please aswer last 2
Consider the two states | w) 3i | ф1) + 3і |...
Consider the two states | w) 3i | ф1) + 3і | ф2)—1 фз) and | x) 3D1 ф1)-i | ф2) + 5i | фз), whcrc I ,2) and | 43) arc orthonormal (a) Calculate ( ), (x I x), (y | x), (x l ) and infer (w + x I x). Are the scalar products ( ) and (x I w) equal? (b) Calculate | x | and | x)(/ \. Are they equal? Calculate their traces. (c) Find the transpose, the complex conjugate and the Hermitian conjugate of | y), | x), wx and | wl. Q2 Ir| ф1), I ф2) аnd | ф3) )i)3i | ¢2)- | ¢3) and | x) =l ¢ı) - il 42) +5i | ¢3) satisfy (a) the triangle incquality (b) the Schwarz inequality orthonormal, show that the states are Q3 Find the constant a so that the states | w) = a | ¢1) 5 | 2) and Ix) 3a | 4| 42) are orthogonal; consider | *t) and | ¢2) to be orthonormal Q4 If ) 1)I p2) and | x) =l ¢ı)- l 42). prove the following relations (a) y I l z) = 2( 1) 2(b2 | ¢b2)