Unformatted text preview: Electromagnetic Theory II Problem Set 6 Due: 21 February 2010 23. J, Problem 7.5. 24. J, Problem 7.13. 25. To understand the behavior of a wave packet, ψ ( r ,t ) = integraldisplay d 3 kf ( k ) e i k · r- iω ( k ) t , (1) it is useful to have an example where all the integrals can be done explicitly for all t . Consider a Gaussian form f ( k ) = Ae- ( k- k ) 2 /δ 2 . Assuming a quadratic behavior ω = k 2 / 2 α for the frequency renders the integrand a Gaussian for all t . a) Calculate the integral that defines ψ in this case, putting the answer in a form K ( t ) e i k · r- (1- itδ 2 / 2 α )( r- r ( t )) 2 / Δ 2 ( t ) and find explicit expressions for K ( t ) , Δ( t ) , r ( t ). b) Show that d r /dt = v g is just the group velocity we have introduced in describing the motion of wave packets with sharply peaked wave number distributions. c) Discuss how the wave function spreads with time. Find the time it takes to double in width. What are the restrictions on an initial wave packet for which there is negli-width....
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- Spring '08
- Wave propagation, 2 2 2 k, propagation direction