Transcript CHM 4412 Chapter 12  University of Illinois at Urbana
Lecture 16
Tunneling
(c) So Hirata, Department of Chemistry, University of Illinois at UrbanaChampaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus TeacherScholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.
Tunneling
We will consider a problem in which a particle of mass
m
and energy
E
hits a potential barrier of height
V
and width
L. V
is greater than
E
. Classically, the particle cannot overcome the barrier, but quantum mechanically it can “tunnel” through it.
Tunneling
The Schr ödinger equation to solve: 2
d
2 Y 2
m dx
2 +
V
Y =
E
Y Energy
E
is
given
(continuous) and assumed
smaller
than
V.
Tunneling
The situation we describe: A particle flies in from the left.
Sometimes
it is bounced back by the left barrier;
sometimes
it passes through it. The particle can be bounced back by the right barrier.
What is the ratio of transmission versus reflection 
A
 2 /
A
’ 2 ?
Tunneling
The shape of the potential naturally divides the space into three regions. In each region, the potential is flat.
For a flat potential,
e ikx
and cos) and
e kx
(or equivalently sin are the most promising functional forms.
e ikx
: 
d
2
dx
2
e ikx e kx
: 
d
2
dx
2 = ( ) 2
e ikx e kx
= 
k
2
e kx
=
k
2
e ikx
Tunneling
Our intuitive picture of the solution:
e ikx
represents an incoming particle, with momentum +
ħk e –ikx
is a reflected particle, with momentum –
ħk e ikx
is a transmitted particle, with momentum +
ħk e –κx , e +κx
is a decaying function which connects the left and right wave functions.
Tunneling
The wave function
within
V
>
E
and
e –κx
and
e +κx
the barrier: Here,
are more suitable.
2 2
m d
2 Y Middle
dx
2 2
d e
x dx
2 2
d e ikx dx
2 2
e
x
2
k e ikx
= (
E

V
) Y Middle Negative Positive
Tunneling
Left
Ae ikx
Be
ikx
2 2
m d
2 Y Left
dx
2 =
k
2 2
m
2 Y Left
E
Middle
Ce
x
De
x
2 2
m d
2 Y Middle
dx
2 +
V
Y Middle = æ
V
ææ k 2 2
m
2 æ ææ Y Middle
= E < V
Right
ikx
2 2
m d
2 Y Right
dx
2 =
k
2 2
m
2 Y Right
E
Boundary conditions
The wave functions must be
continuous
and
smooth
(no infinity in potentials).
Continuous
C D
Smooth
C
D Ce
L
Continuous
De
L
ikL
Ce
L
Smooth
De
L
ikL k
2 2
m
2 =
E
We know
k
and
κ
.
V
k 2 2
m
2 =
E
Tunneling
Energy, thus,
k
and
κ
are known.
We have five unknowns:
A, B, C, D, A
΄.
We have four boundary conditions.
The fifth condition comes from the normalization, determining all five unknowns. We want to determine the transmission probability
A
΄

2 / 
A
 2 . In fact, with the four boundary conditions alone (without normalization), we can find four
ratios
:
B/A, C/A, D/A, A
΄
/A
.
Tunneling
A
¢
A
2 2 µ
e
2 k
L V
k 2 2
m
2 =
E
® k = 2
m
(
V

E
) For a high, wide barrier, the transmission probability
decreases
exponentially with
Thickness of barrier
,
Square root of particle mass
, Square root of energy deficit V
–E
.
Tunneling
Tunneling occurs when the particle does
not
have enough energy to overcome the barrier. Classically, this is impossible. Within the barrier, the kinetic energy would be negative.
Quantum mechanically, this is possible according to the Schrödinger equation. (Some argue that the particle can acquire a higher energy necessary to cross the barrier momentarily thanks to the timeenergy uncertainty principle).
Permeation
We have already seen (in harmonic oscillator problem) that a wave function can permeate in the classically forbidden region. This is by the same mechanism that tunneling occurs.
Ammonia inversion
Scanning tunneling microscope
Gold (100) surface Public image from Wikipedia
Summary
A particle of energy
E
can barrier of potential height
V
tunnel
>
E
.
through a A particle of energy
E
can also
permeate
the potential wall of
V
>
E
.
into The thinner the barrier or the lower the barrier height or the lighter the particle, the more likely the particle can
tunnel
.