Quantum Tunneling Probability Calculator
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Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Quantum Tunneling Probability Calculator
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Formula: T = 1 / (1 + V0^2 * sinh^2(kappa*a) / (4*E*(V0-E)))
Worked example — Exact: T = 1.96 * 10^-6 (0.000196%) | WKB: 5.07 * 10^-7
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 kappa = sqrt(2 * 9.109e-31 * 3.204e-19) / 1.055e-34 = 7.245 * 10^9 m^-1 kappa * a = 7.245e9 * 1e-9 = 7.245 sinh(7.245) = 700.0 T = 1 / (1 + (5^2 * 700^2 * eV^2) / (4 * 3 * 2 * eV^2)) T = 1 / (1 + 25 * 490000 / 24) = 1 / (510,417) = 1.96 * 10^-6 WKB: 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 kappa = sqrt(2 * 6.645e-27 * 4.005e-12) / 1.055e-34 = 2.189 * 10^15 m^-1 a = 15 fm = 15 * 10^-15 m kappa * a = 2.189e15 * 1.5e-14 = 32.84 WKB: 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
What is quantum tunneling and how does it work?
Quantum tunneling is a phenomenon in which a quantum particle passes through a potential energy barrier that it classically does not have enough energy to overcome. In classical physics, a ball rolling toward a hill that is higher than its kinetic energy will always bounce back. In quantum mechanics, particles have wave-like properties described by their wavefunction, and this wavefunction does not abruptly stop at the barrier but instead decays exponentially inside it. If the barrier is thin enough, there is a nonzero probability that the wavefunction extends through to the other side, allowing the particle to appear on the far side of the barrier. Quantum tunneling is essential for nuclear fusion in stars, radioactive alpha decay, and many electronic devices.
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).
What is the WKB approximation and when is it accurate?
The WKB (Wentzel-Kramers-Brillouin) approximation provides a simpler formula for tunneling probability: T = exp(-2*kappa*a), which ignores the pre-exponential factor in the exact result. This approximation is accurate when the barrier is much wider than the decay length 1/kappa, corresponding to the condition kappa*a >> 1. In this regime, the exact transmission coefficient is dominated by the exponential decay and the sinh function can be approximated by an exponential. For thin barriers or when the particle energy is close to the barrier height, the WKB approximation can be significantly inaccurate, overestimating the tunneling probability by factors of 2 or more. The exact formula should always be used when precision matters.
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.
What is alpha decay and how does tunneling explain radioactive half-lives?
Alpha decay occurs when an unstable nucleus emits an alpha particle (two protons and two neutrons bound together). Inside the nucleus, the alpha particle is trapped by the nuclear potential well, bouncing back and forth billions of times per second. Each time it hits the potential barrier at the nuclear surface, there is a small tunneling probability of escaping. The decay rate equals the collision frequency multiplied by the tunneling probability. Since tunneling probability depends exponentially on the barrier parameters, small differences in nuclear size and alpha particle energy produce enormous differences in half-lives. This explains why uranium-238 has a half-life of 4.5 billion years while polonium-212 decays in 0.3 microseconds, a range spanning 23 orders of magnitude explained entirely by tunneling theory.
What happens when the particle energy equals or exceeds the barrier height?
When particle energy E equals or exceeds the barrier height V0, the behavior changes dramatically. Instead of exponential decay inside the barrier, the wavefunction oscillates (like a wave in a medium with different properties). The transmission coefficient oscillates between values less than 1 and exactly 1 as barrier width changes. Perfect transmission (T = 1) occurs at resonance conditions when the barrier width equals an integer number of half-wavelengths of the particle inside the barrier. This resonance tunneling is exploited in semiconductor devices like resonant tunneling diodes. At E = V0, a special case occurs where the transmission formula has a different mathematical form involving a polynomial rather than trigonometric functions.
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 for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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