Classical Physics
According to classical physics, an electron will pass through a potential barrier if it possesses enough kinetic energy to overcome the barrier.If it has less kinetic energy than the height of the potential barrier then it will be unable to pass through the barrier under any conditions.
Quantum Mechanics
Quantum mechanics shows that electrons can be described as waves under certain conditions, and a finite probability exists of an electron tunnelling through a classically forbidden barrier due to its wavelike properties.When a wave meets a potential barrier, the wave does not instantly go to zero, but starts to decay exponentially within the potential barrier. If the wave has not reached zero by the time it has reached the other side of the barrier then there is a finite probability that it will be found on the other side of the barrier - the wave has effectively "tunnelled" through the non-conductive barrier.
"Principles of Quantum Tunnelling: Classical physics and Quantum mechanics" via QTC Science
(Which of course means our current perceptions of the world, down to their very fundamental assumptions - they're all so utterly silly.)
