Download Tunneling through a Barrier

Chapter 8 Chapter 8 Quantum Theory:
Techniques and Applications (Part II)
The Particle in the Box and the Real World,
Phys. Chem. 2nd Ed.
T. Engel, P. Reid (Ch.16)
Objectives
• Importance of the concept for particle in the
box
• Understanding the tunneling of quantum
mechanical particles
Outline
1. The Particle in the Finite Depth Box
2. Pi Electrons in Conjugated Molecules Can Be
Treated as Moving Freely in a Box
3. Tunneling through a Barrier
4. The Scanning Tunneling Microscope
5. Tunneling in Chemical Reactions
8.5 The Particle in the Finite Depth Box
•
•
For a box to be more realistic, we let the box to
have a finite depth.
The potential is defined by
V ( x ) = 0, for a / 2 > x > − a / 2
V ( x ) = V0 , for x ≥ a / 2, x ≤ −a / 2
•
Outside the box,
ψ(x ) = Ae −κx + Be + κx for ∞ ≥ x ≥ a/2
ψ(x ) = A' e − κx + B' e + κx for - ∞ ≤ x ≤ −a/2,
where k =
2m (V0 − E )
h2
8.9 Tunneling through a Barrier
•
•
•
Consider a particle with energy E confined to a
very large box.
A barrier of height V0 separates two regions in
which E < V0.
The particle can escape the barrier and go over
the barrier, called tunneling.
8.9 Tunneling through a Barrier
•
•
To investigate tunneling, finite depth box is
modified by having a finite thickness.
The potential is now
V (x ) = 0, for x < 0
V (x ) = V0 , for 0 ≤ x ≤ a
V (x ) = 0, for x > a
where a = barrier width
Physical Chemistry Fundamentals: Figure 8.8 p.297
Fig. 8.8 A particle
incident on a barrier from
the left has an oscillating
wave function, but inside
the barrier there are no
oscillations (for E < V). If
the barrier is not too thick,
the wavefunction is
nonzero at its opposite
face, and so oscillates
begin again there. (Only
the real component of the
wavefunction is shown.)
Tunnelling
• The particle might be found on the outside of
a container even though according to classical
mechanics it has insufficient energy to escape.
Such leakage by penetration through a
classically forbidden region is called
tunnelling.
• Schrödinger eqn -> the probability of
tunnelling of a particle of mass m incident on
a finite barrier from the left. On the left of the
barrier (for x < 0) the wavefunctions of a
particle with V = 0,
Tunnelling
• Schrödinger equation for the region
representing the barrier (for 0 ≤ x ≤ L), where
the potential energy is the constant V
(E<V)
To the right of the barrier (x > L), where V = 0
again, the wavefunctions are
Tunnelling – Key points
• Tunnelling is the penetration into or through
classically forbidden regions. The transmission
probability decreases exponentially with the
thickness of the barrier and with the squareroot of the mass of the particle.
Physical Chemistry Fundamentals: Figure 8.9
Fig. 8.9. When a particle
is incident on a barrier
from the left, the
wavefunction consists of
a wave representing linear
momentum to the right, a
reflected component
representing momentum
to the left, a varying but
not oscillating component
inside the barrier, and a
(weak) wave representing
motion to the right on the
far side of the barrier.
• The acceptable wavefunctions must be
continuous at the edges of the barrier (at x =
0 and x = L, e0 = 1):
Physical Chemistry Fundamentals: Figure 8.10
The slopes of wavefunctions
(their first derivatives) must also
be continuous at the edges of the
barrier (at x = 0 and x = L)
Fig. 8.10 The wavefunction
and its slope must be
continuous at the edges of the
barrier. The conditions for
continuity enable us to connect
the wavefunctions in the three
zones and hence to obtain
relations between the
coefficients that appear in the
solutions of the Schrödinger
equation.
Tunnelling
• 4 eqns for 6 unknown coefficients. If the particles are
shot towards the barrier from the left, there can be no
particles travelling to the left on the right of the barrier.
Setting B′ = 0 removes 1 more unknown.
• We cannot set B = 0 because some particles may be
reflected back from the barrier toward negative x.
• The probability that a particle is travelling towards
positive x (to the right) on the left of the barrier is
proportional to |A|2, and the probability that it is travelling
to the right on the right of the barrier is |A′|2. The ratio
of these two probabilities is called the transmission
probability, T. (ε=E/V)
Physical Chemistry Fundamentals: Figure 8.11
Fig. 8.11 The transition probabilities for passage through a barrier. The horizontal axis
is the energy of the incident particle expressed as a multiple of the barrier height. The
curves are labelled with the value of L(2mV)1/2/ħ . The graph on the left is for E < V and
that on the right for E > V. Note that T > 0 for E < V whereas classically T would be
zero. However, T < 1 for E > V, whereas classically T would be 1.
Physical Chemistry Fundamentals: Figure 8.12
Fig. 8.12 The
wavefunction of a heavy
particle decays more
rapidly inside a barrier than
that of a light particle.
Consequently, a light
particle has a greater
probability of tunnelling
through the barrier.
Tunnelling
• For high, wide barriers (in the sense that κ L» 1), eqn 8.19a simplifies
to
(8.19b)
• The transmission probability decreases exponentially with the thickness
of the barrier and with m1/2.
• Particles of low mass are more able to tunnel through barriers than
heavy ones (Fig. 8.12).
• Tunnelling is very important for electrons and muons, and moderately
important for protons; for heavier particles it is less important.
• A number of effects in chemistry (e.g., the isotope-dependence of
some reaction rates) depend on the ability of the proton to tunnel
more readily than the deuteron.
• The very rapid equilibration of proton transfer reactions is also a
manifestation of the ability of protons to tunnel through barriers and
transfer quickly from an acid to a base. Tunnelling of protons between
acidic and basic groups is also an important feature of the mechanism
of some enzyme-catalysed reactions.
• Electron tunnelling is one of the factors that determine the rates of
electron transfer reactions at electrodes and in biological systems.
Physical Chemistry Fundamentals: Figure 8.13
The wavefunction penetrates into
the walls, where it decays
exponentially towards zero, and
oscillates within the well. The
wavefunctions and their slopes are
continuous at the edges of the
potential.
Fig. 8.13 A potential well with a finite depth.
Physical Chemistry Fundamentals: Figure 8.14
Finite number of bound
states - always at least
one bound state. N – the
number of levels:
The deeper and wider the
well, the greater the
number of bound states.
Fig. 8.14 The lowest two boundstate wavefunctions for a particle
in the well shown in Fig. 8.13.
Physical Chemistry Fundamentals: Figure 8.15
Fig. 8.15 A scanning
tunnelling microscope
makes use of the
current of electrons that
tunnel between the
surface and the tip.
That current is very
sensitive to the distance
of the tip above the
surface.
Physical Chemistry Fundamentals: Figure 8.16
Fig. 8.16 An STM
image of caesium
atoms on a gallium
arsenide surface.
The Scanning Tunneling Microscope
•
Tunneling through a Barrier
•
Scanning Tunneling Microscope (STM)
allows the imaging of solid surfaces with
atomic resolution with a surprisingly minimal
mechanical complexity.
The STM is used to study the phenomena at
near atomic resolution.
•
8.10 The Scanning Tunneling Microscope
•
Scanning Tunneling Microscope (STM)
Example
As was found for the finite depth well, the wave
function amplitude decays in the barrier according
to ψ (x ) = A exp [− 2m(V0 − E ) / h 2 x] . This result will be used
to calculate the sensitivity of the scanning
tunneling microscope. Assume that the tunneling
current through a barrier of width a is proportional
[
A exp − 2 2m(V0 − E ) / h 2 a
2
]
Example
a. If V0 − E is 4.50 eV, how much larger would the
current be for a barrier width of 0.20 nm than for
0.30 nm?
b. A friend suggests to you that a proton tunneling
microscope would be equally effective as an
electron tunneling microscope. For a 0.20-nm
barrier width, by what factor is the tunneling
current changed if protons are used instead of
electrons?
Solution
a. Putting the numbers into the formula given, we
obtain
(
(
)
)


