Particle in a Box
H n ( x)  En n ( x)
2

Hˆ  K  Vˆ ( x) 
2m
 d2  ˆ
 dx 2   V ( x)


0 xa
0
ˆ
V ( x)  
 x  a & x  0
2

For x  0 or x  a Hˆ 
2m
 d2 
 dx 2     


Hˆ  ( x)   ( x)  E ( x) Has to be finite
  ( x)  0
15_01fig_PChem.jpg
Particle in a Box
2

For 0  x  a Hˆ n ( x) 
2m
 d2 
 dx 2  n ( x)  En n ( x)


 d 2 2mEn 
 d2
2
 dx 2  2  n ( x)  0   dx 2  k  n ( x)  0




k 
2
2
2
k
En 
2m
2mEn
2
 n ( x)  Ae
 ikx
 Be
15_01fig_PChem.jpg
ikx
Particle in a Box
x  0 or x  a  ( x)  0
 n (0)  Aeik 0  Beik 0  0  A   B
 n ( x)  2iB sin(kx)
n
 n (a)  2iB sin(ka)  0  ka  n  k 
a
n 2 2 2 n 2 h 2
En 

2
2ma
8ma 2
n x
 n ( x)  C sin(
)
a
15_01fig_PChem.jpg
Wavefunctions for the Particle in a Box
a
2
2 n x

(
x
)

(
x
)
dx

1

C
sin
n

 ( ) dx
n x
 n ( x)  C sin(
)
a
a
*
n
0
a
0
 sin
2
( sx) dx  

1  cos  2sx 
2
dx
x 1
 sin(2sx)  C
2 4s
a
n x
a  1 n x 1
 n x  
0 sin ( a ) dx   n  2 a  4 sin 2  a 0
a
2

a
2
C 
2
a
2
n x
  n ( x) 
sin(
)
a
a
15_02fig_PChem.jpg
Wavefunctions are Orthonormal
a
*

 n ( x) m ( x)   nm
0
 2
n x 
0  a sin( a )


a
*
2
m x
sin(
) dx
a
a
2
n x
m x
  sin(
) sin(
) dx
a0
a
a
a
a
  n  m x 
  n  m x 
1
  sin 
  sin 
 dx
a0
a
a




15_02fig_PChem.jpg
Wavefunctions are Orthonormal
  n  m x 
1
a
 
cos 

a   n  m  
a

  0
a
  n  m x 
1
a
 
cos 

a   n  m  
a

  0
a
for m  n
 1

1
a
a
 
[0  0]  0
 0  0   
a   n  m
 a  n  m

for m  n



1 a
a
1 

 
cos[n  m]   0  1  1
0  0  limmn  
a   2n  



 a   n  m

15_02fig_PChem.jpg
Particle in a Box Wavefunctions
2
n x
 n ( x) 
sin(
)
a
a
n2 h2
En 
8ma 2
16h 2
E4 
8ma 2
Pn ( x) 
Normalized
n=4
+
-
2 2 n x
sin (
)
a
a
+
Orthogonal
Node
9h 2
E3 
8ma 2
4h 2
E2 
8ma 2
2
# nodes= n-1
+
n=3
n>0
n=2
h
E1 
 0 n=1
8ma 2
Ground state
Wavelength
2a

n
15_03fig_PChem.jpg
+
+
Probabilities
xf
P ( xi , x f )    n* ( x) n ( x) dx 
xi
For 0 <x < a/2
2
P(0, a / 2) 
a
a /2


0
xf
2
2 n x
sin
(
) dx

a xi
a
a /2

sin 2 (
0
n x
) dx
a
a /2
n x
2 a  1 n x 1
 n x  
sin 2 (
)

sin
2


a
a  n  2 a
4
 a 0
 1 n (a / 2  0)


a
2  2


1
n 
 n a / 2 
 n 0   
 sin 2 
  sin 2 
 

4
a
a



 



