Physics 222
David D. Allred
Adapted fromAdapted from
Wavelike properties of
matter
Class 8-1-4: (ThT Q)
Did you complete at least
70% of Chapter 3: 1-3?
A.
Yes B. No
Adapted fromAdapted from
Review
In our reference frame the
beam droops
It happens near stars, too
 http://www.theory.caltech.edu/people/pat
ricia/lclens.html
 And helps show dark matter.
http://apod.nasa.gov/apod/ap080917.html

Adapted fromAdapted from
Gravitational lensing
http://www.nature.com/nature/journal/v417/n6892/fig_tab/417905a_F1.html
Adapted
fromAdapted from
http://www.utahskies.org/HST/Archives/misc.html
The Gravitational Lens G2237 + 0305
The European Space Agency's Faint Object
Camera on board NASA's Hubble Space
Telescope has provided astronomers with the
most detailed image ever taken of the
gravitational lens G2237 + 0305—sometimes
referred to as the "Einstein Cross". The
photograph shows four images of a very distant
quasar which has been multiple-imaged by a
relatively nearby galaxy acting as a gravitational
lens. The angular separation between the upper
and lower images is 1.6 arc seconds.
The quasar seen here is at a distance of
approximately 8 billion light years, whereas the
galaxy at a distance of 400 million light years is
20 times closer. The light from the quasar is bent
in its path by the gravitational field of the galaxy.
This bending has produced the four bright outer
images seen in the photograph. The bright central
region of the galaxy is seen as the diffuse central
Adapted
fromAdapted from
object.
Gravitational lensing
http://www.astronomy.org.nz/aas/MonthlyMeetings/MeetingOct2002.asp
http://www.physics.brown.edu/physics/demopages/Demo/astro/demo/grvlns8.jpg
Adapted fromAdapted from
de Broglie
Photons: p=h/λ
Particle: λ=h/p
Hint: If the particle is
going slow
2
(K=½p /m)
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains why the
Compton effect suggests that light is
quantized.
Adapted fromAdapted from
Puzzles at the Beginning of the
Twentieth Century

Null result of the Michelson-Morley Experiment

Ultraviolet Catastrophe

Photoelectric Effect

Maxwell’s Equations Spell the Demise of Atoms!
 Discrete atomic emission lines

Adapted fromAdapted from
Radiating Atoms
Adapted fromAdapted from
Puzzles at the Beginning of the
Twentieth Century

Null result of the Michelson-Morley Experiment

Ultraviolet Catastrophe

Photoelectric Effect

Maxwell’s Equations Spell the Demise of Atoms!
 Discrete atomic emission lines

Adapted fromAdapted from
The Hydrogen Spectrum
The Balmer Series
1 1 
 R  2 

4 n 
1
Adapted fromAdapted from
Quiz Question
Where did de Broglie get his equation for the wavelength of a
massive particle?
A – From special relativity
B – It is the same equation used for light.
C – It is the same equation used for sound waves traveling
through a medium with mass.
D – From the principle of least action
E – He found it on the internet.
Adapted fromAdapted from
Louis de Broglie
If light, which we
thought of as a
wave, behaves as
a particle, then
maybe things we
think of as
particles behave
as waves…
photo from http://www.aip.org/history/heisenberg/p08.htm
Adapted fromAdapted from
Energy/Frequency and
Momentum/Wavelength Relations for a
Photon
E  hf
or
E
f 
h
Adapted fromAdapted from
p?
Remember from 220 ...
E hf h

p 
c 
c
p
h

or
h

p
Energy/Frequency and
Momentum/Wavelength Relations for an
Electron/Proton/Apple Pie/Ford Taurus
E  hf
Adapted fromAdapted from
p
h

