hws02_r - Homework 2 Solutions TA: Kevin Chiou (original...

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Unformatted text preview: Homework 2 Solutions TA: Kevin Chiou (original solutions by Matt Block) Problem 1 We wrote down and 1 in the notes. Ill reproduce them here, where again m h : ( x ) = 1 / 4 e- 2 x 2 (1) 1 ( x ) = 4 3 ! 1 / 4 xe- 2 x 2 (2) First, we will calculate h p i and h p 1 i for and 1 respectively: h p n i = Z - n ( x ) p n ( x ) dx, p = h i d dx (3) h p i = Z - ( ) 1 2 h i (- x ) e x 2 (4) h p 1 i = Z - ( 4 3 ) 1 2 h i e x 2 ( x- x 3 ) (5) Note that the integrals are all odd over an even interval, and therefore are zero. Therefore: h p i = h p 1 i = 0 (6) To calculate h p 2 i we have the following expressions: h p 2 n i = Z - n ( x ) p 2 n ( x ) dx, p 2 =- h 2 d 2 dx 2 (7) h p 2 i = Z - ( ) 1 2 (- h 2 ) e x 2 (- + 2 x 2 ) (8) h p 2 1 i = Z - ( 4 3 ) 1 2 (- h 2 ) e x 2 x 2 (- 3 + 2 x 2 ) (9) Now, to help us do the integrals, notice the following facts: Z - e- x 2 = r (10) Z - x 2 e- x 2 =- Z - e- x 2 (11)- Z - e- x 2 =- r (12) Z - x 2 e- x 2 = 1 2 r 3 (13) 1 In calculating R - x 4 e- x 2 one can simply do the previous trick twice. This is left as an exercise to the student. We will simply quote the result here: Z - x 4 e- x 2 = 3 4 r...
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hws02_r - Homework 2 Solutions TA: Kevin Chiou (original...

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