4604_Solutions_Set3_fa07 - PHY4604 Fall 2007 Problem Set 3...

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PHY4604 Fall 2007 Problem Set 3 Solutions Department of Physics Page 1 of 15 PHY 4604 Problem Set #3 Solutions 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 . Answer: 4 1 2 = π a A Solution: 1 2 ) 0 , ( ) 0 , ( 2 2 2 2 = = = Ψ Ψ +∞ +∞ a A dx e A dx x x ax π (b) (8 points) Find ) , ( t x Ψ . Answer: ) / 2 1 ( 2 ) , ( ) / 2 1 /( 4 / 1 2 m at i e a t x m at i ax h h + = Ψ + π Solution: ) 4 /( 4 / 1 ) 4 /( ) / ( 2 2 2 ) 2 ( 1 2 2 ) 0 , ( 2 1 ) ( a k a k a ikx x a ikx e a e a A dx e A dx e x k f +∞ + +∞ = = = Ψ = π π π π π and ) 4 /( 4 / 1 ) 4 /( 4 / 1 ) / ( 4 / 1 )) 2 /( ) 4 /( 1 ( 4 / 1 )) 2 /( ( ) 4 /( 4 / 1 ) ) ( ( 2 2 2 2 2 2 2 1 ) 2 ( 1 2 ) 2 ( 1 2 ) 2 ( 1 2 ) 2 ( 1 2 ) 2 ( 1 ) ( 2 1 ) , ( c x c x c ixk k c ikx k m t i a m t k kx i a k t k kx i e c a e c a dx e a dk e e a dk e e a dk e k f t x + + + +∞ +∞ = = = = = = Ψ π π π π π π π π π π π ω h h where ) 2 /( ) 4 /( 1 m t i a c h + = . Thus ) / 2 1 ( 2 ) / 2 / 1 ( 2 ) 2 ( 1 ) , ( ) / 2 1 /( 4 / 1 ) / 2 / 1 /( 4 / 1 2 2 m at i e a e m t i a a t x m at i ax m t i a x h h h h + = = + = Ψ + + π π (c) (8 point) Compute the probability density 2 | ) , ( | ) , ( t x t x Ψ = ρ . Express you answer in terms of the time dependent quantity 2 ) / 2 ( 1 m at a w h + . Sketch ρ (x,t) (as a function of x) at t = 0 , and again at some very large time t .
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