bei48482_FM

Quantum Mechanics 185

necessarily great, but not zero either—of passing through the barrier and emerging

on the other side. The particle lacks the energy to go over the top of the barrier, but

it can nevertheless tunnel through it, so to speak. Not surprisingly, the higher the

barrier and the wider it is, the less the chance that the particle can get through.

The tunnel effectactually occurs, notably in the case of the alpha particles emit-

ted by certain radioactive nuclei. As we shall learn in Chap. 12, an alpha particle whose

kinetic energy is only a few MeV is able to escape from a nucleus whose potential wall

is perhaps 25 MeV high. The probability of escape is so small that the alpha particle

might have to strike the wall 10^38 or more times before it emerges, but sooner or later

it does get out. Tunneling also occurs in the operation of certain semiconductor diodes

(Sec. 10.7) in which electrons pass through potential barriers even though their kinetic

energies are smaller than the barrier heights.

Let us consider a beam of identical particles all of which have the kinetic energy E.

The beam is incident from the left on a potential barrier of height Uand width L,as

in Fig. 5.9. On both sides of the barrier U0, which means that no forces act on the

particles there. The wave function Irepresents the incoming particles moving to the

right and Irepresents the reflected particles moving to the left; IIIrepresents the

transmitted particles moving to the right. The wave function IIrepresents the parti-

cles inside the barrier, some of which end up in region III while the others return to

region I. The transmission probability Tfor a particle to pass through the barrier is

equal to the fraction of the incident beam that gets through the barrier. This proba-

bility is calculated in the Appendix to this chapter. Its approximate value is given by

Te^2 k^2 L (5.60)

where

k 2  (5.61)

and Lis the width of the barrier.

^2 m(U E)





Approximate

transmission

probability

x = 0 x = L

I II III

I+

I–

ΙΙI+

ψII

Energy

E

U

x

Figure 5.9When a particle of energy EUapproaches a potential barrier, according to classical

mechanics the particle must be reflected. In quantum mechanics, the de Broglie waves that correspond

to the particle are partly reflected and partly transmitted, which means that the particle has a finite

chance of penetrating the barrier.

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