Quantum Tunneling Calculator
Free Quantum tunneling Calculator for quantum mechanics. Enter variables to compute results with formulas and detailed steps.
Reviewed by Manoj Kumar, Mathematics Educator
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\nkappa * L = 7.245e9 * 1e-10 = 0.7245\nsinh(0.7245) = 0.7856\nT = 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\nkappa * L = 6.586e12 * 1e-14 = 0.06586\nsinh(0.06586) = 0.06591\nT = 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.
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy