Quantum · Tunneling · Private, in-browser

Quantum Tunneling Probability Calculator

Enter mass, barrier height, width, and particle energy to estimate transmission through a 1D rectangular barrier. Uses the standard T ≈ exp(-2κa) approximation for E < V₀.

Tunneling probability

Particle mass (kg)

Barrier height V₀ (eV)

Particle energy E (eV)

Barrier width (nm)

Displayed digits

Tunneling probability T

—

Decimal form

Tunneling probability

—

Percentage form

Suppression factor

—

Exponent 2κa (unitless)

Barrier region κ

—

κ = √(2m(V₀−E))/ħ (1/m)

Probability bar (capped at 1%)

How this estimate works

Rectangular barrier: For E < V₀, the wave decays inside the barrier with κ = √(2m(V₀−E))/ħ. Transmission is approximated by T ≈ exp(-2κa) where a is the width.
Units: Enter mass in kg, energies in eV, and width in nm. We convert eV → joules and nm → meters automatically.
Above the barrier: If E ≥ V₀, classical transmission is allowed; we report T ≈ 1 (ignoring reflection).
Why it’s tiny: The exponent 2κa grows with mass, barrier height, and width, so probabilities drop exponentially.
Limits: This is the simple 1D, single-barrier textbook approximation—no multiple reflections, no resonance, no 3D effects.

For precise scattering, solve the full Schrödinger matching problem. Here we highlight the scale of exponential suppression for quick insight and coursework.

Quick tunneling facts

Exponent rules everything

Every extra nanometer of width can slash T by orders of magnitude because T ~ exp(-2κa).

Double-edged

Mass matters

Heavier particles tunnel far less. Electrons zip through thin barriers; alphas face huge suppression.

m in κ

Alpha decay is tunneling

An alpha trapped by the nuclear Coulomb barrier sneaks out via tunneling, setting decay lifetimes.

Nuclear

STM relies on it

Scanning tunneling microscopes measure current from electrons tunneling across a tiny vacuum gap.

Instrumentation

Not faster-than-light

Tunneling doesn’t move particles superluminally; group delays can be short without causal signaling.

Causality

Why tunneling is mind-bending (and useful)

Classically, a particle with energy below a barrier stops. Quantum mechanically, its wavefunction decays but does not vanish inside the barrier, leaving a finite chance to emerge on the other side. That probability is the tunneling transmission T. When 2κa is small, T can be sizable; when it is large, T plummets exponentially.

This behavior underpins many phenomena: alpha decay from nuclei, electron transport in tunnel diodes, Josephson junctions in superconducting qubits, and the contrast mechanism of scanning tunneling microscopes. In coursework, it’s the go-to example of how quantum amplitudes differ from classical probabilities.

Use the presets to see both extremes. The electron example shows a modest barrier giving a non-negligible T. The alpha example shows how quickly T collapses when mass and barrier height grow, explaining the long half-lives associated with nuclear tunneling.