I a = 2.0 × 10−10 m
2m(V0 − E )
−10
−10
=
exp
−
2
2
×
10
−
3
.
0
×
10


2
I a = 3.0 × 10−10 m
h


(
(
)
)
(
) (

2 9.11× 10− 31 (4.50) 1. 602× 10−19
−10
= exp − 2
×
−
1
.
0
×
10
2
1.055× 10− 34

= 8. 78
(
)



)
Even a small distance change results in a substantial
change in the tunneling current.
Solution
b. We find that the tunneling current for protons is
appreciably smaller than that for electrons.

2m proton(V0 − E ) 
exp  − 2
a
2
h
I ( proton)


=
I (electron )

2melectron(V0 − E ) 
exp  − 2
a
2
h



2(V0 − E )
= exp  − 2
h2

(
(
m proton − melectron

2(4.50 ) 1.602 ×10 −19
= exp  − 2
−34 2

1.055 ×10
(
)
)(
)a 

1.67 ×10
− 27
− 9.11×10
−31
)× (2.0 × 10
= 1.23 ×10 −79
This result does not make the proton tunneling
microscope look very promising.
−10



)
Tunneling
•
•
•
Most chemical reactions proceed faster as
the temperature of the reaction mixture is
increased.
This is due to energy barrier which must be
overcome in order to transform reactants
into products.
This barrier is referred to as the activation
energy for the reaction.
Tunneling