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Unformatted text preview: Homework Solutions # 1 (Liboff Chapter 3) 3.2 No inverse of D . The integral R x dx /x = + c ; only up to an arbitrary additive constant. No operator which destroys information can have an inverse. I- 1 = I . F- 1 = multiplication by 1 /F ( x ), except where F ( x ) = 0. B- 1 = multiplication by 3. has no inverse. G has no inverse. 3.4 Plug into the eigenvalue equation, getting e ( x + ) g ( x + ) = a e x g ( x ) where a is the eigenvalue. Using g ( x + ) = g ( x ), we get a = e . 3.11 x = ip h- x- x 2 a 2 2 x 2 =- 1 2 a 2 + ip h- x- x 2 a 2 2 h p 2 i =- h 2 Z - dx * 2 x 2 Using, as the book does, = ( x- x ) /a , h p 2 i =- h 2 a | A | 2 Z - d e- 2 / 2- 1 2 a 2- p 2 h 2- ip a h + 2 4 a 2 ! 1 The i term will integrate to zero, as an odd function. The constant coef- ficient terms can be evaluated directly, using...
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