or
or
E
f 
h
h

p
Day 9: Vernal equinox
09-22-2008 @9:39 MDT
3.4 Phase and Group Velocities
p 99
–A group of waves need not have the same velocity
as the waves themselves
3.5 Particle Diffraction p 104
–An experiment that confirms the existence of de
Broglie waves
3.6 Particle in a Box
p 106
–Why the energy of a trapped particle is quantized
(3.2 Waves of What?
Waves of probability)
Public Star Party and
Opening Social
• Get to know the night sky
When: Friday, Sept. 26
8 - 11pm
Where: Big Springs Park
What: Telescopes, dark sky, food
• Look through BIG telescopes
• View spiral structure in galaxies!
• See nebulas of all types
• Watch the moons of Jupiter move
• And a whole lot more (no, really,
we’re not just saying this - there
really is a whole lot more!)
Directions to Big Springs Park:
Head up Provo Canyon. Take a right at Vivian Park. Go through Vivian Park and up the canyon a
little over 3 miles. Look for a sign for the star party.
Adapted fromAdapted from
Dress warm! Bring your friends.
Did you
complete at
least 50%
of Chapter
3: 4-6?
A.
Yes B. No
Review
Adapted fromAdapted from
Draw light cone

For electron accelerated to 511 keV in 20
cm
Adapted fromAdapted from
The Wave Equation:
Which satisfy the Wave equation? pp
A. y = y0sin(kx- ωt)
B. y = y0e-a(x- vt)²; (Gaussian)
C. Any f(kx- ωt)
D. All three
E. Only A and B
Adapted fromAdapted from
Adapted fromAdapted from
3.4 Phase and
Group Velocities
A group of waves need
not have the same
velocity as the waves
themselves
Adapted fromAdapted from
Consider this experiment:

Throwing a pebble in a still pool.

http://en.wikipedia.org/wiki/Group_velocity
Note green dots mark the beginning and
ends of the group of waves but the red
dots mark the top of a given wave
 If you only had one group of waves, what
would it be like?
 Why does this work this way?

Adapted fromAdapted from
Dispersion





Prism: breaks
up white light.
How?
Refractive index
n and velocity
of light?
n= c/v
v(λ) = c/n(λ)
v(k) = c/n(k)
Adapted fromAdapted from
What does dispersion have to do
with Matter-waves?
Dispersion means that the velocity
depends on the wavelength, or k or
frequency.
 Look at space-time diagrams. (On
blackboard)
 Definition of phase and group velocities.
 Problem 6-1 (draw the dispersion
relationship)

Adapted fromAdapted from
(draw the dispersion relationship)

Which one?
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Just a note...
For a photon
E  hf 
hc

and hc  1239.8 eV nm
But...
hv
hc
E  hf 

unless v  c


Adapted fromAdapted from
What Exactly is Waving?

For a photon...
– electric and magnetic fields
– You can measure them if º is small enough.
– For visible light, you can see that it is a wave
indirectly.

For a massive particle
– You can’t measure them --- even in theory!
– They are complex!
– How do we know that there’s really a wave?
Adapted fromAdapted from
How might I verify that my Ford is
a wave?
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Thought Question

Which of the following would be the easiest
particle to use if I wanted to see a matter-wave
diffraction pattern?
1: A car moving at 100 mph
2: A car moving at 1 mph
3: A 1 MeV electron
4: A 1 keV electron
5: What was the question?
Adapted fromAdapted from
Wavelength of a Ford
h h
 
p mv
m  3336 lb  1.5 10 kg
3
v  10 m/s
34
  4.4 10 m
3
Adapted fromAdapted from
Wavelength of a 10 eV Electron
h
h
 
