Quantum Tunneling Probability Calculator
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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.