Lecture 39 Highlights
We are now going to use the WKB approximation to calculate tunneling rates
through odd-shaped barriers. You encountered tunneling before in Phys 401, including:
1) Section 2.5 where tunneling through a delta-function barrier of
height )()( xxV
δ
α
=. The result for the transmission coefficient is EE
T/1
1
0
+
=, where
E
is the energy of the particle and is a characteristic energy in the
problem.
22
02/ h
α
mE =
2) Problem 2.33 Tunneling through a rectangular barrier of height and width
. The transmission coefficient is
0
V
a
() ()
⎟
⎠
⎞
⎜
⎝
⎛−
−
+
=
EVm
a
EVE
V
T
0
2
0
2
02sinh
4
1
1
h
valid for
.
0
VE <
Consider the WKB approximation for particles in the “classically forbidden”
region, . In this region the kinetic energy is negative, which can be
interpreted as resulting from an imaginary momentum
)(xVE <mp 2/
2
(
)
)(2 xVEmp −= . In this case
the WKB solutions we found last time become:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡±= ∫
x
class
class
dxxp
xp
D
x')'(
1
exp
)(
)( h
ψ
,
with
()
)(2 xVEmpclass −= . Notice that the complex exponential has now become a
positive or negative exponential because the solutions are no longer running waves.
Consider a barrier of width awith an arbitrary potential on top, , as discussed
on pages 320-322 of Griffiths. For energies
)(xV
E
less than the minimum of , and for
barriers that are sufficiently tall and thick, the transmission probability is dominated by
the negative exponential term;
)(xV
, where
γ
2−
∝eT ∫
=
a
class dxxp
0
')'(
1
h
γ
Going back to the flat-top barrier of problem 2.33 to test this result, the
transmission probability in the WKB approximation (after doing the integral
with ) is
0
)( VxV =
()
EVm
a
eT −−
∝0
2
2
h. If we expand the exact result given above in the limit
of tall and wide barrier (i.e.
()
12 0>>− EVm
a
h), the result is
()
()
EVm
a
e
V
EVE
T−−
−
≅0
2
2
2
0
0
4h. The pre-factor is on the order of 1, so the exponential
dominates, and is in agreement with the WKB approximate result.
Now consider the problem of cold emission from a metal. A metal can be
modeled as a potential well of depth
Φ
+
F
E, where is the Fermi energy and is the
work function of the metal (see the Figure below). The work function is the energy
F
EΦ
1