p mv
1
.
88

10
m/s
v
7
Adapted fromAdapted from
m  9.110
31
kg
  0.39 nm
3.5 Particle Diffraction
An experiment that
confirms the existence of
de Broglie waves
Adapted fromAdapted from
Adapted fromAdapted from
Davisson and Germer
Adapted fromAdapted from
photo from http://faculty.rmwc.edu/tmichalik/davisson.htm
Quiz Question
Why did Davison and Germer
heat their nickel target?
A – To induce thermal emission of
electrons
B – To remove oxide contamination
C – To study the thermal expansion
coefficient of pure nickel
D – To produce blackbody radiation
E – To keep their graduate students from
sitting on it (ouch, that’s hot!)
Adapted fromAdapted from
Bragg Diffraction
Adapted fromAdapted from
Bragg Diffraction
Why so many peaks?
Adapted fromAdapted from
1. Many orders (n=1,2,3,4,...)
2. Many Bragg planes
Adapted fromAdapted from
Bragg Diffraction
Adapted fromAdapted from
Adapted fromAdapted from
Scanning the Energy of
the Electrons
Adapted fromAdapted from
Adapted fromAdapted from
de Broglie
Photons: p=h/λ
Particle: λ=h/p
Hint: If the particle is
going slow
2
(K=½p /m)
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains how the
Davisson-Germer experiment shows that
electrons are waves.
Adapted fromAdapted from
Cesium Interferometer
f2
f3
p/2
f1
p/2
p
Normalized signal
1
0
-1
-10
-5
0
5
10
15
Rotation rate (x10-5) rad/sec
Adapted fromAdapted from
20
Interference of BEC
Adapted fromAdapted from
C60 Interference
Recent results from Vienna group of Anton Zielinger:
The interfering
particle:
Buckyballs
Apparatus
Interference fringes!
Adapted fromAdapted from
http://www.quantum.univie.ac.at/
Not only more mass,
but more degrees of
freedom too!
3.6 Particle in a Box
p. 106
Why is the energy of a trapped
particle quantized?
Rather talk about this now we should spend
some more time with waves.
Adapted fromAdapted from
(3.2 Waves of What?
Waves of probability)
Extra Credit Activity (2 points) SIM: DavissonGermer: Electron Diffraction
 Run the simulation found at
 http://phet.colorado.edu/new/simulations/sims.p
hp?sim=DavissonGermer_Electron_Diffraction
 Use a variety of parameters in the simulation.
Write a paragraph describing the DavissonGermer experiment and what you have observed
and learned. You may include fugures or a table
if you'd like.

Adapted fromAdapted from
2 Important Properties of Waves
Not
localized
Greater
localization in space
implies poorer localization in
wavelength
Adapted fromAdapted from
Quiz Question
If ψ is the wavefunction of a
particle, where am I most likely
to find the particle?
A: Where ψ is the largest.
B: Where ψ is the smallest.
C: Where ψ2 is the largest.
D: Where |ψ|2 is the largest.
E: It has nothing to do with ψ.
Adapted fromAdapted from
The Wave Function
( x, y , z , t )
||
2
Related to probability of finding the particle
at a given location at a given time.
|  | dV
2
is the probability of finding the
particle in an infinitesimal volume
element dV
|  |
2
Adapted fromAdapted from
dV
The Wave Function in One
Dimension
 ( x, t )
x b
|  |
2
dx
x a
is the probability of finding the particle
between x and x+dx at time t.
|  | dx
2
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains why
|Ψ(x,t)|2 needs to be multiplied by dV to
really be meaningful.
Adapted fromAdapted from
Are you pondering
what I’m pondering?
Yeah, but I don’t think
snails like yodeling.
No you feeble minded rodent, why is ª
not defined such that ª and not |ª|2 is
proportional to probability of finding
the particle?
Adapted fromAdapted from
Thought Question
What tells us about the probability of
finding a photon at a particular location?
A:
B:
C:
D:
E:
The electric field
The magnetic field
The intensity
The wave vector
The Poynting vector
Adapted fromAdapted from
What else does the photon’s
wave function tell us?
The
Poynting vector
(ExB) tells us the
direction it is going.
The
electric field tells us
how it will interfere.
Adapted fromAdapted from
What do Quantum Waves
Represent?
 Everything about the object’s state
– Position, momentum, angular momentum,
excitation energy, dipole moment, etc.
 Does
it express the object’s
temperature?
 What does the amplitude represent?
 Are there other ways to represent the
wave function?
Adapted fromAdapted from
Quantum Probability –
Ensemble Average
 Start
with a million copies of
the same wavefunction.
 Measure the location of each
particle.
 Generate a Histogram
–This plot is essentially ª*ª
Adapted fromAdapted from
P(x,t)/E2
Photon with well defined wavelength:
E=E0sin(kx-!t)
Red=E(x,t)
Blue=P(x,t)
Adapted fromAdapted from
P(x,t)/ª*ª
Electron in free space with well defined wavelength:
ª=Aei(kx-!t)
Red=Re[ª(x,t)]
Green=Im[ª(x,t)]
Blue=P(x,t)
Adapted fromAdapted from
A moving standing wave...
Adapted fromAdapted from
Fourier’s Theorem
Any periodic function can be written as
a sum of sines and cosines...

