Exercise 2.1Consider the two states | y) =i | 1) + 3i | 42) | 3) and | x) = 1) - i |where | 1), 2) and | 3) are orthonormal.(a) Calculate (wy), (x1x), (1x), (x | w), and infer (y+x1+x). Are the scalarproducts (x) and (x 1 ) equal?(b) Calculate y)(x | and | x)(y . Are they equal? Calculate their traces and comparethem.(c) Find the Hermitian conjugates of w),1x), w)(x I, and | x)(y 1.2) + 5i | 03),

(a) We know that, in quantum mechanics, when dealing with a scalar product (or an inner product)…

Consider the two states | y) = i| 1) + 3i | 2)- | 3) and | x) = 1) - i | 2) + 5i | 03), where | 1), 2) and | 3) are orthonormal. (a) Calculate (y | w), (x1x), (1x), (x | w), and infer (y + xy + x). Are the scalar products (x) and (x | w) equal? (b) Calculate y)(x | and | x)(w 1. Are they equal? Calculate their traces and compare them. (c) Find the Hermitian conjugates of y), 1 x), y) (x , and | x)(w ].

Consider the two states | y) = i| 1) + 3i | 2)- | 3) and | x) = 1) - i | 2) + 5i | 03), where | 1), 2) and | 3) are orthonormal. (a) Calculate (y | w), (x1x), (1x), (x | w), and infer (y + xy + x). Are the scalar products (x) and (x | w) equal? (b) Calculate y)(x | and | x)(w 1. Are they equal? Calculate their traces and compare them. (c) Find the Hermitian conjugates of y), 1 x), y) (x , and | x)(w ].