15_01fig_PChem.jpg

Particle in a Box

( ) ( )n n nH x E x

2 2

2ˆ ˆ ˆ( ) ( )

2

dH K V x V x

m dx

0 0ˆ( )& 0

x aV x

x a x

2 2

2ˆFor 0

2

dx or x a H

m dx

ˆ ( ) ( ) ( ) Has to be finiteH x x E x

( ) 0x

15_01fig_PChem.jpg

Particle in a Box

2 22

2 2 2

2( ) 0 ( ) 0n

n n

mEd dx k x

dx dx

( ) ikx ikxn x Ae Be

22

2 nmEk

2 2

2n

kE

m

2 2

2ˆFor 0 ( ) ( ) ( )

2n n n n

dx a H x x E x

m dx

15_01fig_PChem.jpg

Particle in a Box

0 ( ) 0x or x a x

0 0(0) 0ik ikn Ae Be A B

( ) 2 sin( )n x iB kx

( ) 2 sin( ) 0n

na iB ka ka n k

a

( ) sin( )n

n xx C

a

2 2 2 2 2

2 22 8n

n n hE

ma ma

15_02fig_PChem.jpg

Wavefunctions for the Particle in a Box

( ) sin( )n

n xx C

a

*

0

( ) ( ) 1a

n nx x dx 2 2

0

sin ( )a n x

C dxa

2

2

aC

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

sxsx dx dx

1sin(2 )

2 4

xsx C

s

15_02fig_PChem.jpg

Wavefunctions are Orthonormal

*

0

( ) ( )a

n m nmx x *

0

2 2sin( ) sin( )

a n x m xdx

a a a a

0

2sin( )sin( )

a n x m xdx

a a a

0

1sin sin

a n m x n m xdx

a a a

15_02fig_PChem.jpg

Wavefunctions are Orthonormal

0

0

1cos

1cos

a

a

n m xa

a n m a

n m xa

a n m a

1 1

for 0 0 [0 0] 0a a

m na n m a n m

1 1

for 0 0 lim cos[ ] 0 1 12 m n

a am n n m

a n a 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 22sin ( )

f

i

x

x

n xdx

a a

For 0 <x < a/2 /22

0

2(0, / 2) sin ( )

a n xP a dx

a a

/2/22

0 0

2 1 1sin ( ) sin 2

2 4

aa n x a n x n x

a 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

15_02fig_PChem.jpg

Expectation Values

* ˆ( , ) ( ) ( )f

i

x

i f n m

x

O x x x O x dx 2 ˆsin( ) sin( )

f

i

x

x

n x n xO dx

a a a

Average position

0

2ˆsin( ) sin( )

a n x n xx x dx

a a a

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

15_02fig_PChem.jpg

Expectation Values

2

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 2

2

2 3

6

a

22x x x

2 2 2

2

2 3

6 4

a a

2

2

2 3 10.18

6 4a a

15_02fig_PChem.jpg

Expectation Values

0

2 2ˆsin( ) sin( )

a

x x

n x n xp p dx

a a a a

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 xp 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 xp p p odd even

15_02fig_PChem.jpg

Expectation Values

2 2

0

2 2ˆsin( ) sin( )

a

x x

n x n xp p dx

a a a a

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

2ˆ x

d d dp i i

dx dx dx

Uncertainty Principle

2 2 2

2x

n np

a a

2 2

2 2

2 3 2 31 1

6 4 6 4x

nx p a n

a

0.18 0.57 12

n n n

2

2

2 3 10.18

6 4x a a

Free Particle2 2

22 2 2

2( ) ( ) 0

mEd dx k x

dx dx

( ) 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

kE

m

2 2

2ˆ

2

dH

m dx

( ) i tt e

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.

( , ) ikx i t ikx i tx t A e e A e e

Two travelling waves moving in the opposite direction with velocity v.

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 dH V x y

m dx dy

0 0,0 , ,ˆ( , ), , & , 0,0

x y a bV x y

x y a b x y

ˆ ( , ) ( , )H x y E x y

( , ) 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

15_p19_PChem.jpg

For 0 , ,x y a b 2 2 2 2

2 2ˆ ˆ ˆ

2 2x y

d dH H H

m dx m dy

ˆ ˆ ˆ( , ) ( ) ( ) ( ) ( )x yH x y H H x y E x y

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

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 ( , , ) ( , , )H x y z E x y z

2 2 2 2

2 2 2ˆ ˆ ( , , )

2

d d dH V x y z

m dx dy dz

0 0,0,0 , , ,ˆ ( , , ), , , , & , , 0,0,0

x y a b cV x y z

x y z a b c x y z

ˆ ˆ ˆ ˆ( , , ) ( ) ( ) ( )x y zH x y z H H H x y z

( ) ( ) ( )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 nH x E x 2 2

2ˆ ˆ ˆ( ) ( )

2

dH K V x V x

m dx

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

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

mV E E V

( ) for / 2x xx Ae Be x a

( ) for / 2x xx 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’ and C are determined by Vo, m, a, and by the boundary and normalization conditions.

16_03fig_PChem.jpg

Core and Valence Electrons

Weakly bound states - W.Fns. extend beyond boundary.- Delocalized

(valence) - Have high energy.- Overlap with neighboring states of similar energy

Strongly bound states – W.Fns. 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)

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

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

B. Gap Al doped GaAs > B.Gap GaAs

C. Band GaAs < C.Band Al Doped GaAs

e’s in CB of GaAS in energy well.

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