 ( x)   An sin( nk0 x)  Bn cos( nk0 x)
n 0 
 ( x) 


A
(
k
)
sin
kx

f
(
k
)
dk



~
ikx
 ( x)   A(k )e dk

Adapted fromAdapted from
Pure Sine Wave
y=sin(5 x)
Adapted fromAdapted from
Power Spectrum
“Shuttered” Sine Wave
y=sin(5 x)*shutter(x)
Adapted fromAdapted from
Power Spectrum
“Thin” Gaussian
y=exp(-(x/0.2)^2)
Adapted fromAdapted from
Power Spectrum
“Fat” Gaussian
y=exp(-(x/2)^2)
Adapted fromAdapted from
Power Spectrum
Femtosecond Laser Pulse
Et=0=sin(10 x)*exp(-x^2)
Adapted fromAdapted from
Power Spectrum
Quiz Question: Which of the following
best describes “dispersion?”
A – When a wave passing through a slit
spreads out spherically
B – When the phase velocity varies with
wavelength
C – When a wave spontaneously changes its
momentum
D – When a wave only contains a finite
number of frequencies
E – When the police try to get the crowd of
waves to “move along” at the scene of a
crime (“there’s nothing to see here”).
Adapted fromAdapted from
Femtosecond Laser Pulse
Et=0=sin(10 x)*exp(-x^2)
Adapted fromAdapted from
Power Spectrum
What will happen to
my laser pulse as it travels through
empty space?
Thought Question:
A: The pulse will get wider.
B: The pulse will get thinner.
C: The pulse will stay the same.
D: It depends on the spectrum of the
pulse.
E: I have no idea.
Adapted fromAdapted from
Propagation Of Light Pulse
E(x,t)
Adapted fromAdapted from
Power Spectrum
Tracking a Moving Pulse
E(x+vt,t)
Adapted fromAdapted from
Power Spectrum
Laser Pulse in Dispersive Medium
Et=0 = sin(10 x)*exp(-x^2)
Adapted fromAdapted from
Power Spectrum
What will happen
to my laser pulse as it travels
through a piece of glass?
Thought Question:
A: The pulse will get wider.
B: The pulse will get thinner.
C: The pulse will stay the same.
D: It depends on the spectrum of
the pulse.
E: I have no idea.
Adapted fromAdapted from
Time Evolution of Dispersive Pulse
Adapted fromAdapted from
Zooming In on a Dispersive Pulse
E(x+vpt,t)
Adapted fromAdapted from
Tracking a Dispersive Pulse Again
E(x+vgt,t)
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains why a
pulse of light spreads out (or sometimes
shrinks) in length as it travels through
glass.
Adapted fromAdapted from
What happens to
the power spectrum of the pulse
as goes through the glass?
Thought Question:
A: It narrows
B: It broadens
C: It stays the same width
D: It depends on the initial spectrum
of the pulse.
Adapted fromAdapted from
Time Evolution of Power Spectrum
in Dispersive Medium
Adapted fromAdapted from
Phase and Group Velocity

v p   velocity of " wiggles" , important for interferen ce
k
d
vg 
dk
 velocity of " envelope" , related to classical particle velocity
 ( k )  v( k )
Adapted fromAdapted from
Phase vs. Group Velocity: Pure Sine
Wave
For a pure sine wave...
sin( kx  t )
Only vp has meaning

v
k
Adapted fromAdapted from
Phase vs. Group Velocity: Sum
of Two Sines
sin( k1 x  1t )  sin( k2 x  2t )
 k x  1t   k 2 x   2t    k1 x  1t   k 2 x   2t  
 2 cos 1
 sin 

