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Announcements 10/15/12 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

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From warmup Extra time on? a. a.(nothing in particular) Other comments? a. a.Is this related to the Heisenberg uncertainty principle? b. b.Is the average grade on exam 2 typically higher or lower than exam 1?

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Spectrum Lab Software

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Quick Writing We saw that A 1 cos(kx + 1 ) + A 2 cos(kx + 2 ) 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?

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If not all multiples of same k:

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Special Case Centered on a particular k: “Wave packets” HW 21-3 Plot: Explore with Mathematica

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Wave packets, cont. Results: a. a.To localize a wave in space, you need lots of spatial frequencies (k values) b. 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)

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From warmup 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. a.It goes from infinity to infinity so it's kind of at everywhere at once.

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Pure Sine Wave y=sin(5 x) Power Spectrum

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“Shuttered” Sine Wave y=sin(5 x)*shutter(x) Power Spectrum Uncertainty in x = ______ Uncertainty in k = ______ In general: (and technically, = std dev)

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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. a.True b. b.False Clicker question:

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Uncertainty Relationships Position & k-vector Time & Quantum Mechanics: momentum p = k energy E = “ ” = “h bar” = Plank’s constant /(2 )

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Dispersion A dispersive medium: velocity is different for different frequencies a. a.Any real-world examples? Why do we care? a. 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. 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

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Dispersion Review 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 ( values), respectively. Really an infinite number of frequencies spaced infinitely closely together. A dispersive medium: velocity is different for different frequencies.

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Two Different Velocities What happens if a wave pulse is sent through a dispersive medium? Nondispersive? Dispersive wave example: a. a.f(x,t) = cos(x-4t) + cos(2 (x-5t)) – – What is “v”? – – What is v for =4? What is v for =10? What does that wave look like as time progresses? (next slide)

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Mathematica 0.7 seconds1.3 seconds 0.1 seconds What if the two velocities had been the same?

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Time Evolution of Dispersive Pulse Credit: Dr. Durfee Wave moving in time Peak moves at about 13 m/s (on my office computer) How much energy is contained in each frequency component Power spectrum Note: frequencies are infinitely close together

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From warmup: phase & group velocities Examples where v phase v group : http://en.wikipedia.org/wiki/Group_velocity 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

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Phase and Group Velocity Credit: Dr. Durfee Can be different for each frequency component that makes up the wave A property of the wave as a whole Window is moving along with the peak of the pulse 13 m/s 12.5 m/s, for dominant component (peak)

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Transforms A one-to-one correspondence between one function and another function (or between a function and a set of numbers). a. a.If you know one, you can find the other. b. b.The two can provide complementary info. Example: e x = 1 + x + x 2 /2! + x 3 /3! + x 4 /4! + … a. a.If you know the function (e x ), you can find the Taylor’s series coefficients. b. 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 x 0 term there is, the second tells you how much of the x 1 term there is, etc. c. c.Why Taylor’s series? Sometimes they are useful.

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“Fourier” transform The coefficients of the transform give information about what frequencies are present Example: a. a.my car stereo b. b.my computer’s music player c. c.your ear (so I’ve been told)

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Fourier Transform 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 …”

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