2  n 1
 1

0

0
  

n  4 4
 2
15_02fig_PChem.jpg
Independent of n
Expectation Values
xf
xf
2
n x ˆ
n x
ˆ
O( xi , x f )    ( x)O m ( x) dx   sin(
)O sin(
) dx
a xi
a
a
xi
*
n
Average position
2
n x
n x
ˆ
x   sin(
) x sin(
) dx
a0
a
a
a
2
n x
n x
  sin(
) x sin(
) dx
a0
a
a
a
2
2
n

x
2
a
a
  x sin 2 (
) dx 

a0
a
a 4 2
a
Independent of n
15_02fig_PChem.jpg
Expectation Values
2
n x
n x
2
n x 2
n x
ˆ
ˆ
  sin(
) xx sin(
) dx   sin(
) x sin(
) dx
a0
a
a
a0
a
a
a
x2
a
2
n x
  x 2 sin 2 (
) dx
a0
a
a

2
2
2


3
a


x 

a
6 2
x
 x
2
 2
2
2
 3 a 2
6 2
2
2

  3
15_02fig_PChem.jpg
6 2
a2

4
1
  0.18a
4
Expectation Values
px 
px 2  px
a
px  
0
a

0
2i

a
2
2
n x
2
n x
sin(
) pˆ x
sin(
) dx
a
a
a
a
2
n x 
d 2
n x
sin(
)  i
sin(
)dx

a
a 
dx  a
a
n x d
n x
2i
sin(
)
sin(
)
dx

0
a dx
a
a
a
a
 sin(
0
n x  n
)
a  a
n x

cos(
) dx

a

2i n
n x
n x

sin(
)
cos(
) dx  0
2

a
a
a
0
a
odd
even
 px 
15_02fig_PChem.jpg
p x 2  02 
px 2
Expectation Values
a
px
2

0
2
n x 2 2
n x
ˆ
sin(
) px
sin(
) dx
a
a
a
a
d 
d

2
pˆ x  i
i




dx  
dx 

2

a
2

a
n x d 2
n x
sin(
)
sin(
) dx
2
0
a dx
a
2 a
n x  n2 2 
n x
0 sin( a )   a2  sin( a ) dx
2 a
2 2 n 2 2
2 2 n2 2 a
2 n x

sin (
) dx 

3

3
a
a
a
2
0
a
15_02fig_PChem.jpg
2
n2 2
a2
2
d
2
dx 2
Uncertainty Principle
px 
n
n

2
a
a
2 2
2
n
xpx 
a
a
x  a
2
2

  3
6 2
2
2

  3
6 2
1
  n
4
 0.18 n  0.57n 
2
1
  0.18a
4
2
2

  3
6 2
 n 1
1

4
Free Particle
 d2 
 dx 2 


2

Hˆ 
2m
k2 2
E
2m
 d 2 2mE
 dx 2  2

 ( x)  A e
 ( x, t )  A e

 d2
2
 ( x)   dx 2  k  ( x)  0



 ikx
 ikx  it
e
 A e
ikx
 (t )  eit
ikx  it
 A e e
Two travelling waves moving in the opposite direction with velocity v.
k
2mE
2
2m mv2
m2 v2 mv p 2



 
2
2
2

k is determined by the initial velocity of the particle, which can be any
value as there are no constraints imposed on it. This implies that k is a
continuous variable, which further implies that E ,  and  are also continuous.
This is exactly the same as the classical free particle.
Probability Distribution of a Free Particle
 ( x)  A e
ikx
  x  
Wavefunctions cannot be normalized over
Let’s consider the interval
P( x) 
L  x  L
 * ( x) ( x)
L

*

 ( x) ( x)dx

L
A A  e ikx eikx dx
L
L

L
A A e ikx eikx
A eikx A eikx
A e ikx A eikx dx
L

1
L
 dx
1

2L
L
The particle is equally likely to be found anywhere in the interval
Classical Limit
Probability distribution becomes continuous in the limit of infinite n,
and also with limited resolution of observation.
15_04fig_PChem.jpg
Particle in a Two Dimensional Box
Hˆ  ( x, y )  E  ( x, y )
a,b
0,b
y
2

