Quantum mechanical tunneling

Keeping extraneous ideas and postulates to a minimum,

How can we explain the process of quantum-mechanical tunneling?

Answer

Its simple enough. If classical particles encounter a potential step higher than their kinetic energy they cannot penetrate it. Because as soon as they penetrate the barrier they will have a negative kinetic energy, which is not possible.

A wave reacts differently to potential barriers. Waves don't stop abruptly at finite potential barriers. Instead they decay exponentially in such a region. Now, you may wonder why should a wave be able to penetrate a barrier (even to a limited extent) when a particle cannot.(This I cannot answer intuitively. All I can say is that the wave equation admits decaying solutions inside the barrier, and so they can penetrate it).

But aside from that everything is clear. Since a quantum particle acts as a wave, it will have a wavefunction (which is basically a solution to a wave equation) that is decaying exponentially within the barrier. But what if the barrier is so thin that the wavefunction cannot decay to zero before reaching the other end? Then of course, the particle's wavefunction will be non-zero outside the potential barrier and there will be a probability to detect the particle outside the barrier. And hence tunneling.