2
2

 

   2  t  sin  k1  k2  x  1   2  t 
 k  k 
 2 cos 1 2 x  1
 

2
2
2
2

 

envelope
 a b  a b
sin( a )  sin( b)  2 cos
 sin 

 2   2 
Adapted fromAdapted from
wiggles
Phase vs. Group Velocity: Sum
of Two Sines


1   2    k1  k 2 
1   2  
 k1  k 2 
 2 cos
x
t  sin 
x
t
2
2
2
2

 


1   2 

 wiggles

k1  k 2 

k wiggles 
 beat envelope
k beat envelope
2
2
1   2
d
vbeat envelope  vg 

k1  k 2
dk
Adapted fromAdapted from

1   2 


2
k1  k 2   k
2
vwiggles  vp 
 avg
k avg
avg
avg
Tracking a Free Particle
Wavefunction (with stationary
cameras)
Adapted fromAdapted from
Relativistic vs. Non-relativistic
Phase Velocity...
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains why the
phase velocity of a massive particle is
different for relativistic and non-relativistic
calculations (even at low velocities).
Adapted fromAdapted from
Infinite Square Well

Particle in a Box
p 106
– Why the energy of a trapped particle is
quantized
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains how
boundary conditions resulted in quantized
energies for a particle in a box.
Adapted fromAdapted from
Thought Question
How would we have to change the
energy equation for a particle in a
box if the potential in the box were
not zero but some finite positive
value U0?_____
A: We would have to add U0.
B: We would have to subtract U0.
C: We would have to add nU0
D: We would have to subtract nU0.
E: We wouldn’t have to change anything.
Adapted fromAdapted from
3.6 Particle in a Box :Why is the
energy of a trapped particle quantized?
7.4. (2 points) Consider an electron in a 1dimensional infinitely deep well.
(a) If the well is 9 nm wide, what is the groundstate energy of the electron, in eV?
(b) What is the energy of the first excited state?
(c) What is the wavelength of a photon that will
excite the electron from its ground state to it's
third excited state (with n = 4)?
(d) If the well is 2 cm wide, for what value of n
will the electron have an energy of 1 eV?
Adapted fromAdapted from
3.7 – 3.9
3.7 Uncertainty Principle I
We cannot know the future because we cannot know the
present
3.8 Uncertainty Principle II
A particle approach gives the same result
3.9 Applying the Uncertainty Principle
A useful tool, not just a negative statement
Adapted fromAdapted from
Quiz Question
Which type of wave packet has the
minimum uncertainty product? (i.e.
for which one is ¢x¢k the smallest?)
A – A square packet
B – A triangular packet
C – A Gaussian packet
D – A Lorentzian packet
E – I can’t possibly know much about
physics because I understand too much
about social interactions: (¢Ph¢Si¸1/2).
Adapted fromAdapted from
Uncertainty in a Classical Wave
1
 t 
2
Adapted fromAdapted from
1
 x k 
2
Sine Wave
What is it’s wavelength?
What is it’s frequency?
What is it’s location?
When does it occur?
sin(kx ¡ !t)
Adapted fromAdapted from
Beats in Time
What is it’s wavelength?
What is it’s location?
What is it’s frequency?
When does it occur?
sin(kx ¡ !t) + sin(kx ¡ 2wt)
Adapted fromAdapted from
Localization in Time/Frequency
  2  1
1
t  f
beat
2
 t  

t  2
 2  1 
f beat 

2
2
Have we really localized this wave?
Adapted fromAdapted from
Localization in
Position/Wavenumber
What is it’s wavelength?
What is it’s frequency?
What is it’s location?
When does it occur?
sin(kx ¡ !t) + sin(1:1kx ¡ wt)
Adapted fromAdapted from
Localization in Time/Frequency
k  k2  k1
1
x  " f "
beat
k 2  k1 k
" f beat " 

