Consider the two states | y) = i| 0₁) + 3i | P2)-3) and|x) = ₁) - i| 2) + Si | 03), 19 where | 01), 2) and | 3) are orthonormal. (a) Calculate (y | w), (x1x), (y 1x), (x | y), and infer (y + xy + x). Are the scalar products (y | x) and (x | y) equal?
Consider the two states | y) = i| 0₁) + 3i | P2)-3) and|x) = ₁) - i| 2) + Si | 03), 19 where | 01), 2) and | 3) are orthonormal. (a) Calculate (y | w), (x1x), (y 1x), (x | y), and infer (y + xy + x). Are the scalar products (y | x) and (x | y) equal?
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صورة من احمد مرتضى
Transcribed Image Text:2.10 Exercises
Exercise 2.1
Consider the two states | y) = id1) +3i | 42) - | 3) and | x) = 1) - i | 2) + 5i | 3),
where | 01), 2) and | 3) are orthonormal.
(a) Calculate (y | y), (x1x), (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), 1x), y)(x I, and | x)(y 1.
Exercise 2.2
Consider two states [w₁) = 1) + 4i|2) +503) and |y2) = b[ø1) +4|02) – 3i|3), where
11), 12), and 3) 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 | P1),
2), and | 3) are orthonormal, show that the states | y) = i | 1) + 3i | 2)— | 3)
and | x) = 1) - i | 2) + 5i | 3) satisfy
(a) the triangle inequality and
(b) the Schwarz inequality.
Exercise 2.4
Find the constant a so that the states | y) = a | 1) +5 | ₂) and | x) = 3a | 1) - 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):
41) - | 2), prove the following relations (note that | 1)
(a) (wy) + (x | x) = 2(61 | 61) + 2 (22),
(b) (y y) - (x | x) = 2(01 | 2) + 2(021).
..
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