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15_01fig_PChem.jpg
Particle in a Box
( ) ( )n n nx x H
2 2
2ˆ ˆ ˆ( ) ( )
2
dx x
m dx
H K V V
0 0ˆ( )& 0
x aV x
x a x
2 2
2ˆFor 0
2
dx or x a
m dx
H
ˆ ( ) ( ) ( ) Has to be finitex x x H
( ) 0x
Recall
15_01fig_PChem.jpg
Particle in a Box
2
2 2
2( ) 0n
n
mdx
dx
( ) ikx ikxn x Ae Be
22
2 nmEwhere k
2 2
2n
kE
m
2 2
2ˆ ( ) ( ) ( )
2n n n n
dx x x
m dx
H
For 0 x a
22
2( ) 0n
dk x
dx
15_01fig_PChem.jpg
Particle in a Box
For 0 ( ) 0x or x a x
0 0(0) 0ik ikn Ae Be A B
( ) 2 sin( )n x iB kx
( ) 2 sin( ) 0n a iB ka
nka n k
a
( ) sin( )n
n xx N
a
2 2 2 2 2
2 22 8n
n n hE
ma ma
Initial conditions
Recall
15_02fig_PChem.jpg
Wavefunctions for the Particle in a Box*
0
( ) ( ) 1a
n nx x dx 2 2
0
sin ( ) 1a n x
N dxa
2
1 2
2
aN
N a
2( ) sin( )n
n xx
a a
2
0 0
1 1sin ( ) sin 2
2 4
aa n x a n x n xdx
a n a a
2 1 cos 2sin ( )
2
mxmx dx dx
1sin(2 )
2 4
xmx C
m
Normalization
Recall
Thereforea
Recall
Recall
15_02fig_PChem.jpg
Wavefunctions are Orthonormal*
*
0 0
2 2( ) ( ) sin( ) sin( )
a a
n m
n x m xx x dx dx
a a a a
0
2sin( )sin( )
a n x m xdx
a a a
0
1cos cos
a n m x n m xdx
a a a
0
1sin sin
an m x n m xa a
a n m a n m a
+-
+Even
Odd
+-
+Even
Odd
+-
+-
0
1 1 1sin sinn m n m
n m n m
1sin sin cos( ) cos( )
2
0
1 1sin sin sin(0)
sin
n m n mn m n m
n m
n m
15_02fig_PChem.jpg
Wavefunctions are Orthonormal
*
0
1 1( ) ( ) 0 0 0
a
n mx x dx
For m nFor m n
0 0 0
0
0
sinlim
sinlim lim sin lim
lim cos
lim 1
n m
t t t
t
t
n m
n mt dd
t tdtt dt
t
sin 20
2
n
n
*
0
1 1( ) ( ) 0 1
a
n mx x dx
*,
0
( ) ( )a
n m n mx x
AND
sin n m
n m
00
n m
15_03fig_PChem.jpg
Orthogonal
Normalized+
-
Node
# nodes = n-1
n > 0
Wavelength
2a
n
+
+
+
+
2( ) sin( )n
n xx
a a
22
( ) sin ( )n
n xP x
a a
2 2
28n
n hE
ma
2
1 20
8
hE
ma
2
2 2
4
8
hE
ma
2
3 2
9
8
hE
ma
2
4 2
16
8
hE
ma
Ground state
Particle in a Box Wavefunctions
n=1
n=2
n=3
n=4
15_02fig_PChem.jpg
Probabilities
*( , ) ( ) ( )f
i
x
i f n n
x
P x x x x dx 2
*2 2 2
sin ( ) sin ( ) sin ( )f f
i i
x x
x x
n x n x n xdx dx
a a a a a a
/2
0
2 1 1sin 2
2 4
aa n x n x
a n a a
1 ( / 2 0)
221 / 2 0
sin 2 sin 24
n a
a
n a nna a
2 1 10 0
4 4 2
n
n
Independent of n
/22
0
2(0, / 2) sin ( )
a n xP a dx
a a
For 0 <x < a/2 2 1 cos 2sin ( )
2
mxmx dx dx
1sin(2 )
2 4
xmx C
m
Recall
15_02fig_PChem.jpg
Expectation Values* ˆ( , ) ( ) ( )
