Quantum Tunneling Calculator
Free Quantum tunneling Calculator for quantum mechanics. Enter variables to compute results with formulas and detailed steps.
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Adjust values & calculateFormula
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).
Last reviewed: December 2025
Worked Examples
Example 1: Electron Tunneling Through a Thin Oxide Layer
Example 2: Proton Tunneling Through a Nuclear Barrier
Background & Theory
The Quantum Tunneling Calculator applies the following established principles and formulas. Physics is the fundamental natural science concerned with matter, energy, and the interactions between them. Classical mechanics, founded on Newton's three laws of motion, provides the framework for analyzing the motion of objects. The first law states that an object remains at rest or in uniform motion unless acted upon by a net external force. The second law quantifies this relationship: F = ma, where force equals mass times acceleration in SI units of newtons (N = kgยทm/sยฒ). The third law establishes that every action produces an equal and opposite reaction. Kinematics describes motion without reference to its causes. The four fundamental equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t: v = u + at, s = ut + ยฝatยฒ, vยฒ = uยฒ + 2as, and s = ยฝ(u + v)t. These assume constant acceleration and are foundational for solving projectile motion, free fall, and linear dynamics problems. Energy conservation underpins much of physics. Kinetic energy is KE = ยฝmvยฒ, where m is mass in kilograms and v is speed in meters per second. Gravitational potential energy is PE = mgh, where g โ 9.81 m/sยฒ near Earth's surface and h is height in meters. The work-energy theorem states that the net work done on an object equals its change in kinetic energy: W = ฮKE. Electricity and circuits rely on Ohm's law: V = IR, where voltage V is in volts, current I in amperes, and resistance R in ohms. Electrical power is P = IV = IยฒR = Vยฒ/R, measured in watts. Wave mechanics connects frequency f, wave speed v, and wavelength ฮป through f = v/ฮป, with frequency in hertz (Hz). Pressure is defined as force per unit area, P = F/A, in pascals (Pa = N/mยฒ). The ideal gas law PV = nRT links pressure, volume, moles n, the gas constant R = 8.314 J/(molยทK), and absolute temperature in kelvin. Gravitational force between two masses follows Newton's law of universal gravitation: F = Gmโmโ/rยฒ, where G = 6.674ร10โปยนยน Nยทmยฒ/kgยฒ is the gravitational constant.
History
The history behind the Quantum Tunneling Calculator traces back through the following developments. The history of physics spans over two millennia, beginning with the natural philosophy of ancient Greece. Aristotle (384โ322 BCE) proposed that all matter consisted of four elements and that objects moved toward their natural place, with heavier objects falling faster than lighter ones. While largely incorrect, his systematic approach to explaining nature dominated Western thought for nearly 2,000 years. The Scientific Revolution overturned Aristotelian physics. Galileo Galilei (1564โ1642) performed groundbreaking experiments on inclined planes and falling bodies, demonstrating that all objects fall with the same acceleration regardless of mass, and established the principle of inertia. His use of mathematics to describe motion was revolutionary. Isaac Newton synthesized these developments in his landmark Principia Mathematica (1687), laying out the three laws of motion and the law of universal gravitation. Newton's framework unified terrestrial and celestial mechanics, explaining planetary orbits with the same equations governing a falling apple. His calculus provided the mathematical language for expressing rates of change. The 19th century brought two major theoretical achievements. James Clerk Maxwell formulated his equations of electromagnetism between 1861 and 1862, unifying electricity, magnetism, and optics, and predicting the existence of electromagnetic waves traveling at the speed of light. Thermodynamics was developed by Carnot, Clausius, and Kelvin, establishing the laws governing heat, work, and entropy. The 20th century produced two revolutions that fundamentally altered the classical picture. Albert Einstein published the special theory of relativity in 1905, showing that space and time are not absolute but relative to the observer, and that mass and energy are equivalent via E = mcยฒ. His general theory of relativity in 1915 reinterpreted gravity as the curvature of spacetime. Simultaneously, quantum mechanics emerged from the work of Planck, Bohr, Heisenberg, and Schrรถdinger, revealing that at atomic scales energy is quantized and particles exhibit wave-particle duality. These developments culminated in the Standard Model of particle physics, which describes all known fundamental particles and three of the four fundamental forces.
Frequently Asked Questions
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.
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.
References
Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy