Tunnelia

Meet Tunnelia, a quantum particle who gets through walls she could never climb.

In classical physics a ball with energy E cannot cross a hill of height V > E. In quantum mechanics a particle is a wave, and a wave hitting a barrier does not stop dead. Inside the barrier it decays exponentially, falling off with the decay constant κ = √(2m(V − E))/ħ. If the barrier is thin enough, some of the wave is still there on the far side, and the particle has a chance of turning up there. For a thick barrier that chance is roughly e^(−2κL).

Because the dependence is exponential, small changes matter enormously. Doubling the barrier width squares the already small transmission, and heavier particles tunnel far less than light ones like electrons.

Tunnelling is not a curiosity. It explains alpha decay of nuclei, lets the scanning tunnelling microscope image single atoms, and is how flash memory stores and erases charge. In the Sun's core it helps protons get close enough to fuse.

The e^(−2κL) form is an approximation for wide barriers; exact results include a prefactor and depend on the barrier's shape.