Lecture 16
Tunneling
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus
Foundation, Inc. through the Camille Dreyfus Teacher-Scholar 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 Schrödinger equation to solve:
d 2Y
+V Y = EY
2
2m dx
2

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, eikx (or equivalently sin
and cos) and ekx are the most promising
functional forms.
2
2 ikx
d
ikx
ikx
e : - 2 e = - ( ik ) e = k 2eikx
dx
2
d
ekx : - 2 ekx = -k 2ekx
dx
Tunneling

Our intuitive picture of the solution:
eikx represents an
incoming particle,
with momentum +ħk
e–ikx is a reflected
particle, with
momentum –ħk
eikx 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 the barrier: Here,
V > E and e–κx and e+κx are more suitable.
-
2
2m
2

d e
dx
2

d e
dx
d Y Middle
2
dx
2
= (E -V )Y Middle
x
  e
2
2
x
Negative
ikx
k e
2
2
ikx
Positive
Tunneling
 Left  A e
-
2
d 2 Y Left
2m dx
2
ikx
 Be
 ikx
 R ig h t  A  e
k2 2
=
Y Left
2m
-
 M iddle  C e
E
-
2
2m
d 2 Y Middle
dx
2
x
+V Y Middle
 De
2
2m
d 2 Y Right
dx 2
k2 2
=
Y
2m Right
 x
æ
k2 2ö
= çV Y Middle
÷
2m
è
ø
=E<V
ikx
E
Boundary conditions

The wave functions must be continuous and
smooth (no infinity in potentials).
Continuous
AB C D
Continuous
Ce
L
k2 2
= E We know k and κ.
2m
 L
 A e
ikL
Smooth
Smooth
ikA  ikB   C   D
 De
 Ce
V-
L
  De
k2
2
2m
 L
=E
 ikA  e
ikL
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¢
2
A
2
µe
-2k L
V-
k2
2
2m
= E ®k =
2m(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 time-energy
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 tunnel through a
barrier of potential height V > E.
A particle of energy E can also permeate into
the potential wall of V > E.
The thinner the barrier or the lower the barrier
height or the lighter the particle, the more
likely the particle can tunnel.