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Homework 02

# Homework 02

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Unformatted text preview: EEE 434/591 Fall 2010 Homework Solutions No. 2 1. For the wave packet defined by φ(k), shown below, find Ψ(x). What are Δx and Δk? φ(k) -π/a π/a k First, we have to normalize the wave function (the 2a shown in the figure in the book can be ignored—it will automatically be corrected) π /a Ⱥ ka Ⱥ 2 Ⱥ ka Ⱥ 2 2 1 = A ∫ Ⱥ1 − Ⱥ dk = 2 A ∫ Ⱥ1 − Ⱥ dk Ⱥ π Ⱥ π Ⱥ −π / a Ⱥ 0 π /a 2 Ⱥ k 3 a 2 a Ⱥ = 2 A Ⱥ 2 − k 2 + k Ⱥ π Ⱥ 3π Ⱥ 0 2 π /a = 2 A 2 ȹ π ȹ ȹ ȹ 3 ȹ a Ⱥ and € A= 3a . 2π Then, we can Fourier transform the momentum wave function as € 1 Ψ( x ) = 2π = 3a ∞ ∫ φ (k )e ikx dk 2π −∞ π /a ȹ ka ȹ ikx Ⱥ 3a Ⱥ 0 ȹ ka ȹ ikx Ⱥ ∫ ȹ 1 + ȹe dk − ∫ ȹ 1 − ȹe dk Ⱥ...
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