hw03soln - ECE 310 Spring 2005 HW 3 solutions Problem E2.41...

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ECE 310, Spring 2005, HW 3 solutions Problem E2.41 a) Let Ω be the set of all possible arrangements of k indistinguishable photons in N intervals. By Bose-Einstein statistics, k Ω k = ( N - 1+ k k ) . Let A be the set of possible arrangements, at which no two photons occupy the same interval; in other words, each interval contains either one photon or zero photons. Then k A k = ( N k ) and P a = P ( A ) = k A k k Ω k = ( N k ) ( N - 1+ k k ) b) Let B be the event that at least two photons share some unit interval, then B = A c and P b = P ( B ) = 1 - P ( A ) = 1 - ( N k ) ( N - 1+ k k ) c) Let C be the event that two of k photons turned to be in the first interval or, in other words, k - 2 remaining photons were arranged in N - 1 intervals. Hence, k C k = ( N - 4+ k k - 2 ) and P c = P ( C ) = ( N - 4+ k k - 2 ) ( N - 1+ k k ) Problem E2.42 a) According to Bose-Einstein statistics (page 78), there are m = ( 10 - 1+3 3 ) = ( 12 3 ) = 220 distinguishable arrangements. b) The event A corresponds to one of m possible arrangements, thus k A k = 1 and P ( A ) = 1 ( 12 3 ) = 0 . 0045(45) c) The event B is that none of 3 photons are in the cell q 1 , which is the
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