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Consider the two states | y) =i | 1) + 3i | 2)- | 3) and | x) = 1) - i | 42) + 5i | 3), where | 1), | 2) and | 3) are orthonormal. (a) Calculate (y | y), (x1x), (1 x), (x 1 y), and infer (y + xy + x). Are the scalar products (1 x) and (x | y) equal? (b) Calculate | y) (x | and | x)(y 1. Are they equal? Calculate their traces and compare

Consider the two states | y) =i | 1) + 3i | 2)- | 3) and | x) = 1) - i | 42) + 5i | 3), where | 1), | 2) and | 3) are orthonormal. (a) Calculate (y | y), (x1x), (1 x), (x 1 y), and infer (y + xy + x). Are the scalar products (1 x) and (x | y) equal? (b) Calculate | y) (x | and | x)(y 1. Are they equal? Calculate their traces and compare

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2.10 Exercises Exercise 2.1 Consider the two states | y) =i | 1) + 3i | 2)-1 3) and | x) = 1) i | 2) + 5i | 3), where | 1), 2) and | 3) are orthonormal. (a) Calculate (y | y), (x | x), (y | x), (x | y), and infer (y +x | y +x). Are the scalar products (1x) and (x | y) equal? (b) Calculate y)(x | and | x)(y . Are they equal? Calculate their traces and compare them. (c) Find the Hermitian conjugates of y), 1 x), I w)(x I, and | x)(y I. Exercise 2.2 Consider two states |y₁) = 01) + 4i|2) +53) and |y2) = blø1) +4|02) – 3i|3), where 11), 12), andlp3) are orthonormal kets, and where b is a constant. Find the value of b so that ly) and 2) are orthogonal. Exercise 2.3 If | 1), 12), and | 93) are orthonormal, show that the states | y) =i | 1) + 3i | 2) - | 3) and | x) = 1) - i | 2) +5i | Ø3) satisfy (a) the triangle inequality and (b) the Schwarz inequality. Exercise 2.4 Find the constant a so that the states | y) = a | 01) +5 | ₂) and | x) = 3a | 41) - 4 | 2) are orthogonal; consider | 1) and 2) to be orthonormal. Exercise 2.5 If y) =| 1) + | 2) and | x) = and 2) are not orthonormal): (a) (wy) + (x | x) = 2(01 | (b) (y|y) - (x | x) = 2(01 | 1) - | 2), prove the following relations (note that | 1) 1) + 2(22), 2) + 2(021). ..
Transcribed Image Text:2.10 Exercises Exercise 2.1 Consider the two states | y) =i | 1) + 3i | 2)-1 3) and | x) = 1) i | 2) + 5i | 3), where | 1), 2) and | 3) are orthonormal. (a) Calculate (y | y), (x | x), (y | x), (x | y), and infer (y +x | y +x). Are the scalar products (1x) and (x | y) equal? (b) Calculate y)(x | and | x)(y . Are they equal? Calculate their traces and compare them. (c) Find the Hermitian conjugates of y), 1 x), I w)(x I, and | x)(y I. Exercise 2.2 Consider two states |y₁) = 01) + 4i|2) +53) and |y2) = blø1) +4|02) – 3i|3), where 11), 12), andlp3) are orthonormal kets, and where b is a constant. Find the value of b so that ly) and 2) are orthogonal. Exercise 2.3 If | 1), 12), and | 93) are orthonormal, show that the states | y) =i | 1) + 3i | 2) - | 3) and | x) = 1) - i | 2) +5i | Ø3) satisfy (a) the triangle inequality and (b) the Schwarz inequality. Exercise 2.4 Find the constant a so that the states | y) = a | 01) +5 | ₂) and | x) = 3a | 41) - 4 | 2) are orthogonal; consider | 1) and 2) to be orthonormal. Exercise 2.5 If y) =| 1) + | 2) and | x) = and 2) are not orthonormal): (a) (wy) + (x | x) = 2(01 | (b) (y|y) - (x | x) = 2(01 | 1) - | 2), prove the following relations (note that | 1) 1) + 2(22), 2) + 2(021). ..
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