Tunneling Probability Calculator

Quantum tunneling is a phenomenon where a particle passes through a potential energy barrier that it classically should not be able to overcome. This occurs because particles in quantum mechanics behave like waves, and their wave function has a non-zero amplitude on the other side of the barrier, giving them a finite probability of being found there.

The calculator uses the WKB (Wentzel–Kramers–Brillouin) approximation for a rectangular potential barrier: T ≈ e^(−2κa), where κ = √(2m(V₀ − E)) / ℏ is the decay constant, a is the barrier width, m is the particle mass, V₀ is the barrier height, E is the particle energy, and ℏ is the reduced Planck constant (≈ 1.0546×10⁻³⁴ J·s).

If the particle's energy E is greater than or equal to the barrier height V₀, the particle is not in the tunneling regime — it has enough energy to pass over the barrier classically. In this case the transmission coefficient from this formula equals 1 (or is treated as certain passage), and the tunneling approximation no longer applies.

You can enter mass in kilograms (kg) or electron-volts per c² (eV/c²). For an electron, the mass is approximately 9.10938×10⁻³¹ kg or 511,000 eV/c² (0.511 MeV/c²). If you use eV/c², the calculator automatically converts to kg using the relation 1 eV/c² = 1.783×10⁻³⁶ kg.

The tunneling probability depends exponentially on the barrier width (T ∝ e^(−2κa)). This means even a small increase in width causes a dramatic decrease in tunneling probability. For typical atomic-scale barriers in nanometers, a change of just 0.1 nm can reduce probability by orders of magnitude.

Quantum tunneling underlies many modern technologies: scanning tunneling microscopes (STM) exploit electron tunneling to image surfaces at atomic resolution, tunnel diodes and flash memory rely on tunneling for their operation, nuclear fusion in stars is enabled by proton tunneling through Coulomb barriers, and enzyme catalysis in biology is partially explained by tunneling of hydrogen atoms.

The decay constant κ (kappa) describes how rapidly the quantum wave function decays inside the barrier. A larger κ means the wave function attenuates more quickly, resulting in a lower tunneling probability. It depends on both the particle mass and the energy deficit (V₀ − E): κ = √(2m(V₀ − E)) / ℏ.

This calculator applies the WKB approximation for a simple rectangular (square) potential barrier, which is a standard and widely used model in quantum mechanics. It gives excellent results for electrons tunneling through thin insulating layers or vacuum gaps. For more complex barrier shapes, relativistic particles, or multi-dimensional problems, more advanced quantum mechanical treatments would be needed.