Quantum Tunneling Calculator
Compute exact quantum tunneling transmission and reflection through a rectangular barrier for electrons, protons and alpha particles. Free, in-browser.
Quantum Tunneling Probability Density Engine
| Quantity | Real | Imag | Magnitude |
|---|---|---|---|
| r | 0.00000000 | -0.97840731 | 0.97840731 |
| A | 0.01079635 | 0.01079635 | 0.01526834 |
| B | 0.98920365 | -0.98920365 | 1.39894522 |
| t | -0.13125228 | -0.15966209 | 0.20668610 |
A clear path from inputs to results.
Quantum particles can pass through a barrier they classically could not climb. This tool computes that tunneling probability for a particle hitting a rectangular potential barrier: enter the particle energy, the barrier height and width, and the particle type, and it returns the exact transmission and reflection probabilities from matching the Schrödinger-equation wavefunction across the barrier.
How to use
- Choose the particle: electron, proton, or alpha particle (this sets the mass used in the Schrödinger equation).
- Enter the incident energy E of the particle in electron-volts.
- Enter the barrier height V0 in electron-volts and the barrier width L in nanometres.
- Read the transmission probability T (the chance the particle tunnels through) and the reflection probability R = 1 − T.
- Compare against the built-in benchmark and the WKB approximation to see how the exact result behaves as you make the barrier taller, wider, or the energy lower.
The formula
Reference: Griffiths, Introduction to Quantum Mechanics, finite rectangular barrier transmission (exact boundary-matching solution); WKB opaque-barrier limit T ≈ 16·E(V0−E)/V0²·e^(−2κL).
Worked example
Electron through a 12 eV, 0.18 nm barrier at 6 eV
- Particle: electron (mass ratio 1); incident energy E = 6 eV; barrier height V0 = 12 eV; barrier width L = 0.18 nm. The engine uses ħ²/2mₑ = 0.038099821161327 eV·nm².
- 1
Since E = 6 eV is below the 12 eV barrier, the wave is evanescent inside it with decay constant κ = √((V0−E)/(ħ²/2mₑ)) = √(6 / 0.0380998) = 12.5491 nm⁻¹. The dimensionless barrier thickness is κL = 12.5491 × 0.18 = 2.25885, giving sinh(κL) = 4.73378. The exact transmission is T = 1 / [1 + V0²·sinh²(κL) / (4E(V0−E))] = 1 / [1 + 144 × 22.4087 / (4 × 6 × 6)] = 1 / [1 + 22.4087] = 0.042719. Here V0²/(4E(V0−E)) = 144/144 = 1 exactly, so this also equals 1/cosh²(κL), an independent check that returns the same 0.042719.
Transmission T = 0.04272 (about a 4.3% chance of tunneling through) and reflection R = 0.95728. The opaque-barrier WKB estimate 4·e^(−2κL) = 0.04366 is close, confirming the barrier is in the moderately opaque regime.
Interpreting the result
T is the probability that a particle in the incident beam emerges on the far side; R = 1 − T is the probability it is reflected, and the two always sum to one because the barrier neither creates nor absorbs particles. Tunneling falls off roughly exponentially with the barrier's opacity κL, so making the barrier a little wider or taller, or lowering the energy, drops T sharply — which is why scanning tunneling microscopes and alpha decay are so sensitive to distance and energy. When E exceeds V0 the particle is classically allowed over the barrier, and T oscillates toward 1 with resonances rather than decaying.
Assumptions & limitations
- The barrier is a one-dimensional rectangular step of constant height V0 and finite width L with flat potential on both sides; real barriers are smooth and this is an idealization.
- The particle is a non-relativistic, spin-free plane wave described by the time-independent Schrödinger equation.
- Energies and the barrier height are measured in electron-volts and the width in nanometres; the mass enters only through ħ²/2m, scaled by the particle's mass ratio to the electron.
- The closed-form transmission applies for E < V0; the engine switches to the exact over-barrier solution for E ≥ V0, where resonant transmission appears.
- The result is a steady-state scattering probability, not a time-resolved tunneling dynamics or a tunneling-time estimate.
Frequently asked questions
What does the transmission probability actually mean?
It is the fraction of incident particles that make it through the barrier. A transmission of 0.04272 means that out of every thousand electrons that hit this 12 eV, 0.18 nm barrier at 6 eV, about 43 tunnel through and the other 957 are reflected. For a single electron it is the probability of being found on the far side.
How does changing the barrier width affect tunneling?
Very strongly. Transmission falls off roughly as e^(−2κL), so widening the barrier even slightly reduces T sharply. Doubling the 0.18 nm width to 0.36 nm in the worked example would cut the transmission by roughly the square of the current factor. This exponential sensitivity to distance is exactly what makes the scanning tunneling microscope able to resolve single atoms.
Does the particle type change the answer?
Yes, through the mass. The decay constant κ = √(2m(V0−E)/ħ²) grows with mass, so heavier particles tunnel far less. A proton (about 1836 times the electron mass) or an alpha particle (about 7294 times) sees a much larger κL for the same barrier and tunnels with a dramatically smaller probability than an electron — one reason electron tunneling dominates in solid-state devices.
Why can a particle cross a barrier taller than its energy?
In quantum mechanics the particle is a wave, and its wavefunction does not stop abruptly at the barrier — it decays exponentially inside the classically forbidden region. If the barrier is thin enough, a small but non-zero amplitude survives to the far side, giving a finite probability of transmission. This has no classical analogue; a classical ball with less energy than the barrier always bounces back.
How does the WKB approximation compare to the exact result?
The WKB opaque-barrier estimate 4·e^(−2κL) gives 0.04366 against the exact 0.04272 in this example — close but not identical. WKB is a semiclassical approximation that becomes accurate only when the barrier is thick and opaque (large κL). For thin or near-transparent barriers it can be off by a large factor, so the tool reports the exact boundary-matching result as authoritative.
Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.