Quantum Tunneling Calculator
Free Quantum tunneling Calculator for quantum mechanics. Enter variables to compute results with formulas and detailed steps.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Quantum Tunneling Calculator
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Formula: T = 1 / (1 + sinh^2(kappa L) V^2 / (4E(V-E)))
Worked example — Transmission coefficient T = 0.609 (60.9% tunneling probability)
Formula
T = 1 / (1 + sinh^2(kappa L) V^2 / (4E(V-E)))
Where T = transmission coefficient, kappa = sqrt(2m(V-E))/hbar is the decay constant, L = barrier width, V = barrier height in eV, E = particle energy in eV, m = particle mass, and hbar = reduced Planck constant (1.0546e-34 J s).
Worked Examples
Example 1: Electron Tunneling Through a Thin Oxide Layer
Problem:An electron (mass 9.109e-31 kg) encounters a 5 eV barrier that is 1 angstrom (1e-10 m) wide. The electron has 3 eV of kinetic energy. What is the tunneling probability?
Solution:kappa = sqrt(2 * 9.109e-31 * (5-3) * 1.602e-19) / 1.0546e-34 = sqrt(5.833e-49) / 1.0546e-34 = 7.245e9 m^-1 kappa * L = 7.245e9 * 1e-10 = 0.7245 sinh(0.7245) = 0.7856 T = 1 / (1 + (0.7856)^2 * 25 / (4 * 3 * 2)) = 1 / (1 + 0.6171 * 25/24) = 1 / (1 + 0.6428) = 0.609
Result:Transmission coefficient T = 0.609 (60.9% tunneling probability)
Example 2: Proton Tunneling Through a Nuclear Barrier
Problem:A proton (mass 1.673e-27 kg) with 1 MeV energy encounters a 10 MeV nuclear barrier that is 1e-14 m wide. Calculate the tunneling probability.
Solution:kappa = sqrt(2 * 1.673e-27 * 9 * 1.602e-13) / 1.0546e-34 = sqrt(4.825e-39) / 1.0546e-34 = 6.586e12 m^-1 kappa * L = 6.586e12 * 1e-14 = 0.06586 sinh(0.06586) = 0.06591 T = 1 / (1 + (0.06591)^2 * 100 / (4 * 1 * 9)) = 1 / (1 + 0.004344 * 2.778) = 1 / (1.01207) = 0.988
Result:Transmission coefficient T = 0.988 (98.8% tunneling probability for this thin nuclear barrier)
Frequently Asked Questions
What is quantum tunneling and why does it matter?
Quantum tunneling is a phenomenon in quantum mechanics where a particle passes through a potential energy barrier that it classically should not be able to overcome. According to classical physics, a ball rolling toward a hill without enough energy to reach the top would simply bounce back. However, at the quantum scale, particles behave as probability waves, and there is a nonzero probability of finding the particle on the other side of the barrier. This effect is critical in many real-world applications, including nuclear fusion in stars, semiconductor electronics, and scanning tunneling microscopy.
How is the transmission coefficient calculated?
The transmission coefficient T represents the probability that a particle will tunnel through a rectangular potential barrier. For a barrier of height V, width L, and a particle with energy E less than V, the formula involves the hyperbolic sine function: T = 1 / (1 + (sinh(kappa L))^2 V^2 / (4E(V-E))), where kappa = sqrt(2m(V-E)) / hbar. The value of T ranges from 0 to 1, with higher values indicating a greater probability of tunneling. When the particle energy equals or exceeds the barrier height, the transmission coefficient becomes 1, meaning the particle passes through classically.
What factors influence the tunneling probability the most?
Three primary factors determine tunneling probability. First, the barrier width has an exponential effect because the transmission coefficient decreases exponentially with increasing width. Second, the difference between barrier height and particle energy matters significantly since a larger energy deficit means lower tunneling probability. Third, the particle mass plays a crucial role because heavier particles have much lower tunneling probabilities. This is why tunneling is primarily observed for lightweight particles like electrons and protons, not for macroscopic objects like baseballs or people.
