4604_ProblemSet3_fa07

4604_ProblemSet3_fa07 - PHY4604 Fall 2007 Problem Set 3 PHY...

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PHY4604 Fall 2007 Problem Set 3 Department of Physics Page 1 of 2 PHY 4604 Problem Set #3 Due Wednesday October 3, 2007 (in class) (Total Points = 130, Late homework = 50%) Reading: Griffiths Chapter 2 (sections 2.4, 2.5, and 2.6). Useful Math: ) ( 2 1 2 1 2 / ) 1 ( 0 2 + + Γ = n n ax n a dx e x , where Γ (x) is the gamma function and Γ (x+1) = x Γ (x) . Γ (1) = Γ (2) = 1, Γ (n) = (n-1)! if n is a positive integer, and π = Γ ) ( 2 1 . Hence, a dx e ax = +∞ 2 and 4 / 4 / ) 2 / ( ) ( 2 2 2 2 ab ab b x a bx x a e a dx e e dx e = = +∞ + +∞ + . Problem 1 (40 points): A free particle has a Gaussian initial wave function given by 2 ) 0 , ( ax Ae x = Ψ , where A and a are positive real constants. (a) (2 points) Find the value of A that normalizes this wave function such that 1 ) 0 , ( ) 0 , ( = Ψ Ψ +∞ dx x x . (b) (8 points) Find ) , ( t x Ψ . (c) (8 point) Compute the probability density 2 | ) , ( | ) , ( t x t x Ψ = ρ . Express your answer in terms of the time dependent quantity 2 ) / 2 ( 1 m at
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4604_ProblemSet3_fa07 - PHY4604 Fall 2007 Problem Set 3 PHY...

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