f
i
x
i f n n
x
x x x x dx O O2 ˆsin( ) sin( )
f
i
x
x
n x n xdx
a a a
O
Average position
0
2ˆsin( ) sin( )
a n x n xdx
a a a
x x
0
2sin( ) sin( )
a n x n xx dx
a a a
22
0
2 2sin ( )
4 2
a n x a ax dx
a a a
Independent of n
Recall
22
20 0
2
2
2 2
2
sin(2 ) cos(2 )sin ( )
4 4 4
sin(2 ) cos(2 ) 1
4 4 4
0 1 1
4 4 4 4
aa x x cx cxx cx dx
c c
a a ca ca
c c
a a a
c c
as 2ca=2n
From a table of integrals
15_02fig_PChem.jpg
Expectation Values2
0
2ˆˆsin( ) sin( )
a n x n xx xx dx
a a a
2
0
2sin( ) sin( )
a n x n xx dx
a a a
2 2
0
2sin ( )
a n xx dx
a a
2 3
2
2 32
12
a
a
22 x x x 2 2 2
2
2 3
6 4
a a
2
2
2 3 10.18
6 4a a
From a table of integrals or from Maple.
2 2
2
2 3
6
a
15_02fig_PChem.jpg
Expectation Values
0
2 2ˆsin( ) sin( )
a
x x
n x n xdx
a a a a
p p
0
2sin( ) sin( )
ai n x d n xdx
a a dx a
0
2 2sin( ) sin( )
a n x d n xi dx
a a dx a a
22x x x p p p
0
2sin( ) cos( )
ai n x n n xdx
a a a a
20
2sin( )cos( ) 0
ai n n x n xdx
a a a
2 2 20x x x p p podd even
15_02fig_PChem.jpg
Expectation Values
2 2
0
2 2ˆsin( ) sin( )
a
x x
n x n xdx
a a a a
p p
2 2
20
2sin( ) sin( )
a n x d n xdx
a a dx a
2 2 2
20
2sin( ) sin( )
a n x n n xdx
a a a a
2 2 22
30
2sin ( )
an n xdx
a a
2 2 2 2 2 2
3 2
2
2
n a n
a a
22 2
2ˆx
d d di idx dx dx
p
Recall
Uncertainty Principle
2 2 2
2x
n n
a a
p
2 2
2 2
2 3 2 31 1
6 4 6 4x
na n
a
x p
0.18 0.57 12
n n n
2
2
2 3 10.18
6 4a a
x
Free Particle
( ) ikx ikxx A e A e
2 2 2
2 2 2
2 2 v v v 2
2
mE m m m m pk
2 2
2
k
m
2 2
2ˆ
2
d
m dx
H
k is determined by the initial velocity of the particle, which can be any value as there are no constraints imposed on it. Therefore k is a continuous variable, which implies that E , and are also continuous. This is exactly the same as the classical free particle.
Two travelling waves moving in the opposite direction with velocity v.
22
2( ) 0
dk x
dx
Probability Distribution of a Free Particle
( ) ikxx A e
*
*
( ) ( )( )
( ) ( )
ikx ikx
L Likx ikx
L L
A e A ex xP x
x x dx A e A e dx
Wavefunctions cannot be normalized over x
Let’s consider the interval L x L
1 1
2
ikx ikx
L Likx ikx
L L
A A e e
LA A e e dx dx
The particle is equally likely to be found anywhere in the interval
15_04fig_PChem.jpg
Classical LimitProbability distribution becomes continuous in the limit of infinite n, and also with limited resolution of observation.
