Quantum Tunneling Probability Calculator
Estimate the transmission probability of a quantum particle tunneling through a rectangular potential barrier. Useful for understanding tunnel diodes, nuclear fusion rates, and scanning tunneling microscopy.
About this calculator
Quantum tunneling allows a particle to pass through a potential energy barrier that it classically could not surmount. For a rectangular barrier, the WKB (Wentzel–Kramers–Brillouin) approximation gives the transmission probability as T = exp(−2 × d × √(2 m (V − E)) / ħ), where d is the barrier width, m is the particle mass (here the electron mass, 9.109 × 10⁻³¹ kg), V is the barrier height in joules, E is the particle's energy in joules, and ħ ≈ 1.055 × 10⁻³⁴ J·s is the reduced Planck constant. The result T ranges from 0 (no transmission) to 1 (full transmission). Tunneling probability decreases exponentially with barrier width and with the square root of the energy deficit (V − E). This exponential sensitivity makes tunneling extremely width-dependent — even an angstrom of extra barrier width can reduce T by orders of magnitude.
How to use
Consider an electron (m = 9.109 × 10⁻³¹ kg) facing a barrier of height V = 5 × 10⁻¹⁹ J, width d = 1 × 10⁻⁹ m, with particle energy E = 3 × 10⁻¹⁹ J. Enter barrier_height = 5 × 10⁻¹⁹, barrier_width = 1 × 10⁻⁹, particle_energy = 3 × 10⁻¹⁹. Compute: V − E = 2 × 10⁻¹⁹ J. √(2 × 9.109 × 10⁻³¹ × 2 × 10⁻¹⁹) = √(3.644 × 10⁻⁴⁹) ≈ 6.036 × 10⁻²⁵. Exponent = −2 × 1 × 10⁻⁹ × 6.036 × 10⁻²⁵ / 1.055 × 10⁻³⁴ ≈ −11.45. T = e^(−11.45) ≈ 1.06 × 10⁻⁵, meaning roughly a 1-in-100,000 chance per attempt.
Frequently asked questions
What factors most strongly affect quantum tunneling probability?
Tunneling probability is most sensitive to the barrier width d, since T depends on exp(−2d × √(2m(V−E)) / ħ) — doubling the width squares the exponential decay factor. The energy deficit (V − E) also matters strongly: as the particle's energy approaches the barrier height, the argument of the square root approaches zero and T approaches 1. The particle mass plays a key role too — heavier particles (protons, alpha particles) have exponentially lower tunneling probabilities than electrons at the same barrier, which is why electron tunneling dominates in semiconductors and STM, while nuclear tunneling requires extremely thin barriers.
How does quantum tunneling enable scanning tunneling microscopy (STM)?
In an STM, a sharp metal tip is brought within about 0.5–1 nm of a conducting surface without touching it. A small voltage applied between tip and surface allows electrons to tunnel across the vacuum gap. Because tunneling probability is exponentially sensitive to gap width, even a 0.1 nm change in tip–surface distance changes the tunneling current by roughly an order of magnitude. By scanning the tip and recording current variations, the STM maps the surface's electron density with sub-angstrom resolution — enough to image individual atoms. This technique earned Binnig and Rohrer the 1986 Nobel Prize in Physics.
Why is quantum tunneling important in nuclear fusion reactions inside stars?
At the temperatures found in stellar cores (~10⁷ K), the thermal kinetic energies of protons are far below the Coulomb potential barrier they must overcome to fuse. Classically, fusion should not occur at these temperatures. However, quantum tunneling allows protons to penetrate the barrier with a small but non-zero probability. Integrated over the enormous number of collisions occurring every second in a stellar core, this produces the sustained fusion rate that powers stars. Without quantum tunneling, the Sun would be roughly 1000 times cooler at its core before fusion could proceed classically, and life on Earth would not exist.