hw3a - PHYS 507 Answers to Homework III (Fall '05) 1. (a)...

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PHYS 507 Answers to Homework III (Fall ’05) 1. (a) [ X,Y ] = ( XY - Y X ) = Y X - X Y = [ Y ,X ] = [ Y,X ] = - [ X,Y ]. (b) ( A A ) = A ( A ) = A A . Let | φ i be an eigenket of A A with eigenvalue λ . Let | φ i be normalized. In that case λ = h φ | A A | φ i = h φ | A i = h | i = || || 2 , i.e., norm square of A | φ i . Since norm-square cannot be negative (either positive or zero) we have λ 0. 2. ψ ( x 0 ) = h x 0 | ψ i = N exp ± - x 0 2 4 σ 2 + ikx 0 . (a) N = 1 / σ (2 π ) 1 / 4 . (b) ˜ ψ ( p 0 ) = r 2 σ ¯ h 1 (2 π ) 1 / 4 exp ± - σ 2 ¯ h 2 ( p 0 - ¯ hk ) 2 . (c) h p i = ¯ hk , p 2 = ¯ h 2 k 2 + ¯ h 2 4 σ 2 , Δ p = ¯ h 2 σ . 3. (a) Obvious from Fourier transform expression. (b) For any odd function f ( p 0 ) (i.e., f ( - p 0 ) = - f ( p 0 )) we have h f ( p ) i = R f ( p 0 ) ˜ P ( p 0 ) dp 0 = 0 since the integral of any odd function is 0. For this reason h p i = h p 17 i = 0. 4.
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