15_p19_PChem.jpg
2 2 2
2 2ˆ ˆ ( , )
2
d dx y
m dx dy
H V
0 0,0 , ,ˆ ( , ), , & , 0,0
x y a bx y
x y a b x y
V
ˆ ( , ) ( , )x y E x y H
( , ) 0 for , , & , 0,0x y x y a b x y
Particle in a Two Dimensional Box
If ( , ) ( ) ( )x y x y
( ) 0 for & 0x x a x
( ) 0 for & 0y y b y
x
y
0,0 a,0
0,b a,b
Product wavefunction
15_p19_PChem.jpg
For 0 , ,x y a b
ˆ ˆ ˆ( , ) ( ) ( ) ( ) ( )x yx y x y E x y H H H
Particle in a Two Dimensional Box
2 2 2 2
2 2( ) ( ) ( ) ( ) ( ) ( )
2 2
d dx y x y E x y
m dx m dy
2 2 2 2
2 2( ) ( ) ( ) ( ) ( ) ( )
2 2
d dy x x y E x y
m dx m dy
22
2 2 22( )( )
2 ( ) 2 ( )
dd yxdydx E
m x m y
Separable
Particle in a Two Dimensional Box2
2 2 ( )
2 ( ) x
dx
dx Em x
2
2 2 ( )
2 ( ) y
dy
dyE
m y
2 2
2( ) ( )
2 x
dx E x
m dx
2 2
2( ) ( )
2 y
dy E y
m dy
2( ) sin( )
x
xn
n xx
a a
2
( ) sin( )y
yn
n yy
b b
2( , ) sin( )sin( )yx
n yn xx y
a bab
2 2
28x
xn
n hE
ma
2 2
28y
yn
n hE
mb
222
2 28yxnnh
Em a b
2( , ) sin( )sin( )yx
n yn xx y
a a a
22 2
28 x y
hE n n
ma
Particle in a Square Box
1
1
2
3
1
3 2
2
5
1
1
2
0 3
2 2
4 1
2 13
10 8
26 5Quantum Numbers
Number of Nodes
Energy
Particle in a Three Dimensional Box
ˆ ( , , ) ( , , )x y z E x y z H
2 2 2 2
2 2 2ˆ ˆ ( , , )
2
d d dx y z
m dx dy dz
H V
0 0,0,0 , , ,ˆ ( , , ), , , , & , , 0,0,0
x y a b cx y z
x y z a b c x y z
V
ˆ ˆ ˆ ˆ( , , ) ( ) ( ) ( )x y zx y z x y z H H H H
( ) ( ) ( )x y zE E E x y z
Particle in a Three Dimensional Box
2 2
2( ) ( )
2 x
dx E x
m dx
2 2
2( ) ( )
2 y
dy E y
m dy
2( ) sin( )
x
xn
n xx
a a
2( ) sin( )
y
yn
n yy
b b
2 2( , , ) sin( )sin( )sin( )yx z
n yn x n yx y z
a b cabc
2 2
28x
xn
n hE
ma
2 2
28y
yn
n hE
mb
22 22
2 2 28yx znn nh
Em a b c
2 2
2( ) ( )
2 z
dz E z
m dz
2
( ) sin( )z
zn
n zz
c c
2 2
28z
zn
n hE
mc
Free Electron Models
R
R
L
6 electrons
HOMO
LUMO
E
2 2
28n
n hE
mL
2 2 2
28L Hn n h
EmL
2 21 2 1L H L H Hn n n n n
2
2
2 1
8Hn h
EmL
16_01tbl_PChem.jpg
Free Electron Models
2
2
2 1
8Hn h hc
E hmL
nH = 2
234 2
31 12 2
19
2 2 1 6.626 10 /
8(9.11 10 )(723 10 )
5.76 10
kgm sE
kg m
J
34 8
7
19
6.626 10 2.99 10 /3.44 10
6.23 10
Js m shcm
E J
345 nm
375 nm
390 nm
max
nH = 3
nH = 4
Particle in a Finite Well
( ) ( )n n nx E x H 2 2
2ˆ ˆ ˆ( )
2
dV x
m dx
H K V
0 2 2ˆ( )
&2 2
o
a ax
V xV a a
x x
2 2
2
For2 2
( ) ( )2 n n n
a ax
dx E x
m dx
( ) cos( )n
n xx C
a
Inside the box
Particle in a Finite Well
For &2 2
a ax x
2 2
02( ) ( )
2 n n n
dV x E x
m dx
2
02 2
2( ) 0n n
d mV E x
dx
02
2ifn n o
mk V E E V
( ) for / 2kx kxx Ae Be x a
( ) for / 2kx kxx A e B e x a Classically forbidden regionas KE < 0 when Vo > En
Limited number of bound states. WF penetrates deeper into barrier with increasing n.
