Announcements 10/15/12
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Prayer
Saturday: Term project proposals, one proposal per group…
but please CC your partner on the email. See website for
guidelines, grading, ideas, examples.
Colton “Fourier series summary” handout. Notation warning!
xkcd
From warmup
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Extra time on?
a.(nothing in particular)
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Other comments?
a.Is this related to the Heisenberg uncertainty
principle?
b.Is the average grade on exam 2 typically
higher or lower than exam 1?
Spectrum Lab Software
Quick Writing
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We saw that A1cos(kx
+ f1) + A2cos(kx + f2)
gives you a cosine
wave with the same k,
and hence wavelength.
If you add a third,
fourth, fifth, etc., such
cosine wave, you still
get a simple cosine
wave. See 
How can you then add
together cosine waves
to get a more
complicated shape
with same
wavelength? Or can
you?
If not all multiples
of same k:
Special Case
Centered on a particular k: “Wave packets”
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HW 21-3
Plot:
Explore with
Mathematica
Wave packets, cont.
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Results:
a. To localize a wave in space, you need lots of spatial
frequencies (k values)
b. To remove neighboring localized waves (i.e. to make it
non-periodic), you need those frequencies to spaced
close to each other. (infinitely close, really)
From warmup
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PpP states that "a pure sine wave has a
precisely defined frequency... but a completely
undefined position." What does it mean to have
a "completely undefined position"?
a. It goes from infinity to infinity so it's kind of
at everywhere at once.
Pure Sine Wave
y=sin(5 x)
Power Spectrum
“Shuttered” Sine Wave
y=sin(5 x)*shutter(x)
Uncertainty in x = ______
In general:
Power Spectrum
Uncertainty in k = ______
1
xk 
2
(and technically,
 = std dev)
Clicker question:
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The equation that says xk  ½ means that if
you know the precise location of an electron you
cannot know its momentum, and vice versa.
a. True
b. False
Uncertainty Relationships
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Position & k-vector
1
xk 
2
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Time & w
1
t w 
2
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Quantum Mechanics:
momentum p = k
xp 
“” = “h bar”
= Plank’s constant /(2p)
energy E = w
Et 
2
2
Dispersion
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A dispersive medium: velocity is different for different
frequencies
a. Any real-world examples?
Why do we care?
a. Real waves are often not shaped like sine waves.
– Non sine-wave shapes are made up of combinations of
sine waves at different frequencies.
b. Real waves are not infinite in space or in time.
– Finite waves are also made up of combinations of sine
waves at different frequencies.
Focus on (b) for now…
(a) is the main topic of the
“Fourier transform” lectures
Dispersion Review
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Any wave that isn’t 100% sinusoidal contains more
than one frequencies.
To localize a wave in space or time, you need lots of
frequencies--spatial (k values) or angular (w values),
respectively. Really an infinite number of frequencies
spaced infinitely closely together.
A dispersive medium: velocity is different for different
frequencies.
Two Different Velocities
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What happens if a wave pulse is sent through a
dispersive medium? Nondispersive?
Dispersive wave example:
a. f(x,t) = cos(x-4t) + cos(2 (x-5t))
– What is “v”?
– What is v for w=4? What is v for w=10?
What does that wave look like as time progresses?
(next slide)
Mathematica
0.1 seconds
0.7 seconds
What if the two
velocities had
been the same?
1.3 seconds
Time Evolution of Dispersive Pulse
Credit: Dr. Durfee
Power spectrum
Peak
moves at
about 13
m/s (on
my office
computer)
Wave moving in time
Note:
frequencies
are infinitely
close together
How much energy is
contained in each
frequency component
From warmup: phase & group velocities
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Examples where vphase  vgroup: http://en.wikipedia.org/wiki/Group_velocity
For each figure, measure the speed of travel on your monitor (cm/s) for
both the envelope and the ripples.
a. Your results will depend on size of monitor and/or zoom level. But the
ratio of envelope to ripple speed should be the same as me.
–
Top Fig: speed of envelope (green dots) = 0.46 cm/s, speed of
ripples (red dot) = 2x that
–
Second Fig: speed of envelope = 1.50 cm/s; speed of ripples =
-0.33x that
Phase and Group Velocity
Credit: Dr. Durfee
Window is moving
along with the
peak of the pulse
w
vp 
 velocity of "wiggles"
k
Can be different for each
frequency component
that makes up the wave
12.5 m/s, for dominant component
dw
vg 
dk
 velocity of "envelope"
evaluated at k ave
13 m/s
(peak)
A property of the wave
as a whole
Transforms
A one-to-one correspondence between one function and
another function (or between a function and a set of
numbers).
a. If you know one, you can find the other.
b. The two can provide complementary info.
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Example: ex = 1 + x + x2/2! + x3/3! + x4/4! + …
a. If you know the function (ex), you can find the Taylor’s
series coefficients.
b. If you have the Taylor’s series coefficients (1, 1, 1/2!,
1/3!, 1/4!, …), you can re-create the function. The first
number tells you how much of the x0 term there is, the
second tells you how much of the x1 term there is, etc.
c. Why Taylor’s series? Sometimes they are useful.
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“Fourier” transform
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The coefficients of the transform give information
about what frequencies are present
Example:
a. my car stereo
b. my computer’s music player
c. your ear (so I’ve been told)
Fourier Transform
20
10
600
400
200
200
400
600
10
20
Cos 0.9 x
Cos 0.91 x
Cos 0.92 x
Cos 0.93 x
Cos 0.94 x
Cos 0.95 x
Cos 0.96 x
Cos 0.97 x
Cos 0.98 x
Cos 0.99 x
Cos 1. x
Cos 1.03 x
Cos 1.04 x
Cos 1.05 x
Cos 1.06 x
Cos 1.07 x
Cos 1.08 x
Cos 1.09 x
Cos 1.1 x
Cos 1.01 x
Cos 1.02 x
Do the transform (or
have a computer do it)
Answer from computer: “There
are several components at
different values of k; all are
multiples of k=0.01.
k = 0.01: amplitude = 0
k = 0.02: amplitude = 0
…
…
k = 0.90: amplitude = 1
k = 0.91: amplitude = 1
k = 0.92: amplitude = 1
…”