Quantum Tunneling Calculator

Quantum tunneling is the phenomenon where a particle can pass through an energy barrier taller than its own kinetic energy — something classical physics forbids outright. This calculator checks whether your particle energy is below the barrier height; if it is, the particle's wavefunction decays exponentially inside the barrier rather than being reflected with certainty, and some of it leaks through to the other side. The decay constant κ describes how fast that decay happens inside the barrier, computed from the mass, the energy deficit (V − E), and the reduced Planck constant, and it directly sets both the penetration depth (how far the wavefunction reaches before decaying to about a third of its value) and the transmission coefficient, which shrinks exponentially as either the barrier gets wider or κ grows.

When particle energy meets or exceeds the barrier height, the calculator switches to the classical regime: transmission jumps to 100%, reflection to zero, since a particle with enough energy simply goes over the barrier rather than tunneling through it. This is a simplified rectangular-barrier approximation valid for barriers that are thick relative to the decay length — real potential barriers with more complex shapes (like the Coulomb barrier in nuclear fusion) need the same underlying physics integrated over a varying barrier profile rather than the simple exponential formula used here. It's still the right tool for building intuition about why phenomena like scanning tunneling microscopy and flash memory rely on distances measured in single-digit angstroms.