hw2a - PHYS 507 Answers to Homework II (Fall ’05) 1. {| 1...

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Unformatted text preview: PHYS 507 Answers to Homework II (Fall ’05) 1. {| 1 i , | 2 i , | 3 i} is an orthonormal basis for a 3D Hilbert space. | ψ i = 1 √ 2 ( | 1 i + | 2 i ) , | φ i = 1 √ 3 ( | 2 i + (1 + i ) | 3 i ) , A = 2 | ψ ih ψ | + 6 | φ ih φ | . (a) ψ = 1 √ 2 1 1 , φ = 1 √ 3 1 1 + i . (b) ψ † = 1 √ 2 £ 1 1 0 / , φ † = 1 √ 3 £ 0 1 1- i / . (c) M | ψ ih ψ | = ψψ † = 1 2 1 1 0 1 1 0 0 0 0 , M | φ ih φ | == φφ † = 1 3 1 1- i 0 1 + i 2 , (d) M A = 1 1 1 3 2- 2 i 0 2 + 2 i 4 . (e) A 23 = 2- 2 i and A 32 = 2 + 2 i (can be read directly from M A ). (f) A | 3 i = 2 | ψ ih ψ | 3 i + 6 | φ ih φ | 3 i = 6 | φ i 1- i √ 3 = (2- 2 i ) | 2 i + 4 | 3 i . Matrix representation is M A 1 = 2- 2 i 4 . (g) h 3 | A = 2 h 3 | ψ ih ψ | + 6 h 3 | φ ih φ | = 6 1 + i √ 3 h φ | = (2 + 2 i ) h 2 | + 4 h 3 | ....
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.

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hw2a - PHYS 507 Answers to Homework II (Fall ’05) 1. {| 1...

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