2
2
Adapted fromAdapted from
...
Beats in Both...
Adapted fromAdapted from
True Localization...
What does it take to make a
single pulse of wavyness?
Adapted fromAdapted from
Consider two short
pulses of light. Pulse “A” has a
duration of 10 fs. Pulse “B” has a
duration of 50 fs. Which one has
the smallest ¢ω?
Thought Question:
A : Pulse “A”
B : Pulse “B”
C : They are both the same
D : Not enough information has been
given
Adapted fromAdapted from
Time Evolution of Dispersive Pulse
Adapted fromAdapted from
Quick Writing Assignment

In one minute, write a short, clear, and
concise paragraph which explains how the
uncertainty relations allow two waves to
have the same uncertainty in x, but
different uncertainty in k.
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Thought Question
A Gaussian pulse of light strikes a diffraction grating and spreads
out. I let a tiny piece of the diffracting beam through a
pinhole. How will the spectrum of the light coming through the
pinhole compare to the spectrum of the pulse before it hit the
grating?
A : It will be narrower
B : It will be wider
C : It will be the same
D : Not enough information given
E : I have no idea
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Thought Question
A Gaussian pulse of light strikes a diffraction
grating and spreads out. I let a tiny piece of
the diffracting beam through a pinhole. How
will the duration of the pulse coming through
the pinhole compare to the duration of the
pulse before it hit the grating?
A : It will be shorter
B : It will be longer
C : It will be the same
D : Not enough information given
E : I have no idea
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New_sqr60.avi
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“Shuttered” Sine Wave on a
Gratting
y=sin(5 x)*shutter(x)
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Power Spectrum
Atom Emitting “Shuttered” Sine
Wave
y=sin(5 x)*shutter(x)
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Power Spectrum
Something to think about
Can’t we think of a pure sine wave as a
string of pulses too?
 So why doesn’t a pure sine wave contain a
spread of frequencies?


COHERENCE
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Calcium Spectroscopy
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Quantum Uncertainty Relations
Position – Momentum
Energy – Time
Other Dimensions
Angular Momentum
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7.5. (2 points)
(a) How accurately can the position of a
proton with v << c be determined without
giving it more than 1.00 keV of kinetic
energy?
 (b) The position and momentum of a
1.00-keV electron are simultaneously
determined. If its position is located to
within 0.100 nm, what is the percentage
of uncertainty in its momentum?

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Thought Question
Imagine that I measure the location and momentum of an
electron. I measure the location with a precision of 1nm. If I
make a second measurement one second later, about how well
will I be able to predict where I will find the electron with the
second measurement? (me ~ 1e-30 kg, h/4¼ ~ 0.5e-34 Js)
A : To better than 1 nm
C : To within 1 μm
E : Not even to within 1 mm
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B : To within around 1nm
D : To within 1 mm
7.6. (2 points)
(a) How much time is needed to measure
the kinetic energy of an electron whose
speed is 10.0 m/s with an uncertainty no
more than 0.100 percent?
(b) How far will the electron have traveled in
this period of time?
(c) Make the same calculation for a 1.00-g
insect whose speed is the same.
(d) What do these sets of figures indicate?
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Wave-Particle Duality

Things act as wave when propagating
– or, in other words, we use waves to make predictions
as to what we will find when we make our
measurement.

Things act as waves when we measure wavelike properties.

Things act as particles when we measure
particle-like properties

Example: BEC interference --- theorists confused
about “undefined phase”
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Postulates of Quantum Mechanics

Every physically-realizable system is described by a
state function ψ that contains all accessible physical
information about the system in that state

The probability of finding a system within the volume
dv at time t is equal to |ψ|2dv

Every observable is represented by an operator which
is used to obtain information about the observable
from the state function

The time evolution of a state function is determined
by Schrödinger’s Equation
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