# HW2soln - Physics 731 Assignment #2, Solutions 1. (a) To...

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Physics 731 Assignment #2, Solutions 1. (a) To show Tr( XY ) = Tr( Y X ) : Tr( XY ) = X i h a i | XY | a i i = X ij h a i | X | a j ih a j | Y | a i i = X ij h a j | Y | a i ih a i | X | a j i = X j h a j | Y X | a j i = Tr( Y X ) . (b) To show ( XY ) = Y X ) : Y X = X ijk | a i ih a i | Y | a j ih a j | X | a k ih a k | = X ijk | a i ih a j | Y | a i i * h a k | X | a j i * h a k | = X ijk | a i ih a k | X | a j i * h a j | Y | a i i * h a k | = X ijk | a i i ( h a k | X | a j ih a j | Y | a i i ) * h a k | = X ik | a i i ( h a k | XY | a i i ) * h a k | = X ik | a i ih a i | ( XY ) | a k ih a k | = ( XY ) . (c) Let A | a i i = a i | a i i . Then exp[ if ( A )] = X ij | a i ih a i | exp[ if ( A )] | a j ih a j | = X ij | a i ih a i | X m 1 m ! ( if ( A )) m | a j ih a j | = X ij | a i ih a i | X m 1 m ! ( if ( a j )) m δ ij | a j ih a j | = X i | a i i exp[ if ( a i )] h a i | . (d) X a 0 ψ * a 0 ( x 0 ) ψ a 0 ( x 00 ) = X a 0 h a 0 | x 0 ih x 00 | a 0 i = X a 0 h x 00 | a 0 ih a 0 | x 0 i = h x 00 | x 0 i = δ ( x 0 - x 00 ) . 2. (a) The matrix representation of | α ih β | in the basis of {| a 0 i , | a 00 i ,... } is h a 0 | α ih β | a 0 i h a 0 | α ih β | a 00 i ... h a 00 | α ih β | a 0 i h a 00 | α ih β | a 00 i . . . . . . . (b) For

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## This note was uploaded on 12/14/2009 for the course QUANTUM I 731 taught by Professor Everett during the Fall '09 term at Wisconsin.

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HW2soln - Physics 731 Assignment #2, Solutions 1. (a) To...

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