Quantum Tunneling Probability Calculator Formula
Understand the math behind the quantum tunneling probability calculator. Each variable explained with a worked example.
Formulas Used
Transmission Coefficient
transmission = exp(-2 * barrier_width * sqrt(2 * particle_mass * (barrier_height - particle_energy) * 1.602e-19) / 1.0546e-34)Variables
| Variable | Description | Default |
|---|---|---|
barrier_height | Barrier Height (V)(eV) | 5 |
particle_energy | Particle Energy (E)(eV) | 3 |
barrier_width | Barrier Width (d)(m) | 1e-10 |
particle_mass | Particle Mass(kg) | 9.109e-31 |
How It Works
Quantum Tunneling
In quantum mechanics, a particle can pass through a potential barrier even if its energy is below the barrier height.
Approximate Transmission
T = exp(-2d sqrt(2m(V-E)) / hbar)
The probability drops exponentially with barrier width and the square root of the energy deficit.
Worked Example
Electron (m = 9.109e-31 kg) with E = 3 eV hitting a 5 eV barrier, 1 angstrom wide.
- 01V - E = 2 eV = 2 * 1.602e-19 = 3.204e-19 J
- 02sqrt(2m(V-E)) = sqrt(2 * 9.109e-31 * 3.204e-19) = sqrt(5.837e-49) = 7.640e-25
- 03Exponent = -2 * 1e-10 * 7.640e-25 / 1.0546e-34 = -1.449
- 04T = exp(-1.449) = 0.235
Frequently Asked Questions
Is tunneling instantaneous?
The traversal time is debated in physics. Experiments suggest the process is extremely fast, possibly faster than the barrier width divided by the particle speed.
Where does tunneling matter in real life?
Nuclear fusion in stars (protons tunnel through the Coulomb barrier), radioactive alpha decay, scanning tunneling microscopes, and tunnel diodes in electronics.
Can any particle tunnel through any barrier?
In principle yes, but the probability drops exponentially. For macroscopic barriers, the probability is so astronomically small that it effectively never happens.
Ready to run the numbers?
Open Quantum Tunneling Probability Calculator