Appendix to Chapter 5
The Tunnel Effect
W
e consider the situation that was shown in Fig. 5.9 of a particle of energy
EUthat approaches a potential barrier Uhigh and Lwide. Outside
the barrier in regions I and III Schrödinger’s equation for the particle takes
the forms E (^) I 0 (5.73)
E (^) III 0 (5.74)
The solutions to these equations that are appropriate here are
(^) IAeik^1 xBeik^1 x (5.75)
(^) IIIFeik^1 xGeik^1 x (5.76)
where
k 1 (5.77)
is the wave number of the de Broglie waves that represent the particles outside the
barrier.
Because
eicosisin
eicosisin
these solutions are equivalent to Eq. (5.38)—the values of the coefficients are differ-
ent in each case, of course—but are in a more suitable form to describe particles that
are not trapped.
The various terms in Eqs. (5.75) and (5.76) are not hard to interpret. As was shown
schematically in Fig. 5.9, Aeik^1 xis a wave of amplitude A incident from the left on the
barrier. Hence we can write
Incoming wave (^) IAeik^1 x (5.78)
This wave corresponds to the incident beam of particles in the sense that (^) I^2 is their
probability density. If Iis the group velocity of the incoming wave, which equals the
velocity of the particles, then
S (^) I^2 I
2
p
2 mE
Wave number
outside barrier
2 m
2
d^2 III
dx^2
2 m
2
d^2 I
dx^2
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