Page 57 - Science at the nanoscale

RPS: PSP0007 - Science-at-Nanoscale
9:2
June 9, 2009
3.2. Basic Postulates of Quantum Mechanics
Hence we can see that ψ(x) is non-zero inside the potential step
and thus it is possible for a particle to penetrate into the potential
barrier! This is not allowed in classical physics.
3.2.5
Potential Barrier and Quantum Tunneling
As mentioned in the previous section, there is a probability that
the wavefunction can penetrate into the potential step. This sit-
uation becomes very interesting if the potential step is replaced
by a potential barrier. If the potential barrier width W is narrow,
it is possible for a particle to penetrate through the potential bar-
rier and appear on the other side! This phenomenon is known as
quantum tunneling. Let us consider the potential barrier shown
in Fig. 3.8. We divide the system into three region I, II and III as
shown.
For regions I and III with V = 0, the Schr¨odinger equation is
given by Eq. (3.34) hence we can write down the wavefunction as
ikx
ψ I (x) = Pe
+ Qe
and
(3.45)
We have to account for the presence of the reflected wave in
region I while there is no reflected wave in region III.
The intensities of the incident, reflected and transmitted proba-
bility current densities, J, are given by
2
(3.46)
J = v |Q| ,
J = v |P| , ψ III (x) = Se 2 ikx −ikx 2 (3.44) 47 ch03
J = v |S|
hk
where v = represents the magnitude of the velocity of the par-
m
ticle. The reflection coefficient R and the transmission coefficient
II III
I V=0
V=0
V = V o
ψ E < V o
ˇ I
ψ
II ψ
III
x=0 x = W X
Figure 3.8. Potential barrier.