Quantum tunnelling calculator
An electron of 1 eV meets a 2 eV barrier 1 nm wide and tunnels with a probability of about 0.014%. Calculate exact rectangular-barrier transmission, the WKB estimate and the decay length for electrons, protons and alpha particles.
Electron; E = 1 eV; V = 2 eV; a = 1 nm; below the barrier (tunnelling)
- Transmission probability T
- 1.419e-4 (0.01419 %)
- Reflection probability R
- 0.9999
- WKB estimate exp(-2 kappa a)
- 3.549e-5
- Thick-barrier form 16 (E/V)(1 - E/V) exp(-2 kappa a)
- 1.419e-4
- Exponent 2 kappa a
- 10.25
- Decay constant kappa
- 5.123 nm⁻¹
- Decay length 1/kappa
- 0.1952 nm
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Example: What is the probability that a 1 eV electron tunnels through a 2 eV barrier 1 nm wide?
T = [1 + V2 sinh2(κa) / (4E(V – E))]-1, κ = √(2m(V – E)) / ℏ. WKB: T ≈ exp(-2κa).
Transmission through a rectangular barrier
Exact transmission from the closed form, with the WKB estimate and decay length. The resonance row sets the width so that k a = pi.
| Particle | E; V; a | T (exact) | T (WKB) | Decay length (nm) |
|---|---|---|---|---|
| Electron | 1 eV; 2 eV; 1 nm | 1.419e-4 | 3.549e-5 | 0.1952 |
| Electron | 1 eV; 2 eV; 0.5 nm | 0.02355 | 0.005957 | 0.1952 |
| Electron | 1 eV; 2 eV; 0.2 nm | 0.4044 | 0.1288 | 0.1952 |
| Electron | 2 eV; 2 eV; 1 nm | 0.0708 | – | – |
| Electron | 3 eV; 2 eV; 1 nm | 0.7811 | – | – |
| Electron | 3 eV; 2 eV; 0.6132 nm | 1 | – | – |
| Proton | 1 eV; 2 eV; 0.1 nm | 3.420e-19 | 8.549e-20 | 0.004555 |
| Alpha particle | 1 eV; 2 eV; 0.05 nm | 3.975e-19 | 9.939e-20 | 0.002285 |
Checks against published values
| Check | Published | Computed | Relative difference |
|---|---|---|---|
| kappa for a 1 eV barrier, electron | 5.123167 nm^-1 | 5.123167 | 4.3e-8 |
| kappa for a 4.5 eV work function, electron | 10.862 nm^-1 | 10.86788 | 5.4e-4 |
| T at E = V against the closed-form limit | 1 | 1 | 0 |
| T at k a = pi, E above V | 1 | 1 | 0 |
| T over the thick-barrier form | 1 ratio | 1 | 8.9e-14 |
| kappa ratio proton / electron | 42.85035 ratio | 42.85035 | 4.9e-8 |
| T for a barrier of zero width | 1 | 1 | 0 |
Cite this
ScienceShot. Quantum tunnelling calculator. https://scienceshot.com/quantum-tunnelling-calculator.
More formats
ScienceShot. (2026). Quantum tunnelling calculator [Online calculator]. https://scienceshot.com/quantum-tunnelling-calculator
@misc{scienceshot-quantum-tunnelling-calculator-2026,
author = {{ScienceShot}},
title = {{Quantum tunnelling calculator}},
year = {2026},
note = {Online calculator},
url = {https://scienceshot.com/quantum-tunnelling-calculator}
}- kappa for a 1 eV barrier, electron: CODATA 2018 hbar, electron mass and elementary charge give hbar^2/(2 m_e) = 3.80998 eV A^2
- kappa for a 4.5 eV work function, electron: C. J. Chen, Introduction to Scanning Tunneling Microscopy: kappa = 0.512 sqrt(phi) per angstrom for a 4.5 eV barrier
- T at E = V against the closed-form limit: Closed-form limit E = V: T = 1 / (1 + m V a^2 / (2 hbar^2)), from the OpenStax rectangular-barrier solution
- T at k a = pi, E above V: Over-barrier resonance T = 1 when k a = n pi (OpenStax 7.6; Griffiths and Schroeter, chapter 2)
- T over the thick-barrier form: Thick-barrier limit T -> 16 (E/V)(1 – E/V) exp(-2 kappa a) (OpenStax 7.6; Griffiths and Schroeter, chapter 2)
- kappa ratio proton / electron: CODATA 2018 proton-electron mass ratio 1836.152 673 43: kappa scales as the square root of the mass
- T for a barrier of zero width: A barrier of zero width transmits everything (limit a -> 0)
Sources: OpenStax University Physics Volume 3, 7.6; CODATA 2018.
Reviewed by a physicist with CERN experience