Environmental Engineering Reference
In-Depth Information
Figure 2.2 Sketch of
evanescent
light wave
crossing air barrier between two glass plates.
The intensity (electric field squared) of the
evanescent wave decreases exponentially with
air gap spacing t. This can be viewed as quantum
mechanical tunneling of photons, light
particles, with the role of the wave function
being played by the electric field in the light
wave. (Courtesy of M. Medikonda).
exponential function, quite analogous to the exponentially decaying light wave
sketched in Figure 2.2.
In classical physics, the particle will never exist in this region, but nanophysics
allows it in precise terms. If the barrier were constant at
V
B
, the wave function would
be given by exp(
k
r
), with
k
¼
(2
m
(
V
B
E
))
1/2
/
h
, in the range
r
1
¼ r
n
<
r
2
¼ r
tp
.In
this case, the tunneling probability of the particle of energy
E
through the barrier of
height
V
B
>
r
<
E
is exp(
2
k
t
), where
t ¼ r
2
r
1
is the barrier thickness. The
transmission
probability
is de
ned as
2
2
T
¼j ðr
1
Þj
=j ðr
2
Þj
:
ð
2
:
20
Þ
(
r
1
)|
2
(
r
1
)
(
r
1
). In the real case, the height of the barrier follows an 1/
r
dependence as indicated in Figure 1.6. It is dif
cult to solve the Schrodinger equation
in case of an arbitrary barrier shape
V
(
r
) and the practical approach is the simplifying
WKB approximation.This useful approximation, applied to our case, gives [22]
Here, |
¼
T
¼
exp
ð
2
cÞ:
ð
2
:
21
Þ
With
c ¼
h
1
ð
r
2
1
=
2
d
r
f
2
m
r
½V
B
ðrÞEg
:
ð
2
:
22
Þ
r
1
Here,
r
1
and
r
2
are the turning points, where
E¼V
, and
m
r
is the reduced mass
m
r
¼ m
1
m
2
=ðm
1
þm
2
Þ:
ð
2
:
23
Þ
It is clear in physical terms that the value of the tunneling probability T is
independent of which way the particle is going. An incoming wave will be large
