Transcript 28_lecture_acl

Chapter 28: Quantum Physics
• Wave-Particle Duality
• Matter Waves
• The Electron Microscope
• The Heisenberg Uncertainty
Principle
• Wave Functions for a Confined
Particle
•The Hydrogen Atom
•The Pauli Exclusion Principle
•Electron Energy Levels in a Solid
•The Laser
•Quantum Mechanical Tunneling
For Wed recitation:
• Online Qs
• Practice Problems:
# 3, 6, 13, 21, 25
Lab: 2.16 (Atomic Spectra)
• Do Pre-Lab & turn in
• Next week optional 2.03
Final Exam: Tue Dec 11 3:305:30 pm @220 MSC
• 200 pts: Chs.25,27,28,(26)1
• 200 pts: OQ-like on 12,16-24
§28.1 Wave-Particle Duality
Light is both wave-like (interference & diffraction)
and particle-like (photoelectric effect).
Double slit experiment: allow only 1 photon
at a time, but:
• still makes interference pattern!
• can’t determine which slit it will pass thru
• can’t determine where it will hit screen
• can calculate probability:
• higher probability  higher intensity
• IE2, so E2 probability of striking at a
given location; E represents the wave
function.
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§28.2 Matter Waves
If a wave (light) can behave like a particle,
can a particle act like a wave?
Double slit experiment w/ electrons:
•  interference pattern! Wave-like!
Allow only 1 e– at a time:
• still makes interference pattern
• can still calculate probabilities
Add detector to see which slit used:
• one slit or other, not both
• interference pattern goes away!
• wave function “collapses” to particle!!
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Diffraction (waves incident on a crystal sample)
Electrons:
l  m  2dsin 
X-rays:
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Like photons, “matter waves”
have a wavelength:
“de Broglie wavelength”
h

p
p  mv
Momentum:
 by
Electron beam defined
accelerating potential,
gives them Kinetic Energy:

Note: need a relativistic
correction if v~c (Ch.26)
p  2mK
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Example (PP 28.8): What are the de Broglie wavelengths of
electrons with kinetic energy of (a) 1.0 eV and (b) 1.0 keV?
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
§28.3 Electron Microscope
Resolution (see fine detail):
asin  1.22
• visible light microscope
limited by diffraction to
~1/2  (~200 nm).
• much smaller (0.2-10 nm)
using a beam of electrons
(smaller ).
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Fig. 28.06
Transmission
Electr. Micr.
Scanning
Electr. Micr.
Example: We want to image a biological sample at a
resolution of 15 nm using an electron microscope.
(a) What is the kinetic energy of a beam of electrons with a
de Broglie wavelength of 15.0 nm?
(b) Through what potential difference should the electrons
be accelerated to have this wavelength?
-
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§28.4 Heisenberg’s Uncertainty Principle
Sets limits on how precise
measurements of a particle’s
position (x) and momentum (px)
can be:
Uncertainty
in position
& momentum
1
xp x  
2
h
where  
2
The energy-time
1
uncertainty
Et  .
2
principle:
x

Superposition
wave
packet
x
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Example: We send an electron through a very narrow slit
of width 2.010-8 m. What is the uncertainty in the
electron’s y-component momentum?
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Example: An electron is confined to a “quantum wire” of
length 150 nm.
(a)What is the minimum uncertainty in the electron’s
component of momentum along the wire?
(b)In its velocity?
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§28.5 Wave Functions for a Confined Particle

Analogy: standing wave
on a string:
2L
n 
n
Same for electron in a quantum
wire (particle in a 1D box), so

 n 
pn 
  h
n 2L 
h
& particle’s KE is

2 

h
2
E n  n 2

n
E1
2 
8mL 
Conclude: A confined particle
has quantized energy levels
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Electron cloud represents the electron probability density
for an H atom (the electron is confined to its orbit):
E  t 
1
.
2
Energy states
and durations
are “blurred”
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
Example: We want to image a biological sample at a
resolution of 15 nm using an electron microscope.
(a) What is the kinetic energy of a beam of electrons with a
de Broglie wavelength of 15.0 nm?
(b) Through what potential difference should the electrons
be accelerated to have this wavelength?
h
h
 
p
2mK
Square both sides, solve for K:
h2
(6.626 1034 Js) 2
K

=1.07x10-21 J = 0.0067 eV (low E!)
2
31
9
2
2m
2(9.1110 kg)(15 10 m)
(b) K final  U initial  qV  eV
so
V 
K 0.0067eV

e
e
= 0.0067 V (low Voltage, easy desktop machine!)
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Example: We send an electron through a very narrow slit
of width 2.010-8 m. What is the uncertainty in the
electron’s y-component momentum?
Key idea: electron goes through slit; maybe through center, or ±a/2 above/below it,
so use y = a/2! Then H.E.P. says
h so
ypy 
2
6.626 1034 Js
h
h
h
27 kgm
py 
 a 


5.3
10
s
2y 22  2a 2 (2.0 108 m)

Notice: This uncertainty in the electron’s vertical momentum means it can
veer off its straight-line course; many veered electrons  diffraction pattern!!
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Example: An electron is confined to a “quantum wire” of
length 150 nm.
(a)What is the minimum uncertainty in the electron’s
component of momentum along the wire?
(b)In its velocity?
Key idea: electron w/in wire; maybe at center, or ±l/2 from center, so use
x = l/2! Then use H.E.P.
6.626 1034 Js

h
h
h
28 kgm
px 
 l 


7
10
s
2x 22  2l 2 (150 109 m)
(b) Solve for the velocity:


p 6.626 1034 Js
v 

 770 ms  1 kms
31
m
9.1110 kg
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