ch06 - CHAPTER 6 19. (a) We have A|a> = a|a> It...

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Unformatted text preview: CHAPTER 6 19. (a) We have A|a> = a|a> It follows that < a | A | a > = a < a | a> = a if the eigenstate of A corresponding to the eigenvalue a is normalized to unity. The complex conjugate of this equation is < a | A | a >* = < a | A + | a > = a * If A + = A , then it follows that a = a *, so that a is real. 13. We have 〈 ψ | ( AB ) + | ψ 〉 = 〈 ( AB ) ψ | ψ 〉 = 〈 B ψ | A + | ψ 〉 = 〈 ψ | B + A + | ψ 〉 This is true for every ψ , so that ( AB ) + = B + A + 2. TrAB = 〈 n | AB | n 〉 = 〈 n | A 1 B | n 〉 n ∑ n ∑ = 〈 n | A | m 〉〈 m | B | n 〉 = m ∑ n ∑ 〈 m | B | n 〉〈 n | A | m 〉 m ∑ n ∑ = 〈 m | B 1 A | m 〉 = m ∑ 〈 m | BA | m 〉 = m ∑ TrBA 3. We start with the definition of | n > as | n 〉 = 1 n ! ( A + ) n | 0 〉 We now take Eq. (6-47) from the text to see that A | n 〉 = 1 n ! A ( A + ) n | 0 〉 = n n ! ( A + ) n- 1 | 0 〉 = n ( n- 1)! ( A + ) n- 1 | 0 〉 = n | n- 1 〉 4. Let f ( A + ) = C n n = 1 N ∑ ( A + ) n . We then use Eq. (6-47) to obtain Af ( A + ) | 0 〉 = A C n n = 1 N ∑ ( A + ) n | 0 〉 = nC n ( A + ) n- 1 n = 1 N ∑ | 0 〉 = d dA + C n n = 1 N ∑ ( A + ) n | 0 〉 = df ( A + ) dA + | 0 〉 5. We use the fact that Eq. (6-36) leads to x = h 2 m ϖ ( A + A + ) p = i m ϖ h 2 ( A +- A ) We can now calculate 〈 k | x | n 〉 = h 2 m ϖ 〈 k | A + A + | n 〉 = h 2 m ϖ n 〈 k | n- 1 〉 + k 〈 k- 1 | n 〉 ( 29 = h 2 m ϖ n δ k , n- 1 + n + 1 δ k , n + 1 ( 29 which shows that k = n ± 1. 6. In exactly the same way we show that 〈 k | p | n 〉 = i m ϖ h 2 〈 k | A +- A | n 〉 = i m ϖ h 2 ( n + 1 δ k , n + 1- n δ k , n- 1 ) 7. Let us now calculate 〈 k | px | n 〉 = 〈 k | p 1 x | n 〉 = 〈 k | p | q 〉〈 q | x | n 〉 q ∑ We may now use the results of problems 5 and 6. We get for the above i h 2 ( k q ∑ δ k- 1, q- k + 1 δ k + 1, q )( n δ q , n- 1 + n + 1 δ q , n + 1 ) = i h 2 ( kn δ kn- ( k + 1) n δ k + 1, n- 1 + k ( n + 1) δ k- 1, n + 1- ( k + 1)( n + 1) δ k + 1, n + 1 ) = i h 2 (- δ kn- ( k + 1)( k + 2) δ k + 2, n + n ( n + 2) δ k , n + 2 ) To calculate 〈 k | xp | n 〉...
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This note was uploaded on 03/09/2008 for the course PHYS 401 taught by Professor Mokhtari during the Spring '05 term at UCLA.

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ch06 - CHAPTER 6 19. (a) We have A|a> = a|a> It...

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