Quantum Tunneling Probability Calculator
Quantum tunneling allows a particle to pass through a classically forbidden energy barrier. For a rectangular barrier of height V0 and width L, with a particle of mass m and kinetic energy E (where E is less than V0), the transmission probability in the WKB approximation is T ~ exp(-2 kappa L), where kappa = sqrt(2m(V0 - E)) / hbar and hbar = 1.054571817 x 10^-34 J s. This simple exponential form applies when kappa L is large (strong tunneling suppression). For electrons tunneling through thin insulating barriers in tunnel diodes, this formula gives the exponential sensitivity of tunnel current to barrier width that makes STM possible. Enter the mass, particle energy, barrier height, and barrier width.
Quantum tunneling formula (WKB approximation)
kappa = sqrt(2m(V0 - E)) / hbar
T ~ exp(-2 * kappa * L)
hbar = 1.054571817 x 10^-34 J s. E and V0 in joules (1 eV = 1.602 x 10^-19 J). L in meters. This approximation assumes kappa L much greater than 1. The exact rectangular barrier transmission includes prefactors, but the exponential term dominates for thick barriers.
Tunneling examples
- Electron (E = 1 eV) through 5 eV barrier, L = 1 nm: kappa = 10.24 nm^-1, T ~ exp(-20.48) ~ 1.3 x 10^-9.
- Reducing L to 0.5 nm: T ~ exp(-10.24) ~ 3.6 x 10^-5, about 36,000 times larger.
- STM tip at 0.5 nm above surface: small tip displacement changes T by factor ~10, giving atomic resolution.
- Alpha decay: alpha particle tunnels through Coulomb barrier; tunneling probability determines decay rate and half-life.
Quantum tunneling: frequently asked questions
What is quantum tunneling?
Quantum tunneling is the quantum mechanical phenomenon where a particle passes through a potential barrier that it would be classically forbidden to cross (because its energy is less than the barrier height). This arises from the wave nature of quantum particles: the wavefunction has an exponentially decaying tail inside the barrier, giving a nonzero probability of finding the particle on the other side.
What is the WKB approximation for tunneling probability?
For a rectangular barrier of height V0 and width L, with particle mass m and energy E (where E is less than V0), the transmission probability is approximately T ~ exp(-2 kappa L), where kappa = sqrt(2m(V0 - E)) / hbar. This is the simple form of the WKB (Wentzel-Kramers-Brillouin) approximation.
What are practical applications of quantum tunneling?
Tunneling enables: alpha decay in nuclear physics (alpha particles tunnel through the Coulomb barrier); scanning tunneling microscopy (STM) imaging at atomic resolution; tunnel diodes and Josephson junctions in electronics; and fusion reactions in stellar cores (protons tunnel despite insufficient thermal energy to overcome the Coulomb barrier classically).
Why does tunneling probability decrease so rapidly with barrier width?
The exponential factor exp(-2 kappa L) means that doubling the barrier width squares the probability (in the exponent sense). For an electron tunneling through a 1 eV barrier, increasing the barrier width by just 1 nm reduces probability by a factor of exp(-2 * 5.12 nm^-1) = exp(-10.24), roughly a factor of 37,000.
Does tunneling violate energy conservation?
No. Energy is conserved. The particle emerges on the other side of the barrier with the same energy it had before tunneling. It does not gain energy during tunneling; the barrier merely provides a region where the probability amplitude decays rather than oscillates. The uncertainty principle allows temporary violation of the classical energy constraint within the barrier.
Official sources
- NIST CODATA 2018: Reduced Planck Constant.
- OpenStax University Physics Vol. 3: Quantum Tunneling of Particles through Potential Barriers.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.