Quantum Tunneling Probability Calculator
Run Quantum Tunneling Probability calculations instantly — enter your data set to get summary statistics, probability values, and interpretation guidance.
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Where T is the transmission (tunneling) probability, V0 is the barrier height, E is the particle energy, kappa = sqrt(2*m*(V0-E))/hbar is the decay constant, a is the barrier width, and m is the particle mass. The WKB approximation gives T = exp(-2*kappa*a) for thick barriers.
Last reviewed: December 2025
Worked Examples
Example 1: Electron Through a Thin Barrier
Example 2: Alpha Particle Tunneling in Nuclear Decay
Background & Theory
The Quantum Tunneling Probability Calculator applies the following established principles and formulas. Probability theory provides the mathematical foundation for analysing all games of chance. The fundamental measure assigns a probability between 0 and 1 to each outcome by dividing the count of favourable outcomes by the count of equally likely total outcomes. Rolling a standard six-sided die produces a 1/6 probability for each face; the probability that a fair coin lands heads exactly three times in five tosses follows the binomial distribution with parameters n=5 and p=0.5. Expected value (EV) is the probability-weighted average outcome of a random variable: EV equals the sum of each outcome multiplied by its probability. A fair coin flip paying $1 for heads and costing $1 for tails has EV of zero. Casino games are designed with negative expected value for the player; the house edge is the casino's average percentage profit per bet. European roulette with a single zero has a house edge of 2.7 percent, while American roulette's double zero raises it to 5.26 percent. Poker hand probabilities derive from combinatorics. From a 52-card deck, the number of distinct 5-card hands is C(52,5) = 2,598,960. A royal flush can occur in only 4 ways, giving it a probability of approximately 0.000154 percent. Blackjack basic strategy tables, derived from computer simulation of millions of hands, reduce the house edge from roughly 2 percent to below 0.5 percent by specifying the optimal hit, stand, double, or split decision for every player hand against every dealer up-card. Sports betting implied probability converts decimal odds to a probability estimate: implied probability equals 1 divided by decimal odds. Odds of 2.5 imply a 40 percent probability. The Kelly Criterion provides the theoretically optimal bet fraction: f equals (bp minus q) divided by b, where b is the net odds received, p is the probability of winning, and q is the probability of losing. This formula maximises the long-run geometric growth rate of a bankroll.
History
The history behind the Quantum Tunneling Probability Calculator traces back through the following developments. Physical evidence of dice play dates to around 2500 BCE at the Indus Valley city of Mohenjo-daro, where excavators found carved cubic astragali remarkably similar to modern dice. Ancient Egyptian, Greek, and Roman cultures all incorporated dice games into both leisure and religious ritual, suggesting gambling emerged independently across early civilisations as a universal human impulse. The first systematic attempt to mathematically analyse games of chance came from Gerolamo Cardano, the Italian polymath who wrote "Liber de Ludo Aleae" (Book on Games of Chance) around 1564. Cardano derived correct probabilities for dice combinations and introduced the concept of sample space, though his work remained unpublished until 1663. The field transformed into a rigorous discipline through correspondence in 1654 between Blaise Pascal and Pierre de Fermat prompted by a gambling problem posed by the Chevalier de Mere. Their exchange established the rules of probability, including the concept of expected value. Jacob Bernoulli's "Ars Conjectandi" (1713) formalised the law of large numbers, proving that sample frequencies converge to true probabilities as trials increase. The 20th century brought two pivotal developments. Stanislaw Ulam and John von Neumann devised Monte Carlo simulation methods in 1947 while working at Los Alamos, showing that complex probabilistic systems could be analysed by random sampling. Game theory and poker strategy developed in parallel, with John von Neumann's minimax theorem providing early foundations and later work by game theorists formalisingrational play under incomplete information. Online gambling launched in the mid-1990s following the passage of the Free Trade and Processing Act in Antigua in 1994, which issued the first online casino licences. The Unlawful Internet Gambling Enforcement Act of 2006 disrupted US online gambling markets. Esports betting and video game loot box mechanics brought probability and expected value calculations to younger audiences in the 2010s, prompting regulatory scrutiny of randomised virtual reward systems across multiple jurisdictions.
Frequently Asked Questions
Formula
T = 1 / (1 + V0^2 * sinh^2(kappa*a) / (4*E*(V0-E)))
Where T is the transmission (tunneling) probability, V0 is the barrier height, E is the particle energy, kappa = sqrt(2*m*(V0-E))/hbar is the decay constant, a is the barrier width, and m is the particle mass. The WKB approximation gives T = exp(-2*kappa*a) for thick barriers.
Worked Examples
Example 1: Electron Through a Thin Barrier
Problem: Calculate the tunneling probability for an electron (mass = 9.109 * 10^-31 kg) with energy 3 eV through a barrier of height 5 eV and width 1 nm.
Solution: V0 - E = 5 - 3 = 2 eV = 3.204 * 10^-19 J\nkappa = sqrt(2 * 9.109e-31 * 3.204e-19) / 1.055e-34 = 7.245 * 10^9 m^-1\nkappa * a = 7.245e9 * 1e-9 = 7.245\nsinh(7.245) = 700.0\nT = 1 / (1 + (5^2 * 700^2 * eV^2) / (4 * 3 * 2 * eV^2))\nT = 1 / (1 + 25 * 490000 / 24) = 1 / (510,417) = 1.96 * 10^-6\nWKB: T = exp(-2 * 7.245) = exp(-14.49) = 5.07 * 10^-7
Result: Exact: T = 1.96 * 10^-6 (0.000196%) | WKB: 5.07 * 10^-7
Example 2: Alpha Particle Tunneling in Nuclear Decay
Problem: Estimate tunneling probability for an alpha particle (mass = 6.645 * 10^-27 kg) with 5 MeV energy through a 30 MeV barrier of 15 fm (femtometer) width.
