We now consider a situation where a particle is confined by walls of potential energy V, the potential being zero within the walls; the potential does not rise to infinity at the walls. The total energy E of the particle, which arises solely from the motion of the particle within the walls, is such that E < V.

Classically, the particle would be unable to escape from confinement between the walls, as its energy is less than the amount required to get over them.

In previous models, we have considered situations where the confining potential rises instantaneously to infinity. In this instance, the wavefunction decays to zero as soon as it reaches the wall. In our current model, since the potential is not infinite but E is less than V , the wavefunction decays slowly within the barrier. If the barrier is sufficiently thin (the potential drops back to zero after a short enough distance), then the wavefunction may still be nonzero at the far side of the wall. It may then oscillate freely beyond the far side of the wall, outside the box:

Thus the particle may be found outside the box, even though this would be forbidden by classical mechanics. This penetration through a classically forbidden region is called tunnelling. 

Note that the amplitude of the wavefunction is much less outside the box than inside, indicating that the probability of finding the particle outside the box is much less than the probability of finding the particle inside the box.

The tunnelling probability for a particle incident upon such a barrier may be calculated using the Schrodinger equation.

In the regions to the left and right of the barrier, the wavefunctions are those of a particle in free space.

Within the barrier, the Hamiltonian contains a term for the potential, V. This may be subtracted from both sides of the equation, and the resulting ordinary differential equation solved by standard techniques.

We now have three wavefunctions, one for each discrete region in which the particle may be found (to the left of the barrier, within the barrier, and to the right of the barrier). These wavefunctions may be related to one another by observing that they must be continuous and must have a continuous first derivative. i.e. the wavefunction within the box must join up with the wavefunction within the wall and their slopes must be identical at the point where they meet (the edge of the wall) , and the same for the wavefunction within the wall and the wavefunction outside the box.

The relationships derived allow calculation of the transmission probability, which is defined as the ratio of the probability that a particle on the left of the barrier is travelling to the right and the probability that a particle on the right of the barrier is travelling to the right.

It emerges that the transmission probability decreases exponentially with the thickness of the barrier, so tunnelling is only observed to any great degree when the potential barrier is thin. The transmission probability also decreases exponentially with m^½ , where m is the mass of the tunnelling particle. Thus tunnelling is much more important for electrons than for protons, and for particles heavier than a proton is of negligible importance.