Quantum Tunnelling

The particle has total energyand is incident on the barrierfrom the left. The Schrodinger equation in each of regions I, II and III takes the form

In region Iso

The solution iswhere Theterm represents motion to the right and theterm represents motion to the left – indicating reflection of the wavefunction from the barrier.

In region II the particle has total energyand

The solution iswhereere Sincethis represents exponential decay of the wavefunction in this region.

In region IIIso

The solution iswhere

is a measure of finding the partiacle at a point and since this value is greater that 0 in region III there is a non zero probability of finding the particle in region III, which is forbidden by classical mechanics since it has insufficient energy to surmount the barrier.