quantum tunneling explained
In the classical world a ball short of energy cannot clear a hill and simply bounces back. In the quantum world the wave function does not stop inside the wall and only fades as exp(-κx). A small amplitude leaks to the far side, and the chance of getting through is not 0. The first picture shows the oscillation on the left dying inside the wall and coming back small on the right.
A wave arriving from the left does not stop at the wall the way a classical particle would. Inside the wall the oscillation vanishes and the amplitude fades smoothly as exp(-κx), then lives again as a small oscillation on the far side. Make the wall thicker and the leaked amplitude on the right grows visibly smaller. The wave function stays continuous across the wall.
The transmission probability here is a unitless number between 0 and 1, a different quantity from a torque T or a period T. Raise the particle energy toward the height of the wall and that probability climbs toward 1. Roughly T ≈ exp(-2κL), so a taller or thicker wall makes the probability drop steeply. The decay rate is κ = √(2m(V-E))/ℏ, which sets how fast the wave fades inside the wall.
Set the two worlds side by side and the difference is plain. A classical particle with less energy than the wall has T = 0 and must bounce back. A quantum particle in the same place has T > 0 and part of it passes through the wall. Raise the wall and the quantum transmission shrinks but does not touch 0. A region that is closed classically is a region the quantum enters and crosses.
Look closely inside the wall and the amplitude is a smooth tail falling as exp(-κx). Raise the decay, which is the same as making the wall taller, and the tail plunges more steeply. A steeper tail leaves a smaller amplitude at the end of the wall, and less leaks out. A thicker or taller wall makes the decay faster and the transmission much smaller.
This leaking hides all through nature and through technology. Three scenes can be chosen. In alpha decay an alpha particle inside a nucleus comes out through the nuclear barrier, and that transmission probability sets the half-life. One kind of electron microscope draws atoms with the tunnel current that leaks between a tip and a surface. Flash memory stores information by letting electrons tunnel a thin insulating barrier and trap charge.