Chapter 8 1

OutlineReading: Livingston, Chapter 8.1-8.7

Plancks Constant Wave-Particle Duality of Light Wave-Particle Duality of Electrons Wave-Particle Duality: Momentum and Energy Schrodingers Equation Probability Density Case 1: Free Electron Heisenbergs Uncertainty Principle Case 2: Potential Barrier (EV) Case 4: Tunneling Case 5: Infinite Potential Well Case 6: Finite Potential Well

Chapter 8 22

DiffractionWhat if wavelength of light ~ periodic atomic spacing?

Chapter 8 3

Planck: Energy of Electromagnetic Wave

hchE ==

=EPlancks Constant: 6.626 x 10-34 J-s

""2

barhh ==

Relationship between energy and frequency:

Chapter 8 4

Wave-Particle Duality of Light

http://phet.colorado.edu/en/simulation/photoelectric

1. Photoelectric Effect: e- ejected from matl surface when exposed to light

2. Compton Scattering: increase in the wavelength of light scattered by an e-

Effects which are NOT explained by wave properties:

particle-particle collision:electron and photon (light particle)

increase , decrease in E hcE =

hp =

Chapter 8 5

Wave-Particle Duality of ElectronsIf light can act as particles, can electrons act as waves?

de Broglies Hypothesis: e- can have wave-like nature defined by p = mvand p = h/Davisson and Germer Experiment:

sin2dn =Braggs Law of Diffraction: If e- is a wave and ~d, diffraction should be observed.

For an e- with an applied V, the energy is E= eV.

Need to apply correct V.

Chapter 8 6

Wave-Particle Duality: Momentum and Energy

Chapter 8 7

Example: Wave-Particle Duality

Calculate the wavelength of a 50 g golf ball traveling at a velocity of 20 m/s.

Chapter 8 8

Schrodingers Equation: Electron Wave Equation

dtdiV

m

=+ 22

2z

ky

jx

i

+

+

=

)(),,(),,,( tzyxtzyx =Manipulate equation so that it is easier to use:

Remember:

Separate variables:

dtdiV

m =+ 2

2

2

Divide by :dtdiV

m

=+ 22

2

LHS and RHS must equal constant, E:dtdiE

=

= diEdt ti

iEt

eet

==)(

=E

EVm

=+ 22

2 Time-independent Schrodingers Equation

(stationary states)

Multiply by :

m = mass of electronV = potential energy of electron

function waveelectron

Chapter 8 9

Schrodingers Equation: Electron Wave Equation (cont.)

EVm

=+ 22

2 Time-independent Schrodingers Equation

(stationary states)

EEPEK =+ ....

mpEK2

..2

=

= ipMomentum operator:

Please just believe for now

Chapter 8 10

Probability DensityElectron is BOTH particle AND wave = fuzzy in time and space

z

*

Chapter 8 11

Case 1: Free ElectronE

V(z)=0

z

*

1 Dimension

EVm

=+ 22

2

)(),,(),,,( tzyxtzyx = Time-dependent Scrodingers Equation)()(),( kztikzti BeAetx + +=

22* BA +=

Chapter 8 12

Heisenbergs Uncertainty Principle

z

*infinite uncertainty in position

hzp

Electron is BOTH particle AND wave = fuzzy in time and space

mkE

2

22=kp =

Other forms of uncertainty principle:

zp

htE

Chapter 8 13

Example

Laser light is normally monochromatic. However, when the pulse time becomes sufficiently short, the energy range can broaden to cover the entire range of visible light (and a laser beam becomes white). Below what pulse time will this phenomenon occur?

Chapter 8 14

Case 2: Potential Barrier (E

Chapter 8 1515

Case 2: Potential Barrier (E

Chapter 8 16

Case 3: Potential Barrier (E>V)

16

1 Dimension

EVm

=+ 22

2

EV=Vo

z=0

Region 1 Region 2

Region 1:zikzik BeAez 11)(1

+= mEk 21 =

Solutions to time-independent Schrodingers Equation

Region 2:zikzik DeCez 22)(2

+= ( )oVEmk

=

22

Boundary conditions: E>Vo , no reflected wave from right side, D=0zikCez 2)(2 =

K2 is REAL!!!