Hˆ 
2m
 d2
d2  ˆ
 dx 2  dy 2   V ( x, y)


x
0,0
a,0
0, 0  x, y  a, b
0
ˆ
V ( x, y )  
 x, y  a, b & x, y  0, 0
( x, y)  0 for x, y  a, b & x, y  0,0
If ( x, y)   ( x) ( y)
 ( x)  0 for x  a & x  0
 ( y)  0 for y  b & y  0
15_p19_PChem.jpg
Particle in a Two Dimensional Box
2
2
2
2

d

d
Hˆ  Hˆ x  Hˆ y 

2
2m dx
2m dy 2
For 0  x, y  a, b
Hˆ ( x, y)   Hˆ x  Hˆ y  ( x) ( y)  E ( x) ( y)
 2 d2
 2 d2
 ( x) ( y) 
 ( x) ( y)  E ( x) ( y)
2
2
2m dx
2m dy
 2 d2
 2 d2
 ( y)
 ( x )   ( x)
 ( y)  E ( x) ( y)
2
2
2m dx
2m dy
2
d
d
 ( y)

( x)
2
2
2
2
 dx
 dy

E
2m  ( x )
2m  ( y )
2
15_p19_PChem.jpg
Particle in a Two Dimensional Box
d2
 ( x)
2
2
 dx
 Ex
2m  ( x )
d2
 ( y)
2
2
 dy
 Ey
2m  ( y )
 2 d2
 ( x)  Ex ( x)
2
2m dx
 2 d2
 ( y)  E y ( y)
2
2m dy
n y y
2
 n ( y) 
sin(
)
nx x
2
 nx ( x) 
sin(
)
a
a
nx 2 h 2
Enx 
8ma 2
y
Eny 
n y y
nx x
2
 ( x, y ) 
sin(
)sin(
)
a
b
ab
b
b
ny 2 h2
8mb 2
2
2


n
h nx
y
E
 2  2
8m  a
b 
2
Particle in a Square Box
2
13
1
n y y
nx x
2
 ( x, y )  sin(
)sin(
)
a
a
a
0
3
10
h2
2
2


E
n

n
y 
2  x
8ma
3
1
8
2
1
2
2
3
Quantum Numbers
Number of Nodes
2
26
2
5
1
5
Energy
4
1
1
2
Particle in a Three Dimensional Box
Hˆ ( x, y, z )  E ( x, y, z )
2

Hˆ 
2m
 d2
d2 d2  ˆ
 dx 2  dy 2  dz 2   V ( x, y, z )


0, 0, 0  x, y  a, b, c
0
ˆ
V ( x, y , z )  
 x, y, z  a, b, c & x, y, z  0, 0, 0
Hˆ  ( x, y, z )   Hˆ x  Hˆ y  Hˆ z  ( x) ( y ) ( z )
  Ex  E y  Ez  ( x) ( y ) ( z )
Particle in a Three Dimensional Box
 2 d2
 ( x)  Ex ( x)
2
2m dx
nx x
2
 nx ( x) 
sin(
)
a
a
nx 2 h 2
Enx 
8ma 2
n y y
2
 d
)
 ( y)  E y ( y)  n ( y)  sin(
2
b
b
2m dy
ny 2 h2
2
2
y
 d
 ( z )  Ez ( z )
2
2m dz
2
2
nz z
2
 nz ( z ) 
sin(
)
c
c
Eny 
nz 2 h2
Enz 
8mc 2
n y y
nx x
nz  y
 ( x, y , z ) 
sin(
)sin(
)sin(
)
a
b
c
abc
2 2
2
h2  nx 2 ny nz 2 
E
 2  2  2
8m  a
b
c 
8mb 2
Free Electron Models
R
6  electrons
R
L
2
2
nh
En 
8mL2
 n  n  h
E 
8mL2
2
L
2
H
2
LUMO
E
HOMO
nL  nH  1  nL2  nH2  2nH  1
 2nH  1 h2
E 
8mL2
Free Electron Models
 2nH  1 h2
hc
E 
 h 
2
8mL

E 
 2  2  1 6.626  10
34
kgm / s 
2
2
8(9.11 1031 kg )(723  1012 m) 2
 5.76  1019 J
hc  6.626  10 Js  2.99 10


E
 6.23 10 J 
34
8
m / s
19
max
nH = 2
345 nm
nH = 3
375 nm
nH = 4
390 nm
16_01tbl_PChem.jpg
 3.44  107 m
Particle in a Finite Well
2

