Nordheim tunneling); (4) superconducting Josephson junctions; (5) tunnel diodes.
Formula
Transmission coefficient: T = 1/(1 + V²·sinh²(κL)/(4E(V−E)))
Decay constant: κ = √(2m(V−E))/ℏ
Exponential approximation: T ≈ 16E(V−E)/V² · e^(−2κL) (κL≫1)
Schrödinger equation: −ℏ²/(2m)·d²ψ/dx² + Vψ = Eψ
Reduced Planck constant: ℏ = h/(2π) = 1.0546e-34 J·s
$$T = \frac{1}{1 + \frac{V^2 \sinh^2(\kappa L)}{4E(V-E)}}, \quad \kappa = \frac{\sqrt{2m(V-E)}}{\hbar}, \quad T \approx \frac{16E(V-E)}{V^2} e^{-2\kappa L}$$How to Use
- Enter the particle mass m (kg; electron 9.109e-31).
- Enter the barrier height V (J; 1 eV=1.602e-19 J).
- Enter the particle energy E (J; must satisfy E<V).
- Enter the barrier width L (m; atomic scale 1e-10).
- The calculator returns the transmission coefficient T, the decay constant κ, and the probability.
Case Studies
α decay of heavy nuclei
An α particle inside a heavy nucleus faces a Coulomb barrier far above its energy, yet it tunnels out.
Tunneling explains why half-lives span an enormous range: a slightly thicker or higher barrier drops T by orders of magnitude.
This is the same Gamow theory that first connected quantum mechanics to nuclear decay rates.
Scanning tunneling microscopy (STM)
An STM needle reaches within ~1 nm of a conducting surface; electrons tunnel across the gap.
Because T depends exponentially on the gap, sub-angstrom vertical resolution is achievable.
STM earned the 1986 Nobel Prize and images individual atoms.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.