Chapter 8 1

OutlineReading: Livingston, Chapter 8.1-8.7

• Planck’s Constant• Wave-Particle Duality of Light• Wave-Particle Duality of Electrons• Wave-Particle Duality: Momentum and Energy• Schrodinger’s Equation• Probability Density• Case 1: Free Electron• Heisenberg’s Uncertainty Principle• Case 2: Potential Barrier (E<V)• Case 3: Potential Barrier (E>V)• 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

λυ hchE ==

ω=E

Planck’s Constant: 6.626 x 10-34 J-s

""2

barhh −==π

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 mat’l 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 λ, decrease in EλhcE =

λhp =

Chapter 8 5

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

de Broglie’s Hypothesis: e- can have wave-like nature defined by p = mvand p = h/λDavisson and Germer Experiment:

θλ sin2dn =Bragg’s Law of Diffraction:If e- is a wave and λ~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

Schrodinger’s Equation: Electron Wave Equation

dtdiV

mΨ=Ψ+Ψ∇−

22

2z

ky

jx

i∂∂+

∂∂+

∂∂=∇

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

Remember:

Separate variables:

dtdiV

mωψψωψω

=+∇− 22

2

Divide by ωψ:dtdiV

mω

ωψ

ψ =+∇− 2

2

2

LHS and RHS must equal constant, “E”:dtdiE ω

ω=

= ωω

diEdt

tiiEt

eet ωω −−

== )(

ω=E

ψψψ EVm

=+∇− 22

2 Time-independent Schrodinger’s Equation

(stationary states)

Multiply by ψ:

m = mass of electronV = potential energy of electron

Chapter 8 9

Schrodinger’s Equation: Electron Wave Equation (cont.)

ψψψ EVm

=+∇− 22

2 Time-independent Schrodinger’s Equation

(stationary states)

ψψψ 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

ΨΨ *

Chapter 8 11

Case 1: Free ElectronE

V(z)=0

z

ΨΨ *

1 Dimension

ψψψ EVm

=+∇− 22

2

)(),,(),,,( tzyxtzyx ωψ=Ψ Time-dependent Scrodinger’s Equation

)()(),( kztikzti BeAetx +−−− +=Ψ ωω

22* BA +=ΨΨ

Chapter 8 12

Heisenberg’s Uncertainty Principle

z

ΨΨ *infinite uncertainty in position

hzp ≥ΔΔ

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

mkE

2

22=kp =

Other forms of uncertainty principle:

≥ΔΔ zp

htE ≥ΔΔ

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<V)

14

1 Dimension

ψψψ EVm

=+∇− 22

2

EV=Vo

z=0

Region 1 Region 2

Region 1:zikzik BeAex 11)(1

−+=ψ

mEk 21 =

Solutions to time-independent Schrodinger’s Equation

Region 2:zikzik DeCex 22)(2

−+=ψ ( )

oVEmk

−=

22

Boundary conditions:

Chapter 8 1515

Case 2: Potential Barrier (E<V) (cont.)

ψψψ EVm

=+∇− 22

2

EV=Vo

z=0

Region 1 Region 2

Interface Continuity

1 Dimension

Chapter 8 16

Case 3: Potential Barrier (E>V)

16

1 Dimension

ψψψ EVm

=+∇− 22

2

EV=Vo

z=0

Region 1 Region 2

Region 1:zikzik BeAex 11)(1

−+=ψ

mEk 21 =

Solutions to time-independent Schrodinger’s Equation

Region 2:zikzik DeCex 22)(2

−+=ψ ( )

oVEmk

−=

22

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

K2 is REAL!!!

Chapter 8 17

Case 3: Potential Barrier (E>V)

17

1 Dimension

ψψψ EVm

=+∇− 22

2

EV=Vo

z=0

Region 1 Region 2

Interface Continuity

@ x = 0, ψ1=ψ2 CBA =+

@ x = 0,dz

ddz

d 21 ψψ = ( ) CkBAk 21 =−

VEEVEE

kkkk

AB

−+−−=

+−=

21

21

VEEE

kkk

AC

−+=

+= 22

21

1

)*()*(

1

1

AAkBBkR =

)*()*(

1

2

AAkCCkT =

zikzikzik CeBeAe 211 =+ −

zikzikzik eCikeBikeAik 211211 =− −

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<V)

20

1 Dimension

ψψψ EVm

=+∇− 22

2

Region 1:zikzik BeAex 11)(1

−+=ψ

mEk 21 =

Solutions to time-independent Schrodinger’s Equation

Region 2:( )

oVEmk

−=

22

Region 3:zikFex 1)(3 =ψ

mEk 21 =

zikzik DeCex 22)(2−+=ψ

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

ψψψ EVm

=+∇− 22

2

Solutions to time-independent Schrodinger’s 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

Pauli’s 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 Schrodinger’s 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 Heisenberg’s 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

λυ hchE ==ω=E

Planck’s Constant: 6.626 x 10-34 J-s

π2h=

λhp = vmp =

2v2mE =

θλ sin2dn =

dtdiV

mΨ=Ψ+Ψ∇−

22

2ψψψ EV

m=+∇− 2

2

2

tiiEt

eet ωω −−

== )(

mE

2p2

=

ψψ *_ =distribprob

kp =

mkE

2

22=

hzp ≥ΔΔ htE ≥ΔΔ

zikzik BeAex 11)(1−+=ψ

VEEVEE

kkkk

AB

−+−−=

+−=

21

21

VEEE

kkk

AC

−+=

+= 22

21

1

)*()*(

1

1

AAkBBkR =

)*()*(

1

2

AAkCCkT =

mLhnEn 8

22

=

= z

LnAnn

πψ sin ( )2228 fiif nn

mLh

hE −=Δ=ν...3,2,1=n