# PS1 - a Calculate the frequency dependence of the re±ected...

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253a/Schwartz Due September 14, 2010 Problem Set 1 1. Special Relativity Review a) If two protons are approaching each other with 14 TeV center of mass energy, how close are they to the speed of light (in miles/hour)? b) How fast is one proton moving with respect to the other? 2. The GZK bound a) The universe is a blackbody at 2.73 0 K. What is the average energy of the photons in outer space (in electron volts)? b) How much energy would a proton (p) need to collide with a photon ( γ ) in outer space to convert it to a 140 MeV pion ( π 0 )? That is, what is the energy threshold for p + γ p + π 0 ? c) How much energy does the outgoing proton have after this reaction? 3. Lorentz Invariance Review a) Show that i −∞ dk 0 δ ( k 2 m 2 ) θ ( k 0 )= 1 2 ω k (1) where ω k k 2 + m 2 r (2) b) Show that i d 4 k (3) is Lorentz invariant. c) Show that i d 3 k 2 ω k (4) is Lorentz invariant. 4. Compton scattering. Suppose we scatter an X-ray oF of a free electron, but we cannot measure the electron’s momentum, just the re±ected X-ray momentum.
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Unformatted text preview: a) Calculate the frequency dependence of the re±ected X-ray on the scattering angle. Draw a rough plot. b) Why is it ok to treat the electrons as free? c) How would the result change if the electron were massless? d) If you didn’t believe in quantized photon momenta, what kind of distribution would you have expected? (Stuck? Look at Compton’s paper on the isite) 5. Coherent states of the Simple Harmonic Oscillator a) Calculate ∂ z ( e − za † ae za † ) . b) Show that | z a = e za † | a is an eigenstate of a . What is its eigenvalue? c) Calculate A n | z a ? d) Show that these “coherent states” are minimally dispersive: Δ p Δ q = 1 2 , where Δ q 2 = A q 2 a−A q a 2 and Δ p 2 = A p 2 a−A p a 2 , where A q a = a z | q | z A a z | z A and A p a = a z | p | z A a z | z A . e) Why can’t you make an eigenstate of a † ? 1...
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