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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
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Chapter 8 22
DiffractionWhat if wavelength of light ~ periodic atomic
spacing?
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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:
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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 =
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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.
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Chapter 8 6
Wave-Particle Duality: Momentum and Energy
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Chapter 8 7
Example: Wave-Particle Duality
Calculate the wavelength of a 50 g golf ball traveling at a
velocity of 20 m/s.
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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
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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
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Chapter 8 10
Probability DensityElectron is BOTH particle AND wave = fuzzy in
time and space
z
*
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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 +=
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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
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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?
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Chapter 8 14
Case 2: Potential Barrier (E
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Chapter 8 1515
Case 2: Potential Barrier (E
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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!!!
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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
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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.
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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.
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Chapter 8 20
Case 4: Tunneling (E
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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?
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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
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Chapter 8 23
Case 5: Infinite Potential Well (cont.)
2
22
8mLhnEn =
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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?
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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
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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?
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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?
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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?
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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