A,B, A’ B’ & C are determined by Vo, m, a, and by the boundary and normalization conditions.
Note: not ikx!!!
16_03fig_PChem.jpg
Core and Valence Electrons
Weakly bound states - Wavefunctions extend beyond boundary.- Delocalized
(valence) - Have high energy.- Overlap with neighboring states of similar energy
Strongly bound states – Wavefunctioons are confined within the boundary- Localized.
(core) - Have lower energy
Two Free Sodium Atoms
In the lattice
xe-lattice spacing
16_05fig_PChem.jpg
Conduction
Bound States (localized)
Unbound states
Occupied Valence States- Band
Unoccupied Valence States - Band
electrons flow to +
increased occupation of val. states on + side
2
342
2 12 1 (6.03 10 )
8
n hE n J
mL
Consider a sodium crystal sides 1 cm long.Each side is 2x107 atoms long.
Sodium atoms
Energy spacing is very small w.r.t, thermalenergy, kT.
610E
kT
Energy levels form a continuum
Valence States (delocalized)
bias
16_08fig_PChem.jpg
Tunneling
Decay Length = 1/
( ) xx Ae
2
0
1
2 nm V E
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_09fig_PChem.jpg
Scanning Tunneling MicroscopyTip Surface
work functions
no contact
Contact
Contact with Applied Bias
Tunneling occurs from tip to surface
16_11fig_PChem.jpg
Scanning Tunneling Microscopy
16_13fig_PChem.jpg
Tunneling in Chemical Reactions
The electrons tunnel to form the new bond
Small tunnelling distance relatively large barrier
16_14fig_PChem.jpg
Quantum Wells
States AllowedFully occupied
No States allowed
States are allowedEmpty in Neutral X’tal.
Alternating layers of Al doped GaAswith GaAs
3D Box
a = 1 to 10 nm thickb = 1000’s nm long & wide
2 222
2 28y zxn nnh
Em a b
,y z xn n nE E Energy levels for y and z - Continuous
Energy levels for x - Descrete
1D Box along x !!2 22 2 22
2 2 28 1 1000 8y zx xn nn h nh
Ema ma
Band Gap of Al doped GaAs > Band Gap GaAs
Cond. Band GaAs < Cond. Band Al Doped GaAs
e’s in Cond. Band of GaAS in energy well.
Semi Conductor
16_14fig_PChem.jpg
Quantum Wells
finite barrier
22 2, ,28 x CB x VB
hE n n
ma
2 2, ,24 x CB x VB
hn n
ma
QW Devices can be manufactured to have specific frequencies for application in Lasers.
2
2 2, ,
4
x CB x VB
mca
h n n
Eex < Band Gap energy Al doped GaAS
Eex > Band Gap energy GaAS
E
16_16fig_PChem.jpg
Quantum DotsCrystalline spherical particles1 to 10 nm in diameter.
Band gap energy depends on diameter
Easier and cheaper to manufacture
3D PIB !!!
16_18fig_PChem.jpg
Quantum Dots
Quantum Dot Solar Cells
Dye Sensitized Solar Cell
Background
Organic Polymer Solar Cells
Fullerenes(Acceptor)
Organic polymer(Donor)
Organic polymersFullerene(PCBM)
