What is quantum tunneling?

Quantum tunneling is a quantum mechanical effect where a particle passes through a potential barrier even though its energy is lower than the barrier height. Classical physics forbids this, but the wave nature of particles gives them a nonzero probability of appearing on the other side.

What is the WKB approximation?

The Wentzel–Kramers–Brillouin (WKB) approximation is a semiclassical method that gives a simple closed-form estimate for tunneling: T ≈ exp(−2κL), where κ = √(2m(V₀−E))/ħ. It is accurate when the barrier is smooth and wide compared to the de Broglie wavelength.

Why does T drop so fast with barrier width?

Because T depends exponentially on L. Doubling L squares T. Even a 1 nm increase in barrier width can reduce T by many orders of magnitude, which is why STM tips must be held nanometers from the surface.

What happens when particle energy E ≥ V₀?

When E ≥ V₀ the barrier becomes classically accessible and the WKB formula no longer applies in the same way — there is still quantum reflection, but for a simple estimate the transmission is taken as 1 (full transmission).

What units does this calculator use?

Mass is in kilograms (kg), energies in electron-volts (eV, converted internally to joules), and barrier width in nanometers (nm, converted to meters). ħ = 1.054571817 × 10⁻³⁴ J·s is used throughout.

Quantum Tunneling Probability Calculator

Enter particle mass, barrier height, barrier width, and particle energy to compute the WKB tunneling transmission probability.

Particle & Barrier Parameters

Particle preset

Particle mass (kg)

Barrier height V₀ (eV)

Particle energy E (eV)

Barrier width L (nm)

Enter parameters and click Calculate.

Transmission Probability T

Decay constant κ

m⁻¹

Exponent −2κL

dimensionless

Energy deficiency

V₀ − E (eV)

log₁₀(T)

orders of magnitude

Width sensitivity (same mass & energies)

Barrier width L	Transmission T

Summary

Quantum tunneling is one of the most counterintuitive results of quantum mechanics: particles can pass through barriers they classically cannot surmount. This calculator applies the WKB approximation T ≈ exp(−2κL) for a rectangular barrier, where κ = √(2m(V₀−E))/ħ is the decay constant inside the barrier. The exponential dependence on width L makes tunneling probability extremely sensitive to barrier geometry — the basis of scanning tunneling microscopy and tunnel junction devices.

How it works

Enter the particle mass in kg (or select a preset: electron, proton, neutron).
Enter the barrier height V₀ in eV and the particle kinetic energy E in eV.
Enter the barrier width L in nanometers.
For E < V₀ the WKB formula T ≈ exp(−2κL) is used, where κ = √(2m(V₀−E))/ħ.
For E ≥ V₀ the particle classically passes over the barrier; T = 1 is shown.
The decay constant κ and the exponent −2κL are also displayed for inspection.

Use cases

Estimate tunneling rates in scanning tunneling microscopy (STM) gap calculations.
Understand alpha-particle emission rates in nuclear physics.
Explore how barrier width affects transmission probability in tunnel diodes.
Compare tunneling probability for an electron versus a proton at the same energy.
Verify quantum mechanics homework on the WKB approximation.
Demonstrate exponential sensitivity: a 1 nm wider barrier changes T by orders of magnitude.

Frequently Asked Questions

Last updated: 2026-06-15 · Reviewed by Nham Vu