bei48482_FM

is the flux of particles that arrive at the barrier. That is, Sis the number of particles

per second that arrive there.

At x 0 the incident wave strikes the barrier and is partially reflected, with

Reflected wave (^) IBeik^1 x (5.79)
representing the reflected wave. Hence
(^) I (^) I (^) I (5.80)
On the far side of the barrier (xL) there can only be a wave
Transmitted wave (^) IIIFeik^1 x (5.81)
traveling in thexdirection at the velocity IIIsince region III contains nothing that
could reflect the wave. Hence G0 and
(^) III (^) IIIFeik^1 x (5.82)
The transmission probability Tfor a particle to pass through the barrier is the ratio
T (5.83)
between the flux of particles that emerges from the barrier and the flux that arrives at
it. In other words, Tis the fraction of incident particles that succeed in tunneling
through the barrier. Classically T0 because a particle with EUcannot exist inside
the barrier; let us see what the quantum-mechanical result is.
In region II Schrödinger’s equation for the particles is
 (EU) (^) II(UE) (^) II 0 (5.84)
Since UEthe solution is
(^) IICek^2 xDek^2 x (5.85)
where the wave number inside the barrier is
k 2  (5.86)
Since the exponents are real quantities, (^) IIdoes not oscillate and therefore does not
represent a moving particle. However, the probability density  (^) II^2 is not zero, so there
is a finite probability of finding a particle within the barrier. Such a particle may emerge
into region III or it may return to region I.
 2 m(UE)
Wave number
inside barrier
Wave function
inside barrier
2 m
2
d^2 II
dx^2
2 m
2
d^2 II
dx^2
FFIII
AAI
 (^) III^2 III
 (^) I^2 I
Transmission
probability
194 Appendix to Chapter 5
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