Quantum Tunneling Probability Calculator
Estimate the transmission coefficient for a rectangular barrier: T = exp(-2 * d * sqrt(2m(V-E)) / hbar).
Transmission Coefficient
0.2348281293
Transmission Coefficient vs Barrier Height (V)
Fórmula
## 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)** - *V* = barrier height (eV, converted to J) - *E* = particle kinetic energy - *d* = barrier width - *m* = particle mass - *hbar* = reduced Planck constant The probability drops exponentially with barrier width and the square root of the energy deficit.
Ejemplo Resuelto
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
Preguntas Frecuentes
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.
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