hw_solutions_2 - Homework Solutions 2(Liboff Chapter 4...

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Unformatted text preview: Homework Solutions # 2 (Liboff Chapter 4) 4.5 (a) Given in book. (b) | ϕ ih ψ | f i = a | ϕ i , where a = R dx ψ * f . (c) h f | ϕ ih ψ | f i . (d) h f | ϕ ih ψ | ψ i = h f | ϕ i . 4.8 g ( x ) = s 2 a X n a n sin nπx a where a n = s 2 a Z a dx sin nπx a x ( x- a ) e ikx = 1 2 i s 2 a Z a dx ( x 2- ax ) e i ( k + nπ a ) x- e- i ( k + nπ a ) x Now the tedious integration. Use I ( α ) = Z a dx e iαx = 1 iα e iaα- 1 Z a dx x e iαx =- i ∂I ∂α = 1 α 2 e iaα- 1- ia α e iaα Z a dx x 2 e iαx =- ∂ 2 I ∂α 2 = 2 i α 3 e iaα- 1 + 2 a α 2 e iaα- ia 2 α e iaα Afer a bunch of algebra I’m not going to reproduce here, we get a n = s 2 a h (- 1) n e ika- 1 i " 1 ( k + nπ/a ) 3 )- 1 ( k + nπ/a ) 3 ) #- ia (- 1) n e ika " 1 ( k + nπ/a ) 2- 1 ( k- nπ/a ) 2 #- a 2 i h (- 1) n e ika- 1 i " 1 ( k + nπ/a ) 2- 1 ( k- nπ/a ) 2 # Phys 580 HW# 2 Solutions 1 4.9 || ψ + ϕ || 2 = h ψ + ϕ | ψ + ϕ i = ( h ψ | + h ϕ | )( | ψ i + | ϕ i ) = h ψ | ψ i + h ϕ | ψ i...
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This note was uploaded on 11/09/2009 for the course PHY 4604 taught by Professor Mucciolo during the Spring '09 term at University of Central Florida.

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hw_solutions_2 - Homework Solutions 2(Liboff Chapter 4...

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