What is the decay constant kappa in tunneling?
The decay constant kappa (also called the wave vector inside the barrier) characterizes how rapidly the quantum wave function decays as it penetrates the barrier. It is defined as kappa = sqrt(2m(V-E)) / hbar, where m is particle mass, V is barrier height, E is particle energy, and hbar is the reduced Planck constant. A larger kappa means the wave function decays faster, leading to lower transmission probability. The inverse of kappa, known as the penetration depth, gives the characteristic length scale over which the wave function amplitude falls by a factor of e (approximately 2.718).
How does quantum tunneling apply to semiconductor devices?
Quantum tunneling is fundamental to modern semiconductor technology and electronics. In tunnel diodes, electrons tunnel through thin potential barriers, enabling extremely fast switching speeds used in microwave oscillators and high-frequency circuits. Flash memory and EEPROM storage devices rely on Fowler-Nordheim tunneling to program and erase data by moving electrons through thin oxide layers. In modern transistors with gate lengths below 5 nanometers, unwanted tunneling current becomes a significant design challenge. Understanding tunneling physics is therefore essential for engineers designing the next generation of computer chips.
What is the WKB approximation for tunneling?
The WKB (Wentzel-Kramers-Brillouin) approximation is a semiclassical method used to estimate tunneling probabilities for barriers with arbitrary shapes, not just rectangular ones. The transmission coefficient under WKB is given by T approximately equal to exp(-2 times the integral of kappa(x) dx across the barrier), where kappa(x) varies with position for non-rectangular barriers. This approximation works well when the barrier varies slowly compared to the particle wavelength. It is widely used in nuclear physics to calculate alpha decay rates and in field emission calculations for electron tunneling through triangular barriers.
What is the penetration depth in quantum tunneling?
The penetration depth is the distance into a classically forbidden barrier region at which the probability of finding the particle drops to 1/e (about 37%) of its value at the barrier entrance. It equals 1/kappa, or hbar / sqrt(2m(V-E)), and is typically on the order of angstroms (1e-10 meters) for electrons encountering barriers of a few electron volts. A larger penetration depth means the wave function extends further into the barrier, increasing the chance of tunneling if the barrier is thin enough. This concept is directly exploited in scanning tunneling microscopy, where the tunneling current between a sharp tip and a surface is exponentially sensitive to distance.
Can macroscopic objects tunnel through barriers?
Theoretically, quantum tunneling applies to all objects, but for macroscopic objects the probability is so astronomically small that it is effectively zero. The tunneling probability decreases exponentially with both the mass of the object and the width of the barrier. For example, the probability of a baseball tunneling through a wall is roughly 10 to the power of negative billions, a number so small it is meaningless in any practical context. Even over the entire age of the universe, such an event would never occur. This is why quantum effects are only observable at the atomic and subatomic scales where masses and barrier dimensions are tiny.
How does tunneling relate to nuclear fusion in stars?
Nuclear fusion in stars like our Sun relies critically on quantum tunneling. For two protons to fuse, they must overcome their mutual electrostatic (Coulomb) repulsion, which forms a potential energy barrier. At the core temperature of the Sun (about 15 million Kelvin), the thermal energy of protons is far too low to classically surmount this barrier. However, quantum tunneling allows a small fraction of protons to penetrate the Coulomb barrier and fuse, releasing enormous energy. Without tunneling, the Sun would need to be much hotter to sustain fusion, and stars as we know them would not exist.
What is resonant tunneling and how does it differ from regular tunneling?
Resonant tunneling occurs when a particle encounters a double-barrier structure (two barriers with a quantum well between them) and the particle energy matches one of the quasi-bound energy levels of the well. At these resonant energies, the transmission coefficient can reach unity (100%), even though each individual barrier alone would have very low transmission. This phenomenon is fundamentally different from single-barrier tunneling because it exploits constructive interference of the wave function within the well. Resonant tunneling diodes exploit this effect to create negative differential resistance, which is useful in high-frequency oscillators and fast switching circuits.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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