Tunneling is the quantum effect where a particle can appear beyond a barrier it does not have enough energy to classically cross.
Tunneling explains real devices and phenomena: scanning tunneling microscopes, alpha decay in nuclear physics, and the behavior of transistors at small scales. It is one of the most directly observable consequences of quantum mechanics.
In everyday physics, a ball that does not have enough energy to get over a wall simply bounces back. In quantum mechanics, the story is different. The amplitude of the particle does not drop to zero at the barrier. It gradually fades inside the barrier, but some amplitude reaches the other side. That means there is a small but real probability of the particle appearing beyond the barrier. The thicker or taller the barrier, the smaller the transmission probability, but it never reaches exactly zero.
Imagine sound leaking through a thick wall. Most of the sound is blocked, but a small amount gets through. Tunneling is similar: the barrier strongly suppresses the amplitude, but does not block it completely. The crucial difference is that tunneling is a quantum prediction calculated from the wavefunction, not a classical leaking effect.
Solving the Schrodinger equation for a rectangular barrier gives a wavefunction that decays exponentially inside the barrier region but does not become exactly zero. Matching the wavefunction and its derivative at the barrier boundaries gives a nonzero transmission coefficient. The transmission probability decreases exponentially with barrier width and height.
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