Solution: V0 - E = 25 MeV = 25 * 1.602e-13 J = 4.005e-12 J\nkappa = sqrt(2 * 6.645e-27 * 4.005e-12) / 1.055e-34 = 2.189 * 10^15 m^-1\na = 15 fm = 15 * 10^-15 m\nkappa * a = 2.189e15 * 1.5e-14 = 32.84\nWKB: T = exp(-2 * 32.84) = exp(-65.68) = 2.85 * 10^-29
Result: T = ~10^-29 | Extremely low probability per attempt, but high nuclear collision rate produces observable decay
Frequently Asked Questions
How is the tunneling probability calculated for a rectangular barrier?
For a rectangular barrier of height V0 and width a, with a particle of mass m and energy E less than V0, the exact transmission coefficient is T = 1 / (1 + V0^2 * sinh^2(kappa*a) / (4*E*(V0-E))), where kappa = sqrt(2*m*(V0-E))/hbar is the decay constant inside the barrier. This formula comes from solving the time-independent Schrodinger equation in three regions (before, inside, and after the barrier) and applying boundary conditions for continuity of the wavefunction and its derivative. The sinh^2 term grows exponentially for thick barriers, making the tunneling probability decrease exponentially with barrier width. For very thick barriers, this simplifies to the WKB approximation T = exp(-2*kappa*a).
How does particle mass affect tunneling probability?
Particle mass has an enormous effect on tunneling probability because the decay constant kappa is proportional to the square root of the mass: kappa = sqrt(2*m*(V0-E))/hbar. Since the transmission coefficient depends exponentially on kappa, heavier particles have dramatically lower tunneling probabilities. An electron (mass 9.1 times 10^-31 kg) can readily tunnel through barriers that are completely impenetrable to a proton (mass 1.67 times 10^-27 kg, about 1836 times heavier). For the same 1 eV barrier of 0.5 nm width, an electron might have a 10 percent tunneling probability while a proton would have a probability of approximately 10^-20. This mass dependence explains why tunneling effects are most significant for the lightest particles.
What is the role of quantum tunneling in nuclear fusion?
Quantum tunneling is essential for nuclear fusion in stellar cores. For two nuclei to fuse, they must overcome the Coulomb barrier, the electrostatic repulsion between their positive charges. In the Sun core at 15 million Kelvin, the average thermal energy of protons is about 1.3 keV, while the Coulomb barrier for proton-proton fusion is about 550 keV. Classically, fusion should be impossible at these temperatures. However, quantum tunneling allows protons to penetrate the Coulomb barrier with a small but nonzero probability. Combined with the enormous number of collisions per second (about 10^38 per cubic meter), this produces the steady nuclear fusion that powers the Sun. Without quantum tunneling, stars could not shine and life as we know it would not exist.
How does quantum tunneling enable scanning tunneling microscopes?
The scanning tunneling microscope (STM), invented in 1981 by Binnig and Rohrer who received the Nobel Prize for it, exploits quantum tunneling to image surfaces at atomic resolution. A sharp conductive tip is brought within about 1 nanometer of a sample surface. When a voltage is applied, electrons tunnel through the vacuum gap between the tip and surface. The tunneling current depends exponentially on the tip-sample distance, decreasing by roughly a factor of 10 for every 0.1 nm increase in gap width. This extreme sensitivity to distance allows the STM to detect height variations as small as 0.01 nm (one-hundredth of an atomic diameter). By scanning the tip across the surface while maintaining constant tunneling current, a topographic map of individual atoms can be constructed.
How does barrier shape affect tunneling probability in real physical systems?
Real physical barriers are rarely rectangular. The Coulomb barrier in nuclear physics has a 1/r shape, potential barriers in semiconductor devices have trapezoidal or parabolic profiles, and molecular potential barriers have complex shapes determined by the electronic structure. For arbitrary barrier shapes, the WKB approximation generalizes to T = exp(-2 * integral of kappa(x) dx), where the integral is taken over the classically forbidden region. This integral weights wider and taller sections of the barrier more heavily. A triangular barrier (as in field emission) gives a different tunneling probability than a rectangular barrier of the same average height and width. Numerical methods are often required for complex barrier profiles.
What are the practical applications of quantum tunneling in modern technology?
Quantum tunneling is fundamental to numerous technologies. Flash memory and EEPROM storage use Fowler-Nordheim tunneling to program and erase memory cells by moving electrons through thin oxide barriers. Tunnel diodes exploit negative differential resistance from resonant tunneling for high-frequency oscillators and amplifiers. Josephson junctions, where Cooper pairs tunnel between superconductors, are the basis of SQUID magnetometers (the most sensitive magnetic field detectors) and superconducting quantum computers. Tunnel field-effect transistors (TFETs) promise ultra-low-power electronic switches by using tunneling instead of thermal carrier injection. Even biological processes like enzyme catalysis and DNA mutations may involve proton tunneling through hydrogen bonds.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy