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Unformatted text preview: Homework Solutions # 1 (Liboff Chapter 3) 3.2 • No inverse of ˆ D . The integral R x dx ∂φ/∂x = φ + c ; only up to an arbitrary additive constant. No operator which destroys information can have an inverse. • ˆ I 1 = ˆ I . • ˆ F 1 = multiplication by 1 /F ( x ), except where F ( x ) = 0. • ˆ B 1 = multiplication by 3. • ˆ Θ has no inverse. • ˆ G has no inverse. 3.4 Plug φ β into the eigenvalue equation, getting e β ( x + ζ ) g ( x + ζ ) = a β e βx g ( x ) where a β is the eigenvalue. Using g ( x + ζ ) = g ( x ), we get a β = e βζ . 3.11 ∂ψ ∂x = ip ¯ h x x 2 a 2 ψ ∂ 2 ψ ∂x 2 = 1 2 a 2 ψ + ip ¯ h x x 2 a 2 2 ψ h p 2 i = ¯ h 2 Z ∞∞ dx ψ * ∂ 2 ψ ∂x 2 Using, as the book does, η = ( x x ) /a , h p 2 i = ¯ h 2 a  A  2 Z ∞∞ dη e η 2 / 2 1 2 a 2 p 2 ¯ h 2 ip η a ¯ h + η 2 4 a 2 ! 1 The iη term will integrate to zero, as an odd function. The constant coef ficient terms can be evaluated directly, using...
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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.
 Spring '09
 Mucciolo
 mechanics, Work

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