Chapter 8 17

Case 3: Potential Barrier (E>V)

17

1 Dimension

EVm

=+ 22

2

EV=Vo

z=0

Region 1 Region 2

Interface Continuity

@ z = 0, 1=2 CBA =+

@ z = 0, dzd

dzd 21

= ( ) CkBAk 21 =

VEEVEE

kkkk

AB

+

=

+

=

21

21

VEEE

kkk

AC

+=

+=

22

21

1

)*()*(

1

1

AAkBBkR =

)*()*(

1

2

AAkCCkT =

zikzikzik CeBeAe 211 =+

zikzikzik eCikeBikeAik 211 211 =

Ratio of reflected toincident amp

Reflection Coeff

Transmission CoeffRatio of transmitted toincident amp

Chapter 8 18

Example What do you expect

classically? What do you expect based

on quantum mechanics? Calculate the reflection

coefficient for the matter wave.

Chapter 8 19

Example (cont.) What do you expect

classically? What do you expect based

on quantum mechanics? Calculate the reflection

coefficient for the matter wave.

Chapter 8 20

Case 4: Tunneling (E

Chapter 8 21

Review

E

V(z)=0

EV=Vo

z=0

Region 1 Region 2

EV=Vo

z=0

Region 1 Region 2

1. What would you expect to happen classically?2. What happens quantum mechanically?

Chapter 8 22

Case 5: Infinite Potential Well

22

z=0 z=L

V=infinity V=infinityV=0

EVm

=+ 22

2

Solutions to time-independent Schrodingers Equation

Chapter 8 23

Case 5: Infinite Potential Well (cont.)

2

22

8mLhnEn =

Chapter 8 24

Example

1. Sketch a plot of n versus E for an infinite potential well.2. If the width of the well increases by a factor of 2, how

does the energy change?

Chapter 8 25

Paulis Exclusion PrincipleRule of quantum mechanics that allows only two electrons (one spin up and one spin down) to fill each energy level

Chapter 8 26

Case 6: Finite Potential Well (Quantum Well) How do energy wavelength and energy levels change qualitatively if potential barriers are not infinite?

Chapter 8 27

Review Questions

1. Name and explain two experiments that showed photons are particles.

2. Name and explain an experiment that showed electrons are waves.3. What relationship did Planck find between frequency and energy?

Momentum and wavelength?4. How are momentum and energy treated differently for particles and

waves?5. What is the physical meaning of wave function?6. What is Schrodingers Equation (S.E.)? What does each term

represent?7. What are general solutions to the time-dependent and time-

independent S.E.? What is the relationship between E and k?8. Give the general procedure for solving S.E. for a potential profile.9. What is Heisenbergs Uncertainty Principle tell us?

Chapter 8 28

More Review Questions

1. What are the allowed energies for an infinite potential well?

2. What is the ground state energy?3. What are the excited energy levels?4. How many electrons per energy level?5. How to electrons move and down in energy levels?6. If the potential barriers are made finite, how do the

energy levels and wavelength change?

Chapter 8 29

Important Equations

hchE ===E

Plancks Constant: 6.626 x 10-34 J-s

2h

=

hp = vmp =

2v2mE =

sin2dn =

dtdiV

m

=+ 22

2 EV

m=+ 2

2

2

tiiEt

eet

==)(

mE

2p2

=

*_ =distribprob

kp =

mkE

2

22=

hzp htE

zikzik BeAex 11)(1+=

VEEVEE

kkkk

AB

+

=

+

=

21

21

VEEE

kkk

AC

+=

+=

22

21

1

)*()*(

1

1

AAkBBkR =

)*()*(

1

2

AAkCCkT =

mLhnEn 8

22

=

= z

LnAnn sin ( )2228 fiif nnmL

hhE

=

=...3,2,1=n