Particle in a finite 1-D box
For E > V0
Wave functions are oscillatory
functions of x
Sums of exponential functions with
imaginary arguments
Amplitudes are related by boundary
conditions

Physical Chemistry


For E  , it approaches the
behavior of the free particle in one
dimension
Lecture 16
Using the particle in a box to
understand special problems
Oscillatory behavior
Deviation from the free particle
depends on energy relative to the
potential


For  a / 2  x  a / 2
 ( x) 




Very different from predictions
of classical mechanics
Equations in the three regions
look similar
 Solutions are sums of
exponential functions
 Different kinds of solutions
for


 E < V0 and E > V0

Quantum conditions are
determined by boundary
conditions
k ik ' x
e
k'

2mE
2
2m( E  V0 )
2
Compelx matching conditions
For x  a / 2 and x  a / 2
 ( x)  Be  k '|x|
Classically the “external regions are “forbidden”
Quantum mechanically the particle is allowed to
be in this classically forbidden region
The amount of penetration depends on the
energy
Tunneling – appearance in regions where
classically not allowed
For  a / 2  x  a / 2
 ( x)  De ikx
 Ee ikx
where
k
for x  a / 2 and x  a / 2


For " external " regions

E  V0 
2m(V0  E )
2
In the low-energy regime
(E < V0)

 d 

2m dx 2
2mE
2
Particle in a finite 1-D box

 E

k' 
for  a / 2  x  a / 2
2 d 2
2m dx 2
2
The solutions in the box are oscillatory
The solutions in the “external” regions decay
exponentially
Matching at boundaries gives limits on possible
values of E



For  a / 2  x  a / 2
2
 B
Different from classical picture
Particle in a finite 1-D box

k ik ' x
e
k'
Particle in a finite 1-D box
For E < V0
Schroedinger’s equation in the
three regions looks similar
With a finite potential in the
“external” regions, the solutions
– even for E < V0 include the
possibility of the particle being
in those regions
A
where
k' 
With a finite potential in the
“external” regions, the wave
function can be nonzero in
those regions
Must solve Schroedinger’s
equation piecewise in three
regions
 Match the solutions at the
boundaries between regions
 Match derivatives at the
boundaries
Uses symmetry to simplify
problem
 Puts origin at the center of
the box, rather than one
V ( x)  V0
edge
 Solutions can be classified
 0
as even or odd under
inversion through zero
 Be ikx
For " external " regions
 ( x) 
k
Particle in a finite 1-D box
Ae ikx
Only certain allowed energies
Quantum nature apparent for
small boxes
Tunneling into the nearby
external space for finite V0
As the box becomes large,
the spacing between states
becomes small
 Model of translation of an
ideal gas molecule
In the high-energy regime
(E > V0)


Essentially a free particle
with a large number of
available energy states
Often called “the continuum”
1
Metals, semiconductors,
insulators
Two adjacent potential wells
Electrical conductivity
Solve Schroedinger’s equation
subject to boundary conditions
at each discontinuity
Qualitative picture

Solutions depend on the width
of the barrier
Metals conduct electricity when subject to a voltage
Insulators do not conduct electricity, even under a voltage


This phenomenon can be understood by a simple model based on the particle in a box
There are many quantum states closely spaced in energy that form bands of energy levels







 For a low, narrow barrier, the
solution must go over to that
of a single box with a small
perturbation
 For wide enough (or high
enough) barrier, solutions go
over to a wave function that
has separate lobes in each
box


Divided between core and valence states
Some states from different configurations are separated in energy
Whether bands are close in energy determines the ease of conduction

If the valence band is only partially filled, the material is a metal
If the valence band is filled AND there are no bands at nearby energies, the material is an insulator
If the lowest valence band is filled but there are states nearby, the material is a semiconductor
Depends on the band gap of the material
Wave functions are
approximately sums and
differences of such lobal
structures
Wave functions classified
as

Symmetric

Antisymmetric
 electrons
Two adjacent potential wells
Solution to Schroedinger’s
equation for E < V0 approximated
as a sum or difference of wave
functions for the two separate
wells
Qualitative picture
In the barrier region and nearby,
the wave functions overlap


Amount of overlap depends on the
barrier width and height
Overlap provides “spreading” of the
wave function in space to span both
wells
Procedure provides a means of
thinking about the manner in
which electron wave functions on
atoms delocalize to form
molecular orbitals when the two
atoms are brought into contact

Atomic centers at lattice
points
Particle-in-box potential
with a periodic
corrugation
For states at energies
above the corrugation
limit



Considered as a 1  D box
E  E f  Ei


h2
n 2f  ni2
8ma 2

Linear combination of atomic orbitals
(LCAO-MO)
Solid structures

Delocalized over a large range in, e.g.,
aromatic molecules
Particle-in-a-box model approximately
represents the situation of  electrons
Estimate “size” of the box from
spectroscopy of the * transition
Wave functions similar to
those of a particle in a
very large box
Wave functions extend
over the whole box
A huge number of states
close in energy because
the box is large
Virtual continuum of states
Tunneling through a barrier
Earlier model shows that a finite
potential gives the possibility of
finding the particle in the classically
forbidden region
Narrow finite potential: a barrier
Particle entering from left with
energy less than V0



Encounters barrier
Probability in barrier region finite
Probability of finding particle on
right side of barrier is finite
Probability of transmission depends
on



Potential amplitude
Energy of particle
Width of barrier
Characterized by the decay length,
barrier

d

 0 e x
1


2
2m(V0  E )
2
Tunneling junction
Use of the concept of
tunneling through a
barrier
Involves two adjacent
metals that are
separated by a small
distance (< 1 micron)
Applied voltage allows
tunneling of electrons
through the barrier
between the two metals
Summary
Models can be used to approximate many
situations
Particle-in-a-box model useful for
understanding



Conjugated  systems
Metallic conduction
Quantum dots
Tunneling into or through a barrier useful
for



Junctions
Scanning tunneling microscopy
Certain chemical reactions
Scanning tunneling microscopy
Use tunneling to
determine surface
structure
Run in the constanttunneling-current mode


Height adjusted to give
constant tunneling current
and recorded as a
function of position
Display of height versus
position gives an
appearance of the
“surface”
Can “see” atoms in the
surface
Scanning tunneling microscopy
Adsorption of
palladium
phthalocyanine
on graphite
Shows regular
array structure
Resolution to the
atomic level
3