Hˆ  K  Vˆ ( x) 
2m
H n ( x)  En n ( x)
a
a
 x
0
2
2
ˆ
V ( x)  
Vo x  a & x   a
2
2
a
a
For   x 
2
2
 2  d2 
 ( x)  En n ( x)

2 n
2m  dx 
n x
  n ( x)  C cos(
)
a
 d2  ˆ
 dx 2   V ( x)


Particle in a Finite Well
a
a
For x  & x  
2
2
Limited number of bound states. WF penetrates
deeper into barrier with increasing n.
 2 d2

 2m dx 2  V0  n ( x)  En n ( x)


 d 2 2m

 dx 2  2 V0  En  n ( x)  0



2m
2
V0  En  
if En  Vo
 ( x)  Ae x  Be x for x  a / 2
x
 ( x)  Ae  Be
 x
for x  a / 2
Classically forbidden region
as KE < 0 when Vo > En
A,B, A’ B’ and C are determined by Vo, m, a, and by the boundary and normalization conditions.
Core and Valence Electrons
Strongly bound states
(core)
Weakly bound states
(valence)
– W.Fns. are confined within the boundary
- Localized.
- Have lower energy
- W.Fns. extend beyond boundary.
- Delocalized
- Have high energy.
- Overlap with neighboring states of similar energy
Two Free Sodium Atoms
In the lattice
xe-lattice spacing
16_03fig_PChem.jpg
Conduction
Consider a sodium crystal sides 1 cm long.
Each side is 2X107 atoms long.
Unbound states
 2n  1 h2
E 
  2n  1 (6.03  1034 J )
2
8mL
Energy spacing is very small w.r.t, thermal
energy, kT.
E
kT
Valence States
(delocalized)
Bound States
(localized)
106  Energy levels form a continuum
Unoccupied Valence States - Band
Occupied Valence States- Band
electrons flow to +
16_05fig_PChem.jpg
Sodium atoms
increased occupation
of val. states on + side
Tunneling
 ( x)  Ae
 x
1
Decay Length = 1/


2
2m V0  En 
The higher energy
states have longer
decay lengths
The longer the decay
length the more likely
tunneling occurs
The thinner the barrier
the more likely
tunneling occurs
16_08fig_PChem.jpg
Scanning Tunneling Microscopy
Tip
Surface
work functions
no contact
Contact
Tunneling occurs
from tip to surface
Contact with Applied Bias
16_09fig_PChem.jpg
Scanning Tunneling Microscopy
16_11fig_PChem.jpg
Tunneling in Chemical Reactions
16_13fig_PChem.jpg
Quantum Wells
States are allowed
Empty in Neutral X’tal.
B. Gap Al doped
GaAs > B.Gap GaAs
No States allowed
C. Band GaAs < C.
Band Al Doped GaAs
States Allowed
Fully occupied
e’s in CB of GaAS in
energy well.
3D Box
a = 1 to 10 nm thick
b = 1000’s nm long & wide
Alternating layers
of Al doped GaAs
with GaAs
2
2
h 2  nx 2 ny  nz 
E
 2 

8m  a
b 2 
h2
E
8ma 2
 nx 2 ny 2  nz 2 



10002 
 1
 Eny ,nz  Enx
h2 nx 2
8ma 2
1D Box along x !!
Energy levels for y and z - Continuous
Energy levels for x - Descrete
16_14fig_PChem.jpg
Quantum Wells
finite
barrier
E
h2
2
2


E 
n

n
x ,VB 
2  x ,CB
8ma

h
 n
4ma
2
2
x ,CB
n
2
x ,VB
4mca 2

 h  nx2,CB  nx2,VB 

QW Devices can be manufactured to have
specific frequencies for application in Lasers.
Eex>Band Gap energy GaAS
Eex<Band Gap energy Al doped GaAS
16_14fig_PChem.jpg
Quantum Dots
Crystalline spherical particles1 to 10 nm in diameter.
Band gap energy depends on diameter
Easier and cheaper to manufacture
3D PIB
16_16fig_PChem.jpg
Quantum Dots
16_18fig_PChem.jpg