4604_Solutions_Set3

4604_Solutions_Set3 - PHY4604 Problem Set 3 Solutions Department of Physics Page 1 of 15 PHY 4604 Problem Set#3 Solutions Problem 1(40 points A

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Unformatted text preview: PHY4604 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 ) , ( 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 ) , ( ) , ( = Ψ Ψ ∫ +∞ ∞ − ∗ dx x x . Answer: 4 1 2 = π a A Solution: 1 2 ) , ( ) , ( 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 ) , ( 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 . PHY4604 Problem Set 3 Solutions Department of Physics Page 2 of 15 Answer: 2 2 2 2 ) , ( x w we t x − = π ρ Solution: ) / 4 1 ( 2 ) / 2 1 ( ) / 2 1 ( 2 ) , ( ) , ( ) , ( 2 2 2 2 )] / 2 1 /( 1 ) / 2 1 /( 1 [ 2 / 1 ) / 2 1 /( ) / 2 1 /( 2 / 1 2 2 2 m t a e a m at i e m at i e a t x t x t x m at i m at i ax m at i ax m at i ax h h h h h h h + = + − = Ψ Ψ = + + − − + − − − ∗ π π ρ But ) / 4 1 ( 2 ) / 2 1 )( / 2 1 ( 2 ) / 2 1 ( 1 ) / 2 1 ( 1 2 2 2 2 m t a m at i m at i m at i m at i h h h h h + = + − = + + − and hence 2 2 2 2 ) , ( x w we t x − = π ρ . Gaussian Wave Packet-1.0-0.5 0.0 0.5 1.0 x t = 0 t >> 0 As time increases w decreases and the packet spreads out in space. (d) (8 points) Find <x> , <x 2 > , <p x > and <p x 2 > for this wave function. Express you answers in terms of the time dependent quantity 2 ) / 2 ( 1 m at a w h + ≡ ....
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This note was uploaded on 04/19/2008 for the course PHY 4604 taught by Professor Field during the Spring '07 term at University of Florida.

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4604_Solutions_Set3 - PHY4604 Problem Set 3 Solutions Department of Physics Page 1 of 15 PHY 4604 Problem Set#3 Solutions Problem 1(40 points A

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