Scattering and tunnelling
Scattering and tunnelling

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Scattering and tunnelling

4.2 Wave packets and tunnelling in one dimension

Figure 18 shows a sequence of images captured from a wave packet simulation program. The sequence involves a Gaussian wave packet, with energy expectation value 〈E〉 = E0, incident from the left on a finite square barrier of height V0. The sequence is broadly similar to that shown in Figure 6, which involved a similar wave packet and a similar barrier, but with one important difference; in the earlier process E0 was greater than V0, so transmission was classically allowed, but in the case of Figure 18 E0 is less than V0 and transmission is classically forbidden. The bottom image shows that transmission can occur in quantum mechanics.

Figure 18 The passage of a wave packet with 〈E〉 = E0 through a finite square barrier of height V0 when E0 < V0. The probability density |Ψ|2 is shown in a sequence of snapshots with time increasing from top to bottom
Figure 18 The passage of a wave packet with 〈E〉 = E0 through a finite square barrier of height V0 when E0 < V0. The probability density |Ψ|2 is shown in a sequence of snapshots with time increasing from top to bottom

In the case shown in Figure 18, part of the reason for transmission is that the wave packet has a spread of energies, some of which lie above the top of the barrier. However, there is a second reason, which applies even for wave packets with energies wholly below the top of the barrier; there is the possibility that a particle, with insufficient energy to surmount the barrier, may nevertheless tunnel through it. For a given wave packet, the probability of tunnelling decreases with the height of the barrier and it also decreases very markedly with its thickness. We shall now use stationary-state methods to investigate this phenomenon.

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