# CALTECH PH 195 - Problem set number 18 (3 pages)

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## Problem set number 18

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- Pages:
- 3
- School:
- California Institute of Technology
- Course:
- Ph 195 - Advanced Quantum Mechanics

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Physics 195b Problem set number 18 Due 2 PM Thursday March 6 2003 Notes about course Homework should be turned in to the TA s mail slot on the first floor of East Bridge Collaboration policy OK to work together in small groups and to help with each other s understanding Best to first give problems a good try by yourself Don t just copy someone else s work whatever you turn in should be what you think you understand There is a web page for this course which should be referred to for the most up to date information The URL http www hep caltech edu fcp ph195 TA Anura Abeyesinghe anura caltech edu If you think a problem is completely trivial and hence a waste of your time you don t have to do it Just write trivial where your solution would go and you will get credit for it Of course this means you are volunteering to help the rest of the class understand it if they don t find it so simple READING Read the Electromagnetic Interactions course note PROBLEMS 84 Extended boson principle and decays to two pions Do Exercise 2 of the Identical Particles course note 85 Gauge transformation in electromagnetism Do Exercise 1 of the Electromagnetic Interactions course note 86 We discussed the method of stationary phase in class Recall that the problem it addresses is to evaluate integrals of the form I 49 f x ei x dx 150 where f and are real and 0 We showed that in the situation where is very small and has a stationary point at x x0 this integral is approximately I i x0 i 4 sign x0 f x0 e e 2 x0 1 O 151 If there is more than one stationary point then the contributions are to be summed To get a little practice applying this method evaluate the following integral for large t J t 1 0 cos t x3 x dx 152 87 In problem 82 you considered the scattering of particles in a multiplet You determined the total elastic sometimes called scattering cross section and the total inelastic reaction cross sections in terms of the A matrix in the partial wave expansion Consider now the graph in Fig 2 This graph purports to show the allowed and forbidden regions for the total elastic and inelastic cross sections in a given partial wave Derive the formula for the allowed region of this graph Make sure to check the extreme points 50 4 Not Allowed el l TOT 2l 1 k2 2 Allowed 0 0 1 k 2 inel l TOT 2l 1 Figure 2 Made up graph of phase shifts 0 and 1 for elastic p scattering neglecting spin 51

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