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Unformatted text preview: Physics 731 Assignment #1: Solutions 1. (a) We are given that X = a + ~ ~a = a 1 2 2 + X i i a i . By inspection, Tr(1 2 2 ) = 2 . We also have Tr i = 0 and Tr i j = 2 ij . Therefore, Tr X = 2 a = 2 X i =1 X ii , Tr( j X ) = 2 a j = 2 X i,k =1 ( j ) ik X kj . (b) Using the above results, it is straightforward to see that a = ( X 11 + X 22 ) / 2 , a 1 = ( X 12 + X 21 ) / 2 , a 2 = i ( X 12 X 21 ) / 2 , and a 3 = ( X 11 X 22 ) / 2 , 2. The determinant of ~ ~a is det a z a x ia y a x + ia y a z ! = ~ a  2 . Using e i~ n/ 2 = cos( / 2) + i ( ~ n ) sin( / 2) , which is easy to verify through a power series expan sion, by explicit computation one obtains e i~ n/ 2 ~ ~ae i~ n/ 2 = a z ( a x ia y )(cos( / 2) + i sin( / 2)) 2 ( a x + ia y )(cos( / 2) i sin( / 2)) 2 a z ! . Hence, det( e i~ n/ 2 ~ ~ae i~ n/ 2) is equal to ~a  2 . (As a quicker alternative to this, you can use the....
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